9.6B: Viroids
- Relate the structure and replication of a viroid to its ability to cause diseases in plants
Viroids are plant pathogens that consist of a short stretch (a few hundred nucleobases) of highly complementary, circular, single-stranded RNA without the protein coat that is typical for viruses. In comparison, the genome of the smallest known viruses capable of causing an infection by themselves is around 2 kilobases in size. The human pathogen Hepatitis D virus is similar to viroids. Viroids are extremely small in size, ranging from 246 to 467 nucleotide (nt) long genome and consisting of fewer than 10,000 atoms. Viroids were discovered and given this name by Theodor Otto Diener, a plant pathologist at the Agricultural Research Service in Maryland, in 1971.
Viroid RNA does not code for any protein. The replication mechanism involves RNA polymerase II, an enzyme normally associated with synthesis of messenger RNA from DNA, which instead catalyzes “rolling circle” synthesis of new RNA using the viroid’s RNA as template. Some viroids are ribozymes, having catalytic properties which allow self-cleavage and ligation of unit-size genomes from larger replication intermediates. The first viroid to be identified was the potato spindle tuber viroid (PSTVd). Some 33 species have been identified.
There has long been confusion over how viroids are able to induce symptoms in plants without encoding any protein products within their sequences. Evidence now suggests that RNA silencing is involved in the process. First, changes to the viroid genome can dramatically alter its virulence. This reflects the fact that any siRNAs produced would have less complementary base pairing with target messenger RNA. Secondly, siRNAs corresponding to sequences from viroid genomes have been isolated from infected plants. Finally, transgenic expression of the noninfectious hpRNA of potato spindle tuber viroid develops all the corresponding viroid like symptoms. This evidence indicates that when viroids replicate via a double stranded intermediate RNA, they are targeted by a dicer enzyme and cleaved into siRNAs that are then loaded onto the RNA-induced silencing complex. The viroid siRNAs actually contain sequences capable of complementary base pairing with the plant’s own messenger RNAs and induction of degradation or inhibition of translation is what causes the classic viroid symptoms.
Virusoids are circular single-stranded RNAs dependent on plant viruses for replication and encapsidation. The genome of virusoids consists of several hundred nucleotides and only encodes structural proteins. Virusoids are similar to viroids in size, structure, and means of replication. Virusoids, while being studied in virology, are not considered as viruses but as subviral particles. Since they depend on helper viruses, they are classified as satellites.
The Pospiviroidae are a family of viroids, including the first viroid to be discovered, PSTVd. Their secondary structure is key to their biological activity. The classification of this family is based on differences in the conserved central region sequence. Pospiviroidae replication occurs in an asymmetric fashion via host cell RNA polymerase, RNase, and RNA ligase.
The Avsunviroidae are a family of viroids. At present three members are known. They consist of RNA genomes between 246-375 nucleotides in length. They are single stranded covalent circles and have intramolecular base pairing. All members lack a central conserved region. Replication occurs in the chloroplasts of plant cells. Key features of replication include no helper virus required and no proteins are encoded for. Unlike the other family of viroids, Pospiviroidae, Avsunviroidae are thought to replicate via a symmetrical rolling mechanism. It is thought the positive RNA strand acts as a template to form negative strands with the help of an enzyme thought to be RNA polymerase II. The negative RNA strands are then cleaved by ribozyme activity and circularizes. A second rolling circle mechanism forms a positive strand which is also cleaved by ribozyme activity and then ligated to become circular. The site of replication is unknown, but it is thought to be in the chloroplast and in the presence of Mg2+ ions.
Avocado sunblotch viroid (ASBV) is an important disease affecting avocado trees. Infections result in lower yields and poorer quality fruit. ASBV is the smallest known viroid that infects plants and is transmitted by pollen and infected seeds or budwood. Trees infected with the viroid often show no symptoms other than a reduction in yield. However, they are still carriers and can pass the disease onto other plants. Symptoms in more serious infections include depressed longitudinal streaks of yellow in the fruit. The fruit may also become red or white in color. Symptoms in the leaf are uncommon, but include bleached veins and petioles. Rectangular cracking patterns also occur in the bark of older branches. Infected but symptomless trees have a higher concentration of viroid particles than those showing symptoms. Symptomless trees also represent a greater danger in terms of spread of the viroid.
Key Points
- Viroid RNA does not code for any protein. The replication mechanism involves RNA polymerase II, an enzyme normally associated with synthesis of messenger RNA from DNA, which instead catalyzes “rolling circle” synthesis of new RNA using the viroid’s RNA as template.
- The first viroid to be identified was potato spindle tuber viroid (PSTVd). Some 33 species have been identified.
- There has long been confusion over how viroids are able to induce symptoms in plants without encoding any protein products within their sequences. Evidence now suggests that RNA silencing is involved in the process.
- Virusoids are circular single-stranded RNAs dependent on plant viruses for replication and encapsidation. Since they depend on helper viruses, they are classified as satellites.
Key Terms
- Virusoid : Circular single-stranded RNAs dependent on plant viruses for replication and encapsidation. The genome of virusoids consist of several hundred nucleotides and only encodes structural proteins.
- viroid : plant pathogens, of the order Viroidales, that consist of just a short section of RNA but without the protein coat typical of viruses


1.18: The Printing Press
Before we dive into a more detailed exploration of musical concepts and composers, I want us to take a closer look at the advent of the printing press. I don’t think it’s possible to overstate the significance of the printing press on history in general—and music in particular. While this reading focuses primarily on the printing of books, the same process was applied to music. With the rise of the press composers had a new opportunity for income and amateur musicians, many among the rising middle class, provided a new market for their music.
Musicians prior to the advent of the printing press were entirely dependent on either the church or ruling nobles for a living. The printing press didn’t change that system of patronage overnight, but it did provide additional opportunities and freedom for composers. The printing press, by virtue of the volume of copies of musical scores, also greatly enhanced the preservation of the music of this and later periods. As you read about the effect of the printing press on the structure of society, take a moment to reflect on the fact that you are living in a time when a similar leap forward in the distribution of information is happening.
A printing press is a device for applying pressure to an inked surface resting upon a printmedium (such as paper or cloth), thereby transferring the ink. Typically used for texts, the invention of the printing press is widely regarded as one of the most influential events in the second millennium, ushering in the period of modernity.
The printing press was invented in the Holy Roman Empire by Johannes Gutenberg, around 1440. Gutenberg, a goldsmith by profession, devised a hand mould to create metal movable type, and adapted screw presses and other existing technologies, to create a printing system. The mechanization of bookmaking led to the first mass production of books in Europe. A single Renaissance printing press could produce 3,600 pages per workday, compared to about 2,000 by typographic block-printing prevalent in East Asia, and a few by hand-copying. Books of bestselling authors like Luther or Erasmus were sold by the hundreds of thousands in their lifetime.
 By 1500, printing presses in operation throughout Western Europe had produced more than twenty million volumes. In the 16th century, with presses spreading further afield, their output rose to an estimated 150 to 200 million copies. The operation of a printing press became synonymous with the enterprise of printing, and lent its name to a new branch of media, the press. In 1620, the English philosopherFrancis Bacon wrote of printing as one of three inventions that had changed the world.
In the 19th century, the replacement of the hand-operated Gutenberg-style press by steam-powered rotary presses allowed printing on an industrial scale, while Western-style printing was adopted all over the world, becoming practically the sole medium for modern bulk printing, today typically using offset printing techniques.
Follow the link if you would like to read more about the printing press on Wikipedia .
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Printing press. Provided by : Wikpedia. Located at : http://en.Wikipedia.org/wiki/Printing_press . License : CC BY-SA: Attribution-ShareAlike


16.5: The Thyroid Gland
The thyroid gland is situated in the neck, just in front of the windpipe or trachea (see diagram 16.5). It produces the hormone thyroxine , which influences the rate of growth and development of young animals. In mature animals it increases the rate of chemical reactions in the body.
Thyroxine consists of 60% iodine and too little in the diet can cause goitre , an enlargement of the thyroid gland. Many inland soils in New Zealand contain almost no iodine so goitre can be common in stock when iodine supplements are not given. To add to the problem, chemicals called goitrogens that occur naturally in plants like kale that belong to the cabbage family , can also cause goitre even when there is adequate iodine available.
Diagram 16.5 - The thyroid and parathyroid glands


4: Ethical Considerations in Business Writing
Chapter Objectives
The purpose of this chapter is to:
- Explain the meaning of ethics and business ethics
- Describe the role of ethics in a business environment
- Explain what it means to be a professional of integrity
- Demonstrate ethical writing responsibilities
- Distinguish between ethical and legal responsibilities
- Describe four approaches for examining the ethical nature of a decision
The Importance of Ethical Responsibility
Safety, security, and confidentiality are foundational for establishing consumer confidence. In order to build a business relationship of trust, consumers rely on the reputation and identity a company projects. When employees behave unethically, illegal, or immorally, all trust is lost. It is often difficult to regain it.
In May 2024, over 500 million customers of Ticketmaster/Live Nation network were notified that their personal data and credit history were attained by a cyber-criminal group named named ShinyHunters (Baig, 2024).
In September 2017, Equifax, one of the leading credit reporting agencies was hacked, resulting in the private information of approximately 147 million consumers, from around the world, being compromised. Investigations by the Federal Trade Commission and congressional oversight committees concluded that Equifax administrators knew the company’s computer technology and software were vulnerable to encroachment. They had not implemented basic security measures and policies to prevent such a breach. Because of their unethical actions, the company was ordered to pay up to $425 million, as part of a global settlement. The fallout of the breach is anticipated to last several years, as the Federal Trade Commission has set up the opportunity for individuals to file claims against Equifax until the year 2024 (Federal Trade Commission, 2020).
Ethical Considerations
Should we care about ethics in our lives? Should we care about ethics in our practices in business and the professions? Should we care about ethics in all of our business writing media? Your goal is to understand why the answer to these questions is yes . Here are the categories that encompass ethical business writing.
- Authenticity
- Audience-centered
- Documentation
- Citation
- Responsibility
- Credibility
- Integrity
- Trust
- Truth
- Principles
- Transparency
- Justice
- Equity
- Ethical Intelligence
- Outreach
- Regulations
Ethical Writing: Setting a Firm Ethical Tone
Ethics consists of the individual standards of behavior you exhibit in your personal and professional lives. It establishes the levels of honesty, empathy, trustworthiness, and other virtues that identify your personal behavior and your public reputation .
In your personal live, ethics sets norms for the ways in which you interact with family, friends, and colleagues. Ethical acts are generally considered voluntary and personal—often based on your perception or position on what is right or what is wrong. In business, each employee has a choice in making an ethical or unethical decision. Whenever you think about the behavior you expect of yourself in your personal life and as a professional, you are engaging in a philosophical dialog with yourself to establish the standards of behavior you choose to uphold. That is your ethical position.
Business Ethics
Your success at work entails more than simply earning money and promotions. It may also mean treating stakeholders--employees, customers, clients--with honesty and respect. It may come from the sense of pride you feel about engaging in honest transactions, not just because the law or policy demands it, but because you demand it of yourself. It may lie in knowing the profit we make does not come from shortchanging others. Thus, business ethics guides the conduct by which companies and their representatives abide by laws, regulations, and policies. Business ethics means respecting the rights of stakeholders and the communities in which they live. Business ethics centers on principles of good conduct through civil, social, economical, environmental, and lawful actions.
Corporate culture is the acceptance of what is the normal operations of a business. It is a set of values and practices to which employees are expected to follow. Corporate leadership and governance usually sets what is acceptable and what will be tolerated. One way in which this occurs in business is through company policies and procedures. Policies are a set of organizational standards put in place to achieve a desired communication outcome. Procedures are the official ways in which a task is completed and approved.
To see an example of a corporate ethical code or mission statement, visit Johnson & Johnson and read "Our Credo" written by former chair Robert Wood Johnson. (Note: This link was viable at the time of publication; it may be inoperable now.)
World's Most Ethical Companies Ethisphere provides a list of companies recently deemed the most reputable companies for ethical culture according to their standards and research. (Note: This link was viable at the time of publication; it may be inoperable now.)
In our professional lives, ethics guides our interactions. Business ethics consist of businesses having a reputation of composing standards and displaying conduct that exemplifies civil and lawful actions that build trust, confidence, and goodwill amongst the business and its stakeholders.
The size of a company and whether it is public or private often determines its hierarchy. Whether it is a small business, a fortune 500 company, or a governmental entity, the business owner/CEO/manager/supervisor/employee will be the front person to represent the company and to communicate its message. That person’s image, ethics, opinions, and words, even if expressed during personal time, can be intertwined with the business. Stakeholders directly attach the unethical, racist, or otherwise controversial views and actions of an employee to the company’s image. Therefore, ethical codes of conduct and ethical standards must be written in a company's culture.
Ethical Outreach: The Importance of Connecting With the Audience
Clients, customers, suppliers, investors, retailers, employees, the media, the government, members of the surrounding community, competitors, and even the environment are stakeholders in a business; they are the individuals and entities affected by the business’s decisions, as Figure 1.1 illustrates.
Acting With Integrity
Integrity is unity between what we say and what we do. Being a professional of integrity means consistently striving to be the best person you can be in all your interactions with others. It means you practice what you do based upon reasoning, laws, morality, and justice. Integrity in business brings many advantages, not the least of which is that it is a critical factor in allowing business and society to function ethically and responsibly.
Successful corporate leaders and the companies they represent will take pride in their enterprise if they engage in business interactions with honesty and fair play. To treat stakeholders and all those affected by a business with dignity and respect is ethical. Laudable business practices serve the long-term interests of corporations. Why? Because customers, clients, employees, and society at large will much more willingly patronize a business and work hard on its behalf if that business is perceived as caring about the community it serves. And what type of firm has long-term customers and employees? One whose track record gives evidence of honest business practice. Should gaps in integrity appear, companies with an ethical base will react quickly and rectify improprieties that may occur. Company policies and regulations should not only consider the profit margin for the company but also the good the company brings to the communities that it represents.
Although ethics are an individual choice, in business, ethical choices by employees often define the success of a business. To show a connection and relationship with stakeholders, both near and far, successful companies adapt social, economical, legal, and voluntary responsibilities.
Social responsibility represents the interaction and collaboration with stakeholders to reflect the concerns of communities which a business serves. Social responsibility includes:
- Practices that are inclusive and diverse
- Activities that help local communities and stakeholders
An example of a business implementing social responsibility is Warby Parker. Executives at Warby Parker, an online prescription glasses retailer, displayed social responsibility by pledging $1 million to organizations and initiatives that focused on combating systemic racism. This outreach demonstrates that executives understand the importance of addressing inclusion and diversity.
Source: "Work Ethic (1 of 2)" by Brett Jordan is licensed under CC BY 2.0
Economical responsibility represents the obligation businesses have to offer and produce quality goods and services at an equitable price; and to pay employees equitable wages and benefits.
In essence, the first question any company should ask itself is: “Do we have a moral purpose?” Having a moral purpose requires focusing on the common good, which precedes the accumulation of profit and results in economic and social benefits. If companies seek the common good, profits generally will follow.
Legal responsibility represents practices that are in compliance with local, state, federal, and industry regulations and laws.
Many people conflate legal and ethical compliance. They are, however, totally different and call for different standards of behavior. What is legal isn't always ethical. The purpose of a law is to establish and maintain a functioning society. Compliance with these legal standards is strictly mandatory: If we violate these standards, we are subject to punishment. Therefore, compliance in terms of business ethics generally refers to the extent by which a company conducts its business operations in accordance with applicable regulations, statutes, and laws. Yet this represents only a baseline minimum. Companies today need to be focused not only on complying with the letter of the law but also on going above and beyond that basic mandatory requirement to consider ethical practices for their stakeholders and do what is right.
Ethical Considerations for Legal Responsibility
Ethical reasoning often is more topical and reflects the changes in consciousness that individuals and society undergo. Often, ethical thought precedes and sets the stage for changes in the law.
Some professions, such as engineering, journalism, and medicine have traditional codes of ethics. Essentially, a code of ethics is a commitment to treat with honesty and integrity customers, clients, employees, and others affiliated with a profession. The Hippocratic Oath, for example, is embraced by most professionals in health care today as an appropriate standard always owed to patients by physicians, nurses, and others in the field. This obligation traces its etymology to ancient Greece and the physician Hippocrates.
Source: "Code of Ethics for Government Service" by GovdocsGwen is marked with CC PDM 1.0
Voluntary responsibility represents the activities stakeholders would like company employees to partake to improve their local communities. Examples include volunteering at a local food bank or building a community playground.
Ethical Writing Considerations
Aristotle and the Concept of Phronesis, or Practical Wisdom
Phrónēsis (fro-NEE-sis) is a type of practical wisdom that enables us to act virtuously. In “The Big Idea: The Wise Leader,” a Harvard Business Review article on leadership and ethical decision-making, Ikujiro Nonaka, a Japanese organizational theorist, and Hirotaka Takeuchi (2011), a professor of Management Practice at Harvard Business School, discuss the gap between the theory and practice of ethics and the characteristics that make a wise leader. The authors conclude that chief executive officers (CEOs) must incorporate practical wisdom by leading with values and morals. The concept of practical wisdom dates back to Aristotle, who considered phrónēsis , which can also be defined as prudence, to be a key intellectual virtue. Phrónēsis enables people to make ethically sound judgments.
From a business writing perspective, effective writers impart practical wisdom by:
- Providing details and explanations
- Including information that is pertinent to the topic
- Analyzing content to determine its efficacy
- Providing introductions, overviews, and transitions
- Analyzing content to envision the consequences of possible actions or responses
- Including information that reflects the concerns and feedback of stakeholders
- Composing content that is easy to read and follow
- Creating a shared sense of purpose among the document, readers, and stakeholders and inspire people to work together in pursuit of a common goal
- Engaging as many people as possible in conversation and communicate using an audience-centered approach
Plagiarizing is misrepresenting the source or facts, most commonly when you claim the ideas you are writing about are yours. When you are researching professional documents, make sure you are using material with permission. If you are writing about what you’ve researched, make sure you are citing the sources of your information and giving credit to all the necessary researchers.
Artifical Intelligence (AI) and Large Language Models (LLMs)
Rather knowingly or unknowingly, artificial intelligence (AI) is prevalent in our everyday lives. Examples include the algorithms used in social media that push information to you based on your interests, demographics, and search history; face recognition to open your mobile device; spell checkers; and discussions that filter responses. AI is used to produce business documents by programming the of technology and datasets to compile information on a topic. The data that is generated is programmed to mirror human reasoning, thinking, and reactions. LLMs are an extension of AI.
LLMs understand and produce human language. An example of an LLM is Chatbots. The popularity of Chatbots and generative AI in producing documents is unprecedented. In a survey conducted by Forbes, 35% of business owners use or plan to use AI for content creation (Haan, 2023). For examples of how six companies use LLMs, visit this website .
The ethical issues with AI and LLMs are that the algorithms can be biased, producing results that are abstract, incorrect, or irrelevant for user groups. They do not represent authentic individual writing. Additionally, LLMs have little originality and authenticity because they pull from datasets that contain business jargon that have existed in business documents for hundreds of years. An example of this is the phrase, "as per your request." It a phrase that is considered formal business communication because of its staying power. However, it does not bring any new information to the discussion; it is a filler phrase. A better alternative is to delete this phrase and replace it with a detailed overview of the request and a response to it. Doing so will emphasize your writing and the authenticity you bring to the message.
A Clear Writing Style
Ethical writing decisions include word selection, content, organization, and tone. There should be no clichés and idioms in documents because they may pose a problem with comprehension. A cliché is a worn-out or overused expression that tends to sound trite and often doesn’t express what you truly mean to say. Examples include: the bottom line is, per your request, or take it or leave it . Idioms, also known as jargon, are words or expressions that are specialized, usually industry-specific vocabulary. When it is necessary to use jargon, be sure to explain its meaning to increase understandability by various audiences. Opinions and jokes should also be avoided in business documents. Inform and persuade your readers through facts and data instead of opinions and beliefs.
Effective Ways to Produce Authentic Writing
- Compose text that is topic- and audience-specific
- Develop research questions that expand current topics
- Conduct primary research to become the source of the information
- Provide writer’s commentary by explaining the meaning of information from secondary sources
- Write lead-ins that introduce sourced information
- Write lead-outs that explain the importance and implications of sourced information
- Provide examples to illustrate the significance of the information
Attribution refers to acknowledging and recognizing the source of information used in writing, ideas, concepts, and creations. Depending on the creation, the attribution is valid in various forms.
This rule also extends beyond writing to what is referred to as intellectual property. Intellectual property includes the following:
Patents and trademarks are company names ( WalMart), logos (the Target bullseye), processes or slogans (I’m lovin’ it) that belong to a person or company. None of these things can be used without proper recognition of or approval from the appropriate company or individual involved.
A company uses a ™ to show something is trademarked or an ® for something registered with the U.S Patent and Trademark Office. An example would be Nike ™ and their famous swoosh symbol.
The legal consequences are most notable when writing in the professional world. While plagiarizing may give you a failing grade in a class, plagiarizing in the workplace can not only get you fired, but could result in a costly lawsuit or possibly even jail time. It is not only ethical to follow these rules, it is an enforced law. Make sure you properly document all sources so as not to mislead a reader.
Spend a few minutes viewing the following website for clarification on copyright, trademark, and registered trademark: United States Patent and Trademark Office
Copyright law includes items whose distribution is protected by law (books, movies, or software). The copyright symbol is shown with a ©. Copyright is different from plagiarism in that it is a legal issue. Only the copyright holder, the person/organization who owns the protected item, can copy it.
Any written document, once produced, is copyrighted by law. That means if you are borrowing a good idea from a friend at another company, you must cite them as a source. Also, it is a good idea to cite sources from inside your own company, as well. You wouldn’t want someone else taking credit for your ideas. Why should you treat others any differently?
A Business Document is a Legal Document
Most documents written by employees represent the position and commitments of the organization itself. There are always legal issues to consider when writing a professional document and they reflect in writing style and content. Professional documents can serve as evidence in disputes over contracts and in product liability lawsuits. A lawsuit is a civil action brought in court. Today, the average company is involved in 400 lawsuits at any given time. While most companies win their lawsuits, being caught in a lawsuit has many consequences. Lawsuits cost companies time and money. The money spent on lawyers and the time spent in court takes away resources a company could use for improving business and products, or hiring additional employees. Lawsuits also have ramifications for a company’s reputation.
One of the main reasons a company gets involved in a lawsuit is because of unclear writing. When in a lawsuit, all documents may be subpoenaed. This means that any document from memos and emails to proposals and tweets can be subject to review by a court of law.
A Boilerplate for Consistency and Legality
Many companies also like to form a “custom” way of writing. Companies like Microsoft want all documents to be written in the same style and format. The only way to do this is to customize the “correct” way to write in order to represent Microsoft in a consistent manner. This is referred to as boilerplates. Boilerplates are text strings that appear in various company documents, without modification. If every document for a specific purpose is written using the same cadence, the opportunity for readers to understand the entire document may increase. Therefore, a boilerplate is a form of legalese writing in which language from previous documents is used in a new document. Boilerplates are acceptable for legal documents such as applications, contracts, descriptions, and mission statements.
Remember that in business writing, you are trying to persuade the reader using an ethical style. This means avoiding content that will not stand up in court. Make sure that the documents you write for your company are correct, persuasive, and ethical while also preserving your company’s competitive edge.
Ethical Infractions in Business Writing
You have learned what constitutes ethical behavior in business and business writing. In addition to discussing what to do, it is also necessary to discuss ethical infractions to avoid in order to strengthen your ethical knowledge and writing.
An ethical infraction is a decision that results in an ethical and moral breach. In business, the following breaches pertain to business writing and communication.
| Infraction | Definition |
| Defamation |
Causing harm to one's reputation or character by communication false and malicious information. Example: Communicating that the CEO of a company was arrested for drunk driving. |
| Libel | Causing harm to one's reputation or character by writing false and malicious information. |
| Slander | Causing hard to one's good name or character by speaking false and malicious information. |
|
Fraud |
Misrepresenting the truth for personal or professional gain. Example: Someone claiming to be an investor who deposits customers’ funds in her/his bank account instead of the customer's stock account. |
| Invasion of privacy |
Intruding in the business records of employees or stakeholders. Example: Communicating that a colleague is seeking assistance to overcome depression. |
Proprietary refers to company-specific confidential information. Only employees working on the project know the information. They usually sign an agreement stating they will maintain confidentiality, even after they leave the company. Confidentiality is important for companies to maintain a competitive edge.
Confidentiality does not mean that employees should not reveal a company’s unethical behavior. Whistler Blower laws are in place to assist workers who report unethical business behavior.
Ethical Sins of Commission and Omission
Unethical actions are classified into two categories: Sins of commission and sins of omission.
Sins of commission refer to unethical actions that includes manipulating or changing information to benefit a person or company a person represents.
To avoid using misleading or manipulating words and phrases, it is important to be open to alternative viewpoints; conduct an audience analysis. In preparing any type of writing, you may come across conflicting viewpoints. Keep your readers’ needs in mind and consider what lies behind their concerns. To help solve these problems it may also be good to make statements based on not only the primary audience, but also the secondary and gatekeeper audiences. Discussing several points of view and ideas on the subject will make your writing more persuasive and inclusive, resulting in the avoidance of sins of omission. In addition, be aware of biased language; make sure you have not included anything that may point to discriminatory assumptions or stereotypes.
Sins of omission refer to withholding or using inaccurate information in order to benefit a person or company a person represents. Do not use false or skewed facts or argue from such premises because you may deceive readers and cause them to make an uninformed decision.
Ethical Principles
Here are ethical principles specific to business writing.
Transparency
Include an in-depth discussion of content and data; place the information in a document so that it is easy to find and understand.
Neutrality
State information objectively and in a reporting voice.
Comprehensively
Compose information clearly to help your audience understand.
Integrity
Follow the rules of copyright. Attribute ideas, text, and visuals; cite and document sources.
Ethical Considerations for Success
Whatever hopes you have for your future, you almost certainly want to be successful in whatever career you choose. But what does success mean to you, and how will you know you have achieved it? Will you measure it in terms of wealth, status, power, or recognition?
Before blindly embarking on a quest to achieve these goals, which society considers important, stop and think about what a successful career means to you personally. Does it include a stellar reputation, colleagues whose good opinion you value, and the ability to think well of yourself? How might ethics guide your decision-making and contribute to your achievement of these goals? It is important to consider the answer to these questions before you are faced with a decision that you must make in the moment.
Ethical Considerations Checklist
What is social responsibility?
The interaction and collaboration with stakeholders to reflect the concerns of communities which a business serves.
What is ethics?
A principle of right or good conduct; the general nature of morals and of specific moral choices; the individual standards of behavior exhibited in personal and professional lives.
What is business ethics?
The conduct by which companies and their representatives abide by laws, regulations, and policies. Businesses displaying principles of right or good conduct through acting civilly and lawfully, and building goodwill.
What are the categories of ethical responsibility?
Following accepted principles that govern the conduct of an organization, group, person, people, culture, or movement. Ethical responsibility includes:
- Social
- Economical
- Legal
- Voluntary
What are sins of commission?
Changing facts for one’s benefit or the benefit of the company a person represents.
What are sins of omission?
Deleting facts for one’s benefit or the benefit of the company a person represents.
What is in-text citation?
Attributing an idea, quote, or fact to its source in the body of the document.
What is the importance of documentation?
Documentation includes the bibliographic information of a source to lead the reader to the source. Documentation acknowledges the research and writing of others.
What is boilerplating?
A form of legalese writing in which language from previous documents is used in a new document.
What are the four ethical principles?
- Reveal information or data to reveal factual communication
- Present unfavorable data
- Present the information clearly to help your audience understand
- Respect copyright information
What is proprietary information?
Company-specific confidential information.
What is slander?
Defamation is causing injury to one’s reputation by communicating malicious and false information by slander—oral communication.
What is defamation?
Causing injury to one’s reputation by communicating malicious and false information.
What is invasion of privacy?
Unreasonable intrusion into an individual’s personal life.
What is corporate culture?
A set of values and norms practiced within a company that is well established to which each employee is expected to conform.
How can corporate culture influence ethical behavior?
Corporate culture sets the level of social responsibility commitment a company displays.
General Ethical Tips
When you are faced with an ethical dilemma, answer these questions to assist you with determining your next steps.
- What is the nature of the ethical dilemma?
- What are the specific aspects of this dilemma that make me uncomfortable?
- What are my competing obligations in this dilemma?
- What advice does a trusted supervisor or mentor offer?
- Does my company’s code of conduct address this issue?
- Does my professional association’s code of conduct address this issue?
- What am I unwilling to do? What am I willing to do?
- How will I explain or justify my decision?
"General Ethical Tips" is a derivative of "Ethical Decision Checklist:" by ENGL 145: Technical and Report Writing. Bay College. is licensed under the Creative Commons Attribution 4.0 International License. https://www.oercommons.org/courses/b...t-writing/view . Kinonen, Amber, et al.
What Would You Do? Test Your Ethical Decision Making
Ethical decisions are often made in the moment; there may be limited time to ponder a decision. Therefore, it is important to think about what you would do when faced with ethical dilemma. These case studies provide the opportunity to analyze your reactions to the cases. As part of your analyses, answer the following questions in addition to the questions presented in each case study.
Which case are you analyzing?
What is the ethical situation?
What makes the case unethical?
How does the case impact business writing?
What are ethical writing considerations?
Which ethical principles does the case highlight?
What would you do in this situation?
Perhaps you have used or thought about using chatbots to produce documents for you. Identify the ethical concerns for using AI and LLMs.
All links live as of January 2025.
References
Baig, E. (2024, May). Ticketmaster Data Hack Hits Half a Billion Customers: What to Do. AARP. https://www.aarp.org/home-family/per...ta-breach.html
Federal Trade Commission. (2020, January). Equifax Data Breach Settlement. https://www.ftc.gov/enforcement/case...ach-settlement
Haan, K. (2023, April). How Businesses are Using Artificial Intelligence in 2023. Forbes. https://www.forbes.com/advisor/business/software/ai-in-business/#how_businesses_are_using_artificial_intelligence_section
Nonaka, I. & Takeuchi, H. (2011, May). The Big Idea: The Wise Leader. Harvard Business Review. https://hbr.org/2011/05/the-big-idea-the-wise-leader .
This work "Ethical Considerations in Business Writing" is a derivative of “ Business Ethics ” by OpenStax , used under CC BY. “Ethical Considerations in Business Writing” is licensed under CC BY by Valerie A. Gray .


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8.1: Polynomial Addition and Subtraction (and Combining Like Terms)
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To add and subtract polynomials, combine like terms. Like terms have the same variables with the same exponents. The coefficients of the terms may be different. Be careful when subtracting, to distribute the subtraction (think of it as an addition of (−1) times the polynomial).
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8.2: Polynomial Multiplication
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To multiply two monomials, multiply the terms together by adding the exponents and multiplying the numeric coefficients. To multiply a polynomial by a monomial, multiply all terms of the polynomial by the monomial. To multiply two binomials, use the FOIL technique to multiply: first terms, outer terms, inner terms and the last terms. To multiply two polynomials, use the distributive property to multiply every term in the first polynomial by every term in the second polynomial.
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8.3: Factoring and Finding Polynomial Solutions (Zeroes)
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There are several ways to find solutions of polynomials which are trinomials of the form ax^2 + bx + c = 0. These are also called the real zeroes of polynomials. These solutions are Trial and Check Factoring Method, Factor by Grouping Factoring Method, and The Quadratic Formula.


2: Fundamental Equilibrium Concepts
In this chapter, you will learn how to predict the position of the balance and the yield of a product of a reaction under specific conditions, how to change a reaction's conditions to increase or reduce yield, and how to evaluate an equilibrium system's reaction to disturbances.
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- 2.1: Prelude to Equilibrium
- Reactions can occur in both directions simultaneously (reactants to products and products to reactants) and eventually reach a state of balance. Reactions can occur in both directions simultaneously (reactants to products and products to reactants) and eventually reach a state of balance.
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- 2.2: Chemical Equilibria
- A reaction is at equilibrium when the amounts of reactants or products no longer change. Chemical equilibrium is a dynamic process, meaning the rate of formation of products by the forward reaction is equal to the rate at which the products re-form reactants by the reverse reaction.
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- 2.3: Equilibrium Constants
- For any reaction that is at equilibrium, the reaction quotient Q is equal to the equilibrium constant K for the reaction. If a reactant or product is a pure solid, a pure liquid, or the solvent in a dilute solution, the concentration of this component does not appear in the expression for the equilibrium constant. At equilibrium, the values of the concentrations of the reactants and products are constant and the reaction quotient will always equal K.
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- 2.4: Shifting Equilibria - Le Châtelier’s Principle
- Systems at equilibrium can be disturbed by changes to temperature, concentration, and, in some cases, volume and pressure; volume and pressure changes will disturb equilibrium if the number of moles of gas is different on the reactant and product sides of the reaction. The system's response to these disturbances is described by Le Châtelier's principle: The system will respond in a way that counteracts the disturbance. Not all changes to the system result in a disturbance of the equilibrium.
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- 2.5: Equilibrium Calculations
- The ratios of the rate of change in concentrations of a reaction are equal to the ratios of the coefficients in the balanced chemical equation. The sign of the coefficient of X is positive when the concentration increases and negative when it decreases. We learned to approach three basic types of equilibrium problems. When given the concentrations of the reactants and products at equilibrium, we can solve for the equilibrium constant.
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- 2.E: Fundamental Equilibrium Concepts (Exercises)
- These are homework exercises to accompany the Textmap created for "Chemistry" by OpenStax. Complementary General Chemistry question banks can be found for other Textmaps and can be accessed here. In addition to these publicly available questions, access to private problems bank for use in exams and homework is available to faculty only on an individual basis; please contact Delmar Larsen for an account with access permission.
Contributors
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Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


10.10: For Further Exploration
Articles
Venus
Dorminey, B. “Cool Science on a Hot World.” Astronomy (February 2006): 46. Five-page overview of Venus and the Venus Express mission plans.
Kargel, J. “Rivers of Venus.” Sky & Telescope (August 1997): 32. On lava channels.
Robertson, D. “Parched Planet.” Sky & Telescope (April 2008): 26. Overview of our understanding of the planet.
Robinson, C. “Magellan Reveals Venus.” Astronomy (February 1995): 32.
Stofan, E. “The New Face of Venus.” Sky & Telescope (August 1993): 22.
Zimmerman, R. “Taking Venus by Storm.” Astronomy (October 2008): 66. On results from the Venus Express mission.
Mars
Albee, A. “The Unearthly Landscapes of Mars.” Scientific American (June 2003): 44. Results from the Mars Global Surveyor and Mars Odyssey missions and an overview.
Bell, J. “A Fresh Look at Mars.” Astronomy (August 2015): 28. Nice summary of recent spacecraft results and how they are revising our understanding of Mars.
Bell, J. “Uncovering Mars’ Secret Past.” Sky & Telescope (July 2009): 22. How rovers and orbiters are helping us to understand Mars history and the role of water.
Bell, J. “The Red Planet’s Watery Past.” Scientific American (December 2006): 62. Rovers are furnishing proof that ancient Mars was wet.
Burnham, R. “Red Planet Rendezvous.” Astronomy (May 2006): 68. About Mariner Valley and a flyover film constructed from many still images.
Christensen, P. “The Many Faces of Mars.” Scientific American (July 2005): 32. Results from the Rover mission; evidence that Mars was once wet in places.
Lakdawalla, E. “The History of Water on Mars.” Sky & Telescope (September 2013): 16. Clear review of our current understanding of the role of water on Mars in different epochs.
Malin, M. “Visions of Mars.” Sky & Telescope (April 1999): 42. A geological tour of the red planet, with new Mars Global Surveyor images.
McEwen, A. “Mars in Motion.” Scientific American (May 2013): 58. On gullies and other surface changes.
McKay, C. & Garcia, V. “How to Search for Life on Mars.” Scientific American (June 2014): 44. Experiments future probes could perform.
Naeye, R. “Europe’s Eye on Mars.” Sky & Telescope (December 2005): 30. On the Mars Express mission and the remarkable close-up images it is sending.
Talcott, R. “Seeking Ground Truth on Mars.” Astronomy (October 2009): 34. How rovers and orbiters are helping scientists understand the red planet’s surface.
Websites
European Space Agency Mars Express Page: http://www.esa.int/Our_Activities/Space_Science/Mars_Express .
European Space Agency Venus Express Page: http://www.esa.int/Our_Activities/Space_Science/Venus_Express .
High Resolution Imaging Science Experiment: http://hirise.lpl.arizona.edu/ .
Jet Propulsion Lab Mars Exploration Page: http://mars.jpl.nasa.gov/ .
Mars Globe HD app: https://itunes.apple.com/us/app/mars-globe-hd/id376020224?mt=8 .
Mars Rover 360° Panorama: http://www.360cities.net/image/curiosity-rover-martian-solar-day-2#171.10,26.50,70.0 . Interactive.
NASA Center for Mars Exploration: http://www.nasa.gov/mission_pages/mars/main/index.html .
NASA Solar System Exploration Mars Page: http://solarsystem.nasa.gov/planets/mars .
NASA Solar System Exploration Venus Page: http://solarsystem.nasa.gov/planets/venus .
NASA’s Magellan Mission to Venus: www2.jpl.nasa.gov/magellan/.
Russian (Soviet) Venus Missions and Images: http://mentallandscape.com/C_CatalogVenus.htm .
Venus Atlas app: https://itunes.apple.com/us/app/venus-atlas/id317310503?mt=8 .
Venus Express Results Article: www.mpg.de/798302/F002_Focus_026-033.pdf.
Videos
50 Years of Mars Exploration: http://www.jpl.nasa.gov/video/details.php?id=1395 . NASA’s summary of all missions through MAVEN ; good quick overview (4:08).
Being a Mars Rover: What It’s Like to be an Interplanetary Explorer: https://www.youtube.com/watch?v=nRpCOEsPD54 . 2013 talk by Dr. Lori Fenton about what it’s like on the surface of Mars (1:07:24).
Magellan Radar Maps of Venus: https://www.youtube.com/watch?v=dK4JnGTCawM . Fly-overs of six regions on Venus from the Jet Propulsion Labs.
Our Curiosity : https://www.youtube.com/watch?v=XczKXWvokm4 . Mars Curiosity rover 2-year anniversary video narrated by Neil deGrasse Tyson and Felicia Day (6:01).
Planet Venus: The Deadliest Planet, Venus Surface and Atmosphere: https://www.youtube.com/watch?v=HqFVxWfVtoo . Quick tour of Venus’ atmosphere and surface (2:04).
Planetary Protection and Hitchhikers in the Solar System: The Danger of Mingling Microbes: https://www.youtube.com/watch?v=6iGC3uO7jBI . 2009 talk by Dr. Margaret Race on preventing contamination between worlds (1:28:50).


7.6: Sufficient, Complete and Ancillary Statistics
Basic Theory
The Basic Statistical Model
Consider again the basic statistical model, in which we have a random experiment with an observable random variable \(\bs X\) taking values in a set \(S\). Once again, the experiment is typically to sample \(n\) objects from a population and record one or more measurements for each item. In this case, the outcome variable has the form \[ \bs X = (X_1, X_2, \ldots, X_n) \] where \(X_i\) is the vector of measurements for the \(i\)th item. In general, we suppose that the distribution of \(\bs X\) depends on a parameter \(\theta\) taking values in a parameter space \(T\). The parameter \(\theta\) may also be vector-valued. We will sometimes use subscripts in probability density functions, expected values, etc. to denote the dependence on \(\theta\).
As usual, the most important special case is when \(\bs X\) is a sequence of independent, identically distributed random variables. In this case \(\bs X\) is a random sample from the common distribution.
Sufficient Statistics
Let \(U = u(\bs X)\) be a statistic taking values in a set \(R\). Intuitively, \(U\) is sufficient for \(\theta\) if \(U\) contains all of the information about \(\theta\) that is available in the entire data variable \(\bs X\). Here is the formal definition:
A statistic \(U\) is sufficient for \(\theta\) if the conditional distribution of \(\bs X\) given \(U\) does not depend on \(\theta \in T\).
Sufficiency is related to the concept of data reduction . Suppose that \(\bs X\) takes values in \(\R^n\). If we can find a sufficient statistic \(\bs U\) that takes values in \(\R^j\), then we can reduce the original data vector \(\bs X\) (whose dimension \(n\) is usually large) to the vector of statistics \(\bs U\) (whose dimension \(j\) is usually much smaller) with no loss of information about the parameter \(\theta\).
The following result gives a condition for sufficiency that is equivalent to this definition.
Let \(U = u(\bs X)\) be a statistic taking values in \(R\), and let \(f_\theta\) and \(h_\theta\) denote the probability density functions of \(\bs X\) and \(U\) respectively. Then \(U\) is suffcient for \(\theta\) if and only if the function on \( S \) given below does not depend on \( \theta \in T \): \[ \bs x \mapsto \frac{f_\theta(\bs x)}{h_\theta[u(\bs x)]} \]
Proof
The joint distribution of \((\bs X, U)\) is concentrated on the set \(\{(\bs x, y): \bs x \in S, y = u(\bs x)\} \subseteq S \times R\). The conditional PDF of \(\bs X\) given \(U = u(\bs x)\) is \(f_\theta(\bs x) \big/ h_\theta[u(\bs x)]\) on this set, and is 0 otherwise.
The definition precisely captures the intuitive notion of sufficiency given above, but can be difficult to apply. We must know in advance a candidate statistic \(U\), and then we must be able to compute the conditional distribution of \(\bs X\) given \(U\). The Fisher-Neyman factorization theorem given next often allows the identification of a sufficient statistic from the form of the probability density function of \(\bs X\). It is named for Ronald Fisher and Jerzy Neyman.
Fisher-Neyman Factorization Theorem . Let \(f_\theta\) denote the probability density function of \(\bs X\) and suppose that \(U = u(\bs X)\) is a statistic taking values in \(R\). Then \(U\) is sufficient for \(\theta\) if and only if there exists \(G: R \times T \to [0, \infty)\) and \(r: S \to [0, \infty)\) such that \[ f_\theta(\bs x) = G[u(\bs x), \theta] r(\bs x); \quad \bs x \in S, \; \theta \in T \]
Proof
Let \( h_\theta \) denote the PDF of \( U \) for \( \theta \in T \). If \( U \) is sufficient for \( \theta \), then from the previous theorem, the function \( r(\bs x) = f_\theta(\bs x) \big/ h_\theta[u(\bs x)] \) for \( \bs x \in S\) does not depend on \( \theta \in T \). Hence \( f_\theta(\bs x) = h_\theta[u(\bs x)] r(\bs x) \) for \( (\bs x, \theta) \in S \times T \) and so \((\bs x, \theta) \mapsto f_\theta(\bs x) \) has the form given in the theorem. Conversely, suppose that \( (\bs x, \theta) \mapsto f_\theta(\bs x) \) has the form given in the theorem. Then there exists a positive constant \( C \) such that \( h_\theta(y) = C G(y, \theta) \) for \( \theta \in T \) and \( y \in R \). Hence \( f_\theta(\bs x) \big/ h_\theta[u(x)] = r(\bs x) / C\) for \( \bs x \in S \), independent of \( \theta \in T \).
Note that \(r\) depends only on the data \(\bs x\) but not on the parameter \(\theta\). Less technically, \(u(\bs X)\) is sufficient for \(\theta\) if the probability density function \(f_\theta(\bs x)\) depends on the data vector \(\bs x\) and the parameter \(\theta\) only through \(u(\bs x)\).
If \(U\) and \(V\) are equivalent statistics and \(U\) is sufficient for \(\theta\) then \(V\) is sufficient for \(\theta\).
Minimal Sufficient Statistics
The entire data variable \(\bs X\) is trivially sufficient for \(\theta\). However, as noted above, there usually exists a statistic \(U\) that is sufficient for \(\theta\) and has smaller dimension, so that we can achieve real data reduction. Naturally, we would like to find the statistic \(U\) that has the smallest dimension possible. In many cases, this smallest dimension \(j\) will be the same as the dimension \(k\) of the parameter vector \(\theta\). However, as we will see, this is not necessarily the case; \(j\) can be smaller or larger than \(k\). An example based on the uniform distribution is given in (38) .
Suppose that a statistic \(U\) is sufficient for \(\theta\). Then \(U\) is minimally sufficient if \(U\) is a function of any other statistic \(V\) that is sufficient for \(\theta\).
Once again, the definition precisely captures the notion of minimal sufficiency, but is hard to apply. The following result gives an equivalent condition.
Let \(f_\theta\) denote the probability density function of \(\bs X\) corresponding to the parameter value \(\theta \in T\) and suppose that \(U = u(\bs X)\) is a statistic taking values in \(R\). Then \(U\) is minimally sufficient for \(\theta\) if the following condition holds: for \(\bs x \in S\) and \(\bs y \in S\) \[ \frac{f_\theta(\bs x)}{f_\theta(\bs{y})} \text{ is independent of } \theta \text{ if and only if } u(\bs x) = u(\bs{y}) \]
Proof
Suppose that the condition in the theorem is satisfied. Then the PDF \(f_\theta\) of \( \bs X \) must have the form given in the factorization theorem (3) so \(U\) is sufficient for \(\theta\). Next, suppose that \(V = v(\bs X)\) is another sufficient statistic for \( \theta \), taking values in \( R \). From the factorization theorem, there exists \( G: R \times T \to [0, \infty) \) and \( r: S \to [0, \infty) \) such that \( f_\theta(\bs x) = G[v(\bs x), \theta] r(\bs x) \) for \( (\bs x, \theta) \in S \times T \). Hence if \( \bs x, \bs y \in S \) and \( v(\bs x) = v(\bs y) \) then \[\frac{f_\theta(\bs x)}{f_\theta(\bs{y})} = \frac{G[v(\bs x), \theta] r(\bs x)}{G[v(\bs{y}), \theta] r(\bs{y})} = \frac{r(\bs x)}{r(\bs y)}\] does not depend on \( \theta \in \Theta \). Hence from the condition in the theorem, \( u(\bs x) = u(\bs y) \) and it follows that \( U \) is a function of \( V \).
If \(U\) and \(V\) are equivalent statistics and \(U\) is minimally sufficient for \(\theta\) then \(V\) is minimally sufficient for \(\theta\).
Properties of Sufficient Statistics
Sufficiency is related to several of the methods of constructing estimators that we have studied.
Suppose that \(U\) is sufficient for \(\theta\) and that there exists a maximum likelihood estimator of \(\theta\). Then there exists a maximum likelihood estimator \(V\) that is a function of \(U\).
Proof
From the factorization theorem (3) , the log likelihood function for \( \bs x \in S \) is \[\theta \mapsto \ln G[u(\bs x), \theta] + \ln r(\bs x)\] Hence a value of \(\theta\) that maximizes this function, if it exists, must be a function of \(u(\bs x)\).
In particular, suppose that \(V\) is the unique maximum likelihood estimator of \(\theta\) and that \(V\) is sufficient for \(\theta\). If \(U\) is sufficient for \(\theta\) then \(V\) is a function of \(U\) by the previous theorem. Hence it follows that \(V\) is minimally sufficient for \(\theta\). Our next result applies to Bayesian analysis.
Suppose that the statistic \(U = u(\bs X)\) is sufficient for the parameter \(\theta\) and that \( \theta \) is modeled by a random variable \( \Theta \) with values in \( T \). Then the posterior distribution of \( \Theta \) given \( \bs X = \bs x \in S \) is a function of \( u(\bs x) \).
Proof
Let \( h \) denote the prior PDF of \( \Theta \) and \( f(\cdot \mid \theta) \) the conditional PDF of \( \bs X \) given \( \Theta = \theta \in T \). By the factorization theorem (3) , this conditional PDF has the form \( f(\bs x \mid \theta) = G[u(\bs x), \theta] r(\bs x) \) for \( \bs x \in S \) and \( \theta \in T \). The posterior PDF of \( \Theta \) given \( \bs X = \bs x \in S \) is \[ h(\theta \mid \bs x) = \frac{h(\theta) f(\bs x \mid \theta)}{f(\bs x)}, \quad \theta \in T \] where the function in the denominator is the marginal PDF of \( \bs X \), or simply the normalizing constant for the function of \( \theta \) in the numerator. Let's suppose that \( \Theta \) has a continuous distribution on \( T \), so that \( f(\bs x) = \int_T h(t) G[u(\bs x), t] r(\bs x) dt \) for \( \bs x \in S \). Then the posterior PDF simplifies to \[ h(\theta \mid \bs x) = \frac{h(\theta) G[u(\bs x), \theta]}{\int_T h(t) G[u(\bs x), t] dt} \] which depends on \(\bs x \in S \) only through \( u(\bs x) \).
Continuing with the setting of Bayesian analysis, suppose that \( \theta \) is a real-valued parameter. If we use the usual mean-square loss function, then the Bayesian estimator is \( V = \E(\Theta \mid \bs X) \). By the previous result, \( V \) is a function of the sufficient statistics \( U \). That is, \( \E(\Theta \mid \bs X) = \E(\Theta \mid U) \).
The next result is the Rao-Blackwell theorem , named for CR Rao and David Blackwell. The theorem shows how a sufficient statistic can be used to improve an unbiased estimator.
Rao-Blackwell Theorem . Suppose that \(U\) is sufficient for \(\theta\) and that \(V\) is an unbiased estimator of a real parameter \(\lambda = \lambda(\theta)\). Then \(\E_\theta(V \mid U)\) is also an unbiased estimator of \( \lambda \) and is uniformly better than \(V\).
Proof
This follows from basic properties of conditional expected value and conditional variance. First, since \(V\) is a function of \(\bs X\) and \(U\) is sufficient for \(\theta\), \(\E_\theta(V \mid U)\) is a valid statistic; that is, it does not depend on \(\theta\), in spite of the formal dependence on \(\theta\) in the expected value. Next, \(\E_\theta(V \mid U)\) is a function of \(U\) and \(\E_\theta[\E_\theta(V \mid U)] = \E_\theta(V) = \lambda\) for \(\theta \in \Theta\). Thus \(\E_\theta(V \mid U)\) is an unbiased estimator of \(\lambda\). Finally \(\var_\theta[\E_\theta(V \mid U)] = \var_\theta(V) - \E_\theta[\var_\theta(V \mid U)] \le \var_\theta(V)\) for any \(\theta \in T\).
Complete Statistics
Suppose that \(U = u(\bs X)\) is a statistic taking values in a set \(R\). Then \(U\) is a complete statistic for \(\theta\) if for any function \(r: R \to \R\) \[ \E_\theta\left[r(U)\right] = 0 \text{ for all } \theta \in T \implies \P_\theta\left[r(U) = 0\right] = 1 \text{ for all } \theta \in T \]
To understand this rather strange looking condition, suppose that \(r(U)\) is a statistic constructed from \(U\) that is being used as an estimator of 0 (thought of as a function of \(\theta\)). The completeness condition means that the only such unbiased estimator is the statistic that is 0 with probability 1.
If \(U\) and \(V\) are equivalent statistics and \(U\) is complete for \(\theta\) then \(V\) is complete for \(\theta\).
The next result shows the importance of statistics that are both complete and sufficient; it is known as the Lehmann-Scheffé theorem , named for Erich Lehmann and Henry Scheffé.
Lehmann-Scheffé Theorem . Suppose that \(U\) is sufficient and complete for \(\theta\) and that \(V = r(U)\) is an unbiased estimator of a real parameter \(\lambda = \lambda(\theta)\). Then \(V\) is a uniformly minimum variance unbiased estimator (UMVUE) of \(\lambda\).
Proof
Suppose that \(W\) is an unbiased estimator of \(\lambda\). By the Rao-Blackwell theorem (10) , \(\E(W \mid U)\) is also an unbiased estimator of \(\lambda\) and is uniformly better than \(W\). Since \(\E(W \mid U)\) is a function of \(U\), it follows from completeness that \(V = \E(W \mid U)\) with probability 1.
Ancillary Statistics
Suppose that \(V = v(\bs X)\) is a statistic taking values in a set \(R\). If the distribution of \(V\) does not depend on \(\theta\), then \(V\) is called an ancillary statistic for \(\theta\).
Thus, the notion of an ancillary statistic is complementary to the notion of a sufficient statistic. A sufficient statistic contains all available information about the parameter; an ancillary statistic contains no information about the parameter. The following result, known as Basu's Theorem and named for Debabrata Basu, makes this point more precisely.
Basu's Theorem . Suppose that \(U\) is complete and sufficient for a parameter \(\theta\) and that \(V\) is an ancillary statistic for \( \theta \). Then \(U\) and \(V\) are independent.
Proof
Let \(g\) denote the probability density function of \(V\) and let \(v \mapsto g(v \mid U)\) denote the conditional probability density function of \(V\) given \(U\). From properties of conditional expected value, \(\E[g(v \mid U)] = g(v)\) for \(v \in R\). But then from completeness, \(g(v \mid U) = g(v)\) with probability 1.
If \(U\) and \(V\) are equivalent statistics and \(U\) is ancillary for \(\theta\) then \(V\) is ancillary for \(\theta\).
Applications and Special Distributions
In this subsection, we will explore sufficient, complete, and ancillary statistics for a number of special distributions. As always, be sure to try the problems yourself before looking at the solutions.
The Bernoulli Distribution
Recall that the Bernoulli distribuiton with parameter \(p \in (0, 1)\) is a discrete distribution on \( \{0, 1\} \) with probability density function \( g \) defined by \[ g(x) = p^x (1 - p)^{1-x}, \quad x \in \{0, 1\} \] Suppose that \(\bs X = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the Bernoulli distribution with parameter \(p\). Equivalently, \(\bs X\) is a sequence of Bernoulli trials , so that in the usual langauage of reliability, \(X_i = 1\) if trial \(i\) is a success, and \(X_i = 0\) if trial \(i\) is a failure. The Bernoulli distribution is named for Jacob Bernoulli and is studied in more detail in the chapter on Bernoulli Trials
Let \(Y = \sum_{i=1}^n X_i\) denote the number of successes. Recall that \(Y\) has the binomial distribution with parameters \(n\) and \(p\), and has probability density function \( h \) defined by \[ h(y) = \binom{n}{y} p^y (1 - p)^{n-y}, \quad y \in \{0, 1, \ldots, n\} \]
\(Y\) is sufficient for \(p\). Specifically, for \( y \in \{0, 1, \ldots, n\} \), the conditional distribution of \(\bs X\) given \(Y = y\) is uniform on the set of points \[ D_y = \left\{(x_1, x_2, \ldots, x_n) \in \{0, 1\}^n: x_1 + x_2 + \cdots + x_n = y\right\} \]
Proof
The joint PDF \( f \) of \( \bs X \) is defined by \[ f(\bs x) = g(x_1) g(x_2) \cdots g(x_n) = p^y (1 - p)^{n-y}, \quad \bs x = (x_1, x_2, \ldots, x_n) \in \{0, 1\}^n \] where \( y = \sum_{i=1}^n x_i \). Now let \( y \in \{0, 1, \ldots, n\} \). Given \( Y = y \), \( \bs X \) is concentrated on \( D_y \) and \[ \P(\bs X = \bs x \mid Y = y) = \frac{\P(\bs X = \bs x)}{\P(Y = y)} = \frac{p^y (1 - p)^{n-y}}{\binom{n}{y} p^y (1 - p)^{n-y}} = \frac{1}{\binom{n}{y}}, \quad \bs x \in D_y \] Of course, \( \binom{n}{y} \) is the cardinality of \(D_y\).
This result is intuitively appealing: in a sequence of Bernoulli trials, all of the information about the probability of success \(p\) is contained in the number of successes \(Y\). The particular order of the successes and failures provides no additional information. Of course, the sufficiency of \(Y\) follows more easily from the factorization theorem (3) , but the conditional distribution provides additional insight.
\(Y\) is complete for \(p\) on the parameter space \( (0, 1) \).
Proof
If \(r: \{0, 1, \ldots, n\} \to \R\), then \[\E[r(Y)] = \sum_{y=0}^n r(y) \binom{n}{k} p^y (1 - p)^{n-y} = (1 - p)^n \sum_{y=0}^n r(y) \binom{n}{y} \left(\frac{p}{1 - p}\right)^y\] The last sum is a polynomial in the variable \(t = \frac{p}{1 - p} \in (0, \infty)\). If this polynomial is 0 for all \(t \in (0, \infty)\), then all of the coefficients must be 0. Hence we must have \( r(y) = 0 \) for \( y \in \{0, 1, \ldots, n\} \).
The proof of the last result actually shows that if the parameter space is any subset of \( (0, 1) \) containing an interval of positive length, then \( Y \) is complete for \( p \). But the notion of completeness depends very much on the parameter space. The following result considers the case where \(p\) has a finite set of values.
Suppose that the parameter space \( T \subset (0, 1) \) is a finite set with \( k \in \N_+ \) elements. If the sample size \(n \) is at least \( k \), then \(Y\) is not complete for \(p\).
Proof
Suppose that \( r: \{0, 1, \ldots, n\} \to \R \) and that \( \E[r(Y)] = 0 \) for \( p \in T \). Then we have \[ \sum_{y=0}^n \binom{n}{y} p^y (1 - p)^{n-y} r(y) = 0, \quad p \in T \] This is a set of \( k \) linear, homogenous equations in the variables \( (r(0), r(1), \ldots, r(n)) \). Since \( n \ge k \), we have at least \( k + 1 \) variables, so there are infinitely many nontrivial solutions.
The sample mean \(M = Y / n\) (the sample proportion of successes) is clearly equivalent to \( Y \) (the number of successes), and hence is also sufficient for \( p \) and is complete for \(p \in (0, 1)\). Recall that the sample mean \( M \) is the method of moments estimator of \( p \), and is the maximum likelihood estimator of \( p \) on the parameter space \( (0, 1) \).
In Bayesian analysis, the usual approach is to model \( p \) with a random variable \( P \) that has a prior beta distribution with left parameter \( a \in (0, \infty) \) and right parameter \( b \in (0, \infty) \). Then the posterior distribution of \( P \) given \( \bs X \) is beta with left parameter \( a + Y \) and right parameter \( b + (n - Y) \). The posterior distribution depends on the data only through the sufficient statistic \( Y \), as guaranteed by theorem (9) .
The sample variance \( S^2 \) is an UMVUE of the distribution variance \( p (1 - p) \) for \( p \in (0, 1) \), and can be written as \[ S^2 = \frac{Y}{n - 1} \left(1 - \frac{Y}{n}\right) \]
Proof
Recall that the sample variance can be written as \[S^2 = \frac{1}{n - 1} \sum_{i=1}^n X_i^2 - \frac{n}{n - 1} M^2\] But \(X_i^2 = X_i\) since \(X_i\) is an indicator variable, and \(M = Y / n\). Substituting gives the representation above. In general, \(S^2\) is an unbiased estimator of the distribution variance \(\sigma^2\). But in this case, \(S^2\) is a function of the complete, sufficient statistic \(Y\), and hence by the Lehmann Scheffé theorem (13) , \(S^2\) is an UMVUE of \(\sigma^2 = p (1 - p)\).
The Poisson Distribution
Recall that the
Poisson distribution
with parameter \(\theta \in (0, \infty)\) is a discrete distribution on \( \N \) with probability density function \( g \) defined by \[ g(x) = e^{-\theta} \frac{\theta^x}{x!}, \quad x \in \N \] The Poisson distribution is named for Simeon Poisson and is used to model the number of
random points
in region of time or space, under certain ideal conditions. The parameter \(\theta\) is proportional to the size of the region, and is both the mean and the variance of the distribution. The Poisson distribution is studied in more detail in the chapter on Poisson process.
Suppose now that \(\bs X = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the Poisson distribution with parameter \(\theta\). Recall that the sum of the scores \(Y = \sum_{i=1}^n X_i\) also has the Poisson distribution, but with parameter \(n \theta\).
The statistic \(Y\) is sufficient for \(\theta\). Specifically, for \( y \in \N \), the conditional distribution of \( \bs X \) given \( Y = y \) is the multinomial distribution with \( y \) trials, \( n \) trial values, and uniform trial probabilities.
Proof
The joint PDF \( f \) of \( \bs X \) is defined by \[ f(\bs x) = g(x_1) g(x_2) \cdot g(x_n) = \frac{e^{-n \theta} \theta^y}{x_1! x_2! \cdots x_n!}, \quad \bs x = (x_1, x_2, \ldots, x_n) \in \N^n \] where \( y = \sum_{i=1}^n x_i \). Given \( Y = y \in \N \), random vector \( \bs X \) takes values in the set \(D_y = \left\{\bs x = (x_1, x_2, \ldots, x_n) \in \N^n: \sum_{i=1}^n x_i = y\right\}\). Moreover, \[\P(\bs X = \bs x \mid Y = y) = \frac{\P(\bs X = \bs x)}{\P(Y = y)} = \frac{e^{-n \theta} \theta^y / (x_1! x_2! \cdots x_n!)}{e^{-n \theta} (n \theta)^y / y!} = \frac{y!}{x_1! x_2! \cdots x_n!} \frac{1}{n^y}, \quad \bs x \in D_y\] The last expression is the PDF of the multinomial distribution stated in the theorem. Of course, the important point is that the conditional distribution does not depend on \( \theta \).
As before, it's easier to use the factorization theorem to prove the sufficiency of \( Y \), but the conditional distribution gives some additional insight.
\(Y\) is complete for \(\theta \in (0, \infty)\).
Proof
If \(r: \N \to \R\) then \[\E\left[r(Y)\right] = \sum_{y=0}^\infty e^{-n \theta} \frac{(n \theta)^y}{y!} r(y) = e^{-n \theta} \sum_{y=0}^\infty \frac{n^y}{y!} r(y) \theta^y\] The last sum is a power series in \(\theta\) with coefficients \( n^y r(y) / y! \) for \( y \in \N \). If this series is 0 for all \(\theta\) in an open interval, then the coefficients must be 0 and hence \( r(y) = 0 \) for \( y \in \N \).
As with our discussion of Bernoulli trials, the sample mean \( M = Y / n \) is clearly equivalent to \( Y \) and hence is also sufficient for \( \theta \) and complete for \( \theta \in (0, \infty) \). Recall that \( M \) is the method of moments estimator of \( \theta \) and is the maximum likelihood estimator on the parameter space \( (0, \infty) \).
An UMVUE of the parameter \(\P(X = 0) = e^{-\theta}\) for \( \theta \in (0, \infty) \) is \[ U = \left( \frac{n-1}{n} \right)^Y \]
Proof
The probability generating function of \(Y\) is \[ P(t) = \E(t^Y) = e^{n \theta(t - 1)}, \quad t \in \R \] Hence \[ \E\left[\left(\frac{n - 1}{n}\right)^Y\right] = \exp \left[n \theta \left(\frac{n - 1}{n} - 1\right)\right] = e^{-\theta}, \quad \theta \in (0, \infty) \] So \( U = [(n - 1) / n]^Y \) is an unbiased estimator of \( e^{-\theta} \). Since \( U \) is a function of the complete, sufficient statistic \( Y \), it follows from the Lehmann Scheffé theorem (13) that \( U \) is an UMVUE of \( e^{-\theta} \).
The Normal Distribution
Recall that the normal distribution with mean \(\mu \in \R\) and variance \(\sigma^2 \in (0, \infty)\) is a continuous distribution on \( \R \) with probability density function \( g \) defined by \[ g(x) = \frac{1}{\sqrt{2 \, \pi} \sigma} \exp\left[-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2\right], \quad x \in \R \] The normal distribution is often used to model physical quantities subject to small, random errors, and is studied in more detail in the chapter on Special Distributions. Because of the central limit theorem, the normal distribution is perhaps the most important distribution in statistics.
Suppose that \(\bs X = (X_1, X_2, \ldots, X_n)\) is a random sample from the normal distribution with mean \(\mu\) and variance \(\sigma^2\). Then each of the following pairs of statistics is minimally sufficient for \( (\mu, \sigma^2) \)
- \((Y, V)\) where \(Y = \sum_{i=1}^n X_i\) and \(V = \sum_{i=1}^n X_i^2\).
- \(\left(M, S^2\right)\) where \(M = \frac{1}{n} \sum_{i=1}^n X_i\) is the sample mean and \(S^2 = \frac{1}{n - 1} \sum_{i=1}^n (X_i - M)^2\) is the sample variance.
- \( (M, T^2) \) where \( T^2 = \frac{1}{n} \sum_{i=1}^n (X_i - M)^2 \) is the biased sample variance.
Proof
- The joint PDF \( f \) of \( \bs X \) is given by \[ f(\bs x) = g(x_1) g(x_2) \cdots g(x_n) = \frac{1}{(2 \pi)^{n/2} \sigma^n} \exp\left[-\frac{1}{2 \sigma^2} \sum_{i=1}^n (x_i - \mu)^2\right], \quad \bs x = (x_1, x_2 \ldots, x_n) \in \R^n \] After some algebra, this can be written as \[ f(\bs x) = \frac{1}{(2 \pi)^{n/2} \sigma^n} e^{-n \mu^2 / \sigma^2} \exp\left(-\frac{1}{2 \sigma^2} \sum_{i=1}^n x_i^2 + \frac{2 \mu}{\sigma^2} \sum_{i=1}^n x_i \right), \quad \bs x = (x_1, x_2 \ldots, x_n) \in \R^n\] It follows from the factorization theorem (3) that \( (Y, V) \) is sufficient for \(\left(\mu, \sigma^2\right)\). Minimal sufficiency follows from the condition in theorem (6) .
- Note that \( M = \frac{1}{n} Y, \; S^2 = \frac{1}{n - 1} V - \frac{n}{n - 1} M^2\). Hence \(\left(M, S^2\right)\) is equivalent to \( (Y, V) \) and so \(\left(M, S^2\right)\) is also minimally sufficient for \(\left(\mu, \sigma^2\right)\).
- Similarly, \( M = \frac{1}{n} Y \) and \( T^2 = \frac{1}{n} V - M^2 \). Hence \( (M, T^2) \) is equivalent to \( (Y, V) \) and so \( (M, T^2) \) is also minimally sufficient for \( (\mu, \sigma^2) \).
Recall that \( M \) and \( T^2 \) are the method of moments estimators of \( \mu \) and \( \sigma^2 \), respectively, and are also the maximum likelihood estimators on the parameter space \( \R \times (0, \infty) \).
Run the normal estimation experiment 1000 times with various values of the parameters. Compare the estimates of the parameters in terms of bias and mean square error.
Sometimes the variance \( \sigma^2 \) of the normal distribution is known, but not the mean \( \mu \). It's rarely the case that \( \mu \) is known but not \( \sigma^2 \). Nonetheless we can give sufficient statistics in both cases.
Suppose again that \( \bs X = (X_1, X_2, \ldots, X_n) \) is a random sample from the normal distribution with mean \( \mu \in \R \) and variance \( \sigma^2 \in (0, \infty)\). If
- If \( \sigma^2 \) is known then \( Y = \sum_{i=1}^n X_i \) is minimally sufficient for \( \mu \).
- If \( \mu \) is known then \( U = \sum_{i=1}^n (X_i - \mu)^2 \) is sufficient for \( \sigma^2 \).
Proof
- This results follow from the second displayed equation for the PDF \( f(\bs x) \) of \( \bs X \) in the proof of the previous theorem.
- This result follows from the first displayed equation for the PDF \( f(\bs x) \) of \( bs X \) in the proof of the previous theorem.
Of course by equivalence, in part (a) the sample mean \( M = Y / n \) is minimally sufficient for \( \mu \), and in part (b) the special sample variance \( W = U / n \) is minimally sufficient for \( \sigma^2 \). Moreover, in part (a), \( M \) is complete for \( \mu \) on the parameter space \( \R \) and the sample variance \( S^2 \) is ancillary for \( \mu \) (Recall that \( (n - 1) S^2 / \sigma^2 \) has the chi-square distribution with \( n - 1 \) degrees of freedom.) It follows from Basu's theorem (15) that the sample mean \( M \) and the sample variance \( S^2 \) are independent. We proved this by more direct means in the section on special properties of normal samples, but the formulation in terms of sufficient and ancillary statistics gives additional insight.
The Gamma Distribution
Recall that the gamma distribution with shape parameter \(k \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\) is a continuous distribution on \( (0, \infty) \) with probability density function \( g \) given by \[ g(x) = \frac{1}{\Gamma(k) b^k} x^{k-1} e^{-x / b}, \quad x \in (0, \infty) \] The gamma distribution is often used to model random times and certain other types of positive random variables, and is studied in more detail in the chapter on Special Distributions.
Suppose that \(\bs X = (X_1, X_2, \ldots, X_n)\) is a random sample from the gamma distribution with shape parameter \(k\) and scale parameter \(b\). Each of the following pairs of statistics is minimally sufficient for \((k, b)\)
- \((Y, V)\) where \(Y = \sum_{i=1}^n X_i\) is the sum of the scores and \(V = \prod_{i=1}^n X_i\) is the product of the scores.
- \((M, U)\) where \(M = Y / n\) is the sample (arithmetic) mean of \(\bs X\) and \(U = V^{1/n}\) is the sample geometric mean of \(\bs X\).
Proof
- The joint PDF \( f \) of \( \bs X \) is given by \[ f(\bs x) = g(x_1) g(x_2) \cdots g(x_n) = \frac{1}{\Gamma^n(k) b^{nk}} (x_1 x_2 \ldots x_n)^{k-1} e^{-(x_1 + x_2 + \cdots + x_n) / b}, \quad \bs x = (x_1, x_2, \ldots, x_n) \in (0, \infty)^n \] From the factorization theorem (3) , \( (Y, V) \) is sufficient for \( (k, b) \). Minimal sufficiency follows from condition (6) .
- Clearly \( M = Y / n \) is equivalent to \( Y \) and \( U = V^{1/n} \) is equivalent to \( V \). Hence \( (M, U) \) is also minimally sufficient for \( (k, b) \).
Recall that the method of moments estimators of \( k \) and \( b \) are \( M^2 / T^2 \) and \( T^2 / M \), respectively, where \( M = \frac{1}{n} \sum_{i=1}^n X_i \) is the sample mean and \( T^2 = \frac{1}{n} \sum_{i=1}^n (X_i - M)^2 \) is the biased sample variance. If the shape parameter \( k \) is known, \( \frac{1}{k} M \) is both the method of moments estimator of \( b \) and the maximum likelihood estimator on the parameter space \( (0, \infty) \). Note that \( T^2 \) is not a function of the sufficient statistics \( (Y, V) \), and hence estimators based on \( T^2 \) suffer from a loss of information.
Run the gamma estimation experiment 1000 times with various values of the parameters and the sample size \( n \). Compare the estimates of the parameters in terms of bias and mean square error.
The proof of the last theorem actually shows that \( Y \) is sufficient for \( b \) if \( k \) is known, and that \( V \) is sufficient for \( k \) if \( b \) is known.
Suppose again that \(\bs X = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the gamma distribution with shape parameter \( k \in (0, \infty) \) and scale parameter \(b \in (0, \infty)\). Then \(Y = \sum_{i=1}^n X_i\) is complete for \(b\).
Proof
\( Y \) has the gamma distribution with shape parameter \( n k \) and scale parameter \( b \). Hence, if \(r: [0, \infty) \to \R\), then \[\E\left[r(Y)\right] = \int_0^\infty \frac{1}{\Gamma(n k) b^{n k}} y^{n k-1} e^{-y/b} r(y) \, dy = \frac{1}{\Gamma(n k) b^{n k}} \int_0^\infty y^{n k - 1} r(y) e^{-y / b} \, dy\] The last integral can be interpreted as the Laplace transform of the function \( y \mapsto y^{n k - 1} r(y) \) evaluated at \( 1 / b \). If this transform is 0 for all \(b\) in an open interval, then \( r(y) = 0 \) almost everywhere in \( (0, \infty) \).
Suppose again that \(\bs X = (X_1, X_2, \ldots, X_n)\) is a random sample from the gamma distribution on \( (0, \infty) \) with shape parameter \( k \in (0, \infty) \) and scale parameter \(b \in (0, \infty)\). Let \( M = \frac{1}{n} \sum_{i=1}^n X_i \) denote the sample mean and \( U = (X_1 X_2 \ldots X_n)^{1/n} \) the sample geometric mean, as before. Then
- \( M / U \) is ancillary for \( b \).
- \( M \) and \( M / U \) are independent.
Proof
- We can take \( X_i = b Z_i \) for \( i \in \{1, 2, \ldots, n\} \) where \( \bs{Z} = (Z_1, X_2, \ldots, Z_n) \) is a random sample of size \( n \) from the gamma distribution with shape parameter \( k \) and scale parameter 1 (the standard gamma distribution with shape parameter \( k \)). Then \[ \frac{M}{U} = \frac{1}{n} \sum_{i=1}^n \frac{X_i}{(X_1 X_2 \cdots X_n)^{1/n}} = \frac{1}{n} \sum_{i=1}^n \left(\frac{X_i^n}{X_1 X_2 \cdots X_n}\right)^{1/n} = \frac{1}{n} \sum_{i=1}^n \left(\prod_{j \ne i} \frac{X_i}{X_j}\right)^{1/n} \] But \( X_i / X_j = Z_i / Z_j\) for \( i \ne j \), and the distribution of \( \left\{Z_i / Z_j: i, j \in \{1, 2, \ldots, n\}, \; i \ne j\right\} \) does not depend on \( b \). Hence the distribution of \( M / U \) does not depend on \( b \).
- This follows from Basu's theorem (15) , since \( M \) is complete and sufficient for \( b \) and \( M / U \) is ancillary for \( b \).
The Beta Distribution
Recall that the beta distribution with left parameter \(a \in (0, \infty)\) and right parameter \(b \in (0, \infty)\) is a continuous distribution on \( (0, 1) \) with probability density function \( g \) given by \[ g(x) = \frac{1}{B(a, b)} x^{a-1} (1 - x)^{b-1}, \quad x \in (0, 1)\] where \( B \) is the beta function. The beta distribution is often used to model random proportions and other random variables that take values in bounded intervals. It is studied in more detail in the chapter on Special Distribution
Suppose that \(\bs X = (X_1, X_2, \ldots, X_n)\) is a random sample from the beta distribution with left parameter \(a\) and right parameter \(b\). Then \((P, Q)\) is minimally sufficient for \((a, b)\) where \(P = \prod_{i=1}^n X_i\) and \(Q = \prod_{i=1}^n (1 - X_i)\).
Proof
The joint PDF \( f \) of \( \bs X \) is given by \[ f(\bs x) = g(x_1) g(x_2) \cdots g(x_n) = \frac{1}{B^n(a, b)} (x_1 x_2 \cdots x_n)^{a - 1} [(1 - x_1) (1 - x_2) \cdots (1 - x_n)]^{b-1}, \quad \bs x = (x_1, x_2, \ldots, x_n) \in (0, 1)^n \] From the factorization theorem (3) , it follows that \( (U, V) \) is sufficient for \( (a, b) \). Minimal sufficiency follows from condition (6) .
The proof also shows that \( P \) is sufficient for \( a \) if \( b \) is known, and that \( Q \) is sufficient for \( b \) if \( a \) is known. Recall that the method of moments estimators of \( a \) and \( b \) are \[ U = \frac{M\left(M - M^{(2)}\right)}{M^{(2)} - M^2}, \quad V = \frac{(1 - M)\left(M - M^{(2)}\right)}{M^{(2)} - M^2} \] respectively, where \( M = \frac{1}{n} \sum_{i=1}^n X_i \) is the sample mean and \( M^{(2)} = \frac{1}{n} \sum_{i=1}^n X_i^2 \) is the second order sample mean. If \( b \) is known, the method of moments estimator of \( a \) is \( U_b = b M / (1 - M) \), while if \( a \) is known, the method of moments estimator of \( b \) is \( V_a = a (1 - M) / M \). None of these estimators is a function of the sufficient statistics \( (P, Q) \) and so all suffer from a loss of information. On the other hand, if \( b = 1 \), the maximum likelihood estimator of \( a \) on the interval \( (0, \infty) \) is \( W = -n / \sum_{i=1}^n \ln X_i \), which is a function of \( P \) (as it must be).
Run the beta estimation experiment 1000 times with various values of the parameters. Compare the estimates of the parameters.
The Pareto Distribution
Recall that the Pareto distribution with shape parameter \(a \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\) is a continuous distribution on \( [b, \infty) \) with probability density function \( g \) given by \[ g(x) = \frac{a b^a}{x^{a+1}}, \quad b \le x \lt \infty \] The Pareto distribution, named for Vilfredo Pareto, is a heavy-tailed distribution often used to model income and certain other types of random variables. It is studied in more detail in the chapter on Special Distribution.
Suppose that \(\bs X = (X_1, X_2, \ldots, X_n)\) is a random sample from the Pareto distribution with shape parameter \(a\) and scale parameter \( b \). Then \( \left(P, X_{(1)}\right) \) is minimally sufficient for \( (a, b) \) where \(P = \prod_{i=1}^n X_i\) is the product of the sample variables and where \( X_{(1)} = \min\{X_1, X_2, \ldots, X_n\} \) is the first order statistic.
Proof
The joint PDF \( f \) of \( \bs X \) at \( \bs x = (x_1, x_2, \ldots, x_n) \) is given by \[ f(\bs x) = g(x_1) g(x_2) \cdots g(x_n) = \frac{a^n b^{n a}}{(x_1 x_2 \cdots x_n)^{a + 1}}, \quad x_1 \ge b, x_2 \ge b, \ldots, x_n \ge b \] which can be rewritten as \[ f(\bs x) = g(x_1) g(x_2) \cdots g(x_n) = \frac{a^n b^{n a}}{(x_1 x_2 \cdots x_n)^{a + 1}} \bs{1}\left(x_{(n)} \ge b\right), \quad (x_1, x_2, \ldots, x_n) \in (0, \infty)^n \] So the result follows from the factorization theorem (3) . Minimal sufficiency follows from condition (6) .
The proof also shows that \( P \) is sufficient for \( a \) if \( b \) is known (which is often the case), and that \( X_{(1)} \) is sufficient for \( b \) if \( a \) is known (much less likely). Recall that the method of moments estimators of \( a \) and \( b \) are \[U = 1 + \sqrt{\frac{M^{(2)}}{M^{(2)} - M^2}}, \quad V = \frac{M^{(2)}}{M} \left( 1 - \sqrt{\frac{M^{(2)} - M^2}{M^{(2)}}} \right)\] respectively, where as before \( M = \frac{1}{n} \sum_{i=1}^n X_i \) is the sample mean and \( M^{(2)} = \sum_{i=1}^n X_i^2 \) the second order sample mean. These estimators are not functions of the sufficient statistics and hence suffers from loss of information. On the other hand, the maximum likelihood estimators of \( a \) and \( b \) on the interval \( (0, \infty) \) are \[W = \frac{n}{\sum_{i=1}^n \ln X_i - n \ln X_{(1)}}, \quad X_{(1)}\] respectively. These are functions of the sufficient statistics, as they must be.
Run the Pareto estimation experiment 1000 times with various values of the parameters \( a \) and \( b \) and the sample size \( n \). Compare the method of moments estimates of the parameters with the maximum likelihood estimates in terms of the empirical bias and mean square error.
The Uniform Distribution
Recall that the
continuous uniform distribution
on the interval \( [a, a + h] \), where \( a \in \R \) is the location parameter and \( h \in (0, \infty) \) is the scale parameter, has probability density function \( g \) given by \[ g(x) = \frac{1}{h}, \quad x \in [a, a + h] \] Continuous uniform distributions are widely used in applications to model a number chosen
at random
from an interval. Continuous uniform distributions are studied in more detail in the chapter on Special Distributions. Let's first consider the case where both parameters are unknown.
Suppose that \(\bs X = (X_1, X_2, \ldots, X_n)\) is a random sample from the uniform distribution on the interval \([a, a + h]\). Then \(\left(X_{(1)}, X_{(n)}\right)\) is minimally sufficient for \((a, h)\), where \( X_{(1)} = \min\{X_1, X_2, \ldots, X_n\} \) is the first order statistic and \( X_{(n)} = \max\{X_1, X_2, \ldots, X_n\} \) is the last order statistic.
Proof
The PDF \( f \) of \( \bs X \) is given by \[ f(\bs x) = g(x_1) g(x_2) \cdots g(x_n) = \frac{1}{h^n}, \quad \bs x = (x_1, x_2, \ldots x_n) \in [a, a + h]^n \] We can rewrite the PDF as \[ f(\bs x) = \frac{1}{h^n} \bs{1}[x_{(1)} \ge a] \bs{1}[x_{(n)} \le a + h], \quad \bs x = (x_1, x_2, \ldots, x_n) \in \R^n \] It then follows from the factorization theorem (3) that \(\left(X_{(1)}, X_{(n)}\right) \) is sufficient for \( (a, h) \). Next, suppose that \( \bs x, \, \bs y \in \R^n \) and that \( x_{(1)} \ne y_{(1)} \) or \( x_{(n)} \ne y_{(n)} \). For a given \( h \in (0, \infty) \), we can easily find values of \( a \in \R \) such that \( f(\bs x) = 0 \) and \( f(\bs y) = 1 / h^n \), and other values of \( a \in \R \) such that \( f(\bs x) = f(\bs y) = 1 / h^n \). By condition (6) , \(\left(X_{(1)}, X_{(n)}\right) \) is minimally sufficient.
If the location parameter \( a \) is known, then the largest order statistic is sufficient for the scale parameter \( h \). But if the scale parameter \( h \) is known, we still need both order statistics for the location parameter \( a \). So in this case, we have a single real-valued parameter, but the minimally sufficient statistic is a pair of real-valued random variables.
Suppose again that \( \bs X = (X_1, X_2, \ldots, X_n) \) is a random sample from the uniform distribution on the interval \( [a, a + h] \).
- If \( a \in \R \) is known, then \( X_{(n)} \) is sufficient for \( h \).
- If \( h \in (0, \infty) \) is known, then \( \left(X_{(1)}, X_{(n)}\right) \) is minimally sufficient for \( a \).
Proof
Both parts follow easily from the analysis given in the proof of the last theorem.
Run the uniform estimation experiment 1000 times with various values of the parameter. Compare the estimates of the parameter.
Recall that if both parameters are unknown, the method of moments estimators of \( a \) and \( h \) are \( U = 2 M - \sqrt{3} T \) and \( V = 2 \sqrt{3} T \), respectively, where \( M = \frac{1}{n} \sum_{i=1}^n X_i \) is the sample mean and \( T^2 = \frac{1}{n} \sum_{i=1}^n (X_i - M)^2 \) is the biased sample variance. If \( a \) is known, the method of moments estimator of \( h \) is \( V_a = 2 (M - a) \), while if \( h \) is known, the method of moments estimator of \( h \) is \( U_h = M - \frac{1}{2} h \). None of these estimators are functions of the minimally sufficient statistics, and hence result in loss of information.
The Hypergeometric Model
So far, in all of our examples, the basic variables have formed a random sample from a distribution. In this subsection, our basic variables will be dependent.
Recall that in the hypergeometric model , we have a population of \( N \) objects, and that \( r \) of the objects are type 1 and the remaining \( N - r \) are type 0 . The population size \( N \) is a positive integer and the type 1 size \( r \) is a nonnegative integer with \( r \le N \). Typically one or both parameters are unknown. We select a random sample of \( n \) objects, without replacement from the population, and let \( X_i \) be the type of the \( i \)th object chosen. So our basic sequence of random variables is \( \bs X = (X_1, X_2, \ldots, X_n) \). The variables are identically distributed indicator variables with \( \P(X_i = 1) = r / N \) for \( i \in \{1, 2, \ldots, n\} \), but are dependent. Of course, the sample size \( n \) is a positive integer with \( n \le N \).
The variable \( Y = \sum_{i=1}^n X_i \) is the number of type 1 objects in the sample. This variable has the hypergeometric distribution with parameters \( N \), \( r \), and \( n \), and has probability density function \( h \) given by \[ h(y) = \frac{\binom{r}{y} \binom{N - r}{n - y}}{\binom{N}{n}} = \binom{n}{y} \frac{r^{(y)} (N - r)^{(n - y)}}{N^{(n)}}, \quad y \in \{\max\{0, N - n + r\}, \ldots, \min\{n, r\}\} \] (Recall the falling power notation \( x^{(k)} = x (x - 1) \cdots (x - k + 1) \)). The hypergeometric distribution is studied in more detail in the chapter on Finite Sampling Models.
\( Y \) is sufficient for \( (N, r) \). Specifically, for \( y \in \{\max\{0, N - n + r\}, \ldots, \min\{n, r\}\} \), the conditional distribution of \( \bs X \) given \( Y = y \) is uniform on the set of points \[ D_y = \left\{(x_1, x_2, \ldots, x_n) \in \{0, 1\}^n: x_1 + x_2 + \cdots + x_n = y\right\} \]
Proof
By a simple application of the multiplication rule of combinatorics, the PDF \( f \) of \( \bs X \) is given by \[ f(\bs x) = \frac{r^{(y)} (N - r)^{(n - y)}}{N^{(n)}}, \quad \bs x = (x_1, x_2, \ldots, x_n) \in \{0, 1\}^n \] where \( y = \sum_{i=1}^n x_i \). If \( y \in \{\max\{0, N - n + r\}, \ldots, \min\{n, r\}\} \), the conditional distribution of \( \bs X \) given \( Y = y \) is concentrated on \( D_y \) and \[ \P(\bs X = \bs x \mid Y = y) = \frac{\P(\bs X = \bs x)}{\P(Y = y)} = \frac{r^{(y)} (N - r)^{(n-y)}/N^{(n)}}{\binom{n}{y} r^{(y)} (N - r)^{(n - y)} / N^{(n)}} = \frac{1}{\binom{n}{y}}, \quad \bs x \in D_y \] Of course, \( \binom{n}{y} \) is the cardinality of \( D_y \).
There are clearly strong similarities between the hypergeometric model and the Bernoulli trials model above. Indeed if the sampling were with replacement, the Bernoulli trials model with \( p = r / N \) would apply rather than the hypergeometric model. It's also interesting to note that we have a single real-valued statistic that is sufficient for two real-valued parameters.
Once again, the sample mean \( M = Y / n \) is equivalent to \( Y \) and hence is also sufficient for \( (N, r) \). Recall that the method of moments estimator of \( r \) with \( N \) known is \( N M \) and the method of moment estimator of \( N \) with \( r \) known is \( r / M \). The estimator of \( r \) is the one that is used in the capture-recapture experiment.
Exponential Families
Suppose now that our data vector \(\bs X\) takes values in a set \(S\), and that the distribution of \(\bs X\) depends on a parameter vector \(\bs{\theta}\) taking values in a parameter space \(\Theta\). The distribution of \(\bs X\) is a \(k\)-parameter exponential family if \(S\) does not depend on \(\bs{\theta}\) and if the probability density function of \(\bs X\) can be written as
\[ f_\bs{\theta}(\bs x) = \alpha(\bs{\theta}) r(\bs x) \exp\left(\sum_{i=1}^k \beta_i(\bs{\theta}) u_i(\bs x) \right); \quad \bs x \in S, \; \bs{\theta} \in \Theta \]
where \(\alpha\) and \(\left(\beta_1, \beta_2, \ldots, \beta_k\right)\) are real-valued functions on \(\Theta\), and where \(r\) and \(\left(u_1, u_2, \ldots, u_k\right)\) are real-valued functions on \(S\). Moreover, \(k\) is assumed to be the smallest such integer. The parameter vector \(\bs{\beta} = \left(\beta_1(\bs{\theta}), \beta_2(\bs{\theta}), \ldots, \beta_k(\bs{\theta})\right)\) is sometimes called the natural parameter of the distribution, and the random vector \(\bs U = \left(u_1(\bs X), u_2(\bs X), \ldots, u_k(\bs X)\right)\) is sometimes called the natural statistic of the distribution. Although the definition may look intimidating, exponential families are useful because they have many nice mathematical properties, and because many special parametric families are exponential families. In particular, the sampling distributions from the Bernoulli , Poisson , gamma , normal , beta , and Pareto considered above are exponential families. Exponential families of distributions are studied in more detail in the chapter on special distributions.
\(\bs U\) is minimally sufficient for \(\bs{\theta}\).
Proof
That \( U \) is sufficient for \( \theta \) follows immediately from the factorization theorem. That \( U \) is minimally sufficient follows since \( k \) is the smallest integer in the exponential formulation.
It turns out that \(\bs U\) is complete for \(\bs{\theta}\) as well, although the proof is more difficult.


3.3.1: Introduction (2015)
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- Robin DeRosa, Abby Goode et al.
- Plymouth State University/
Jonathan Edwards is most commonly known as a philosopher, Puritan theologian, and Protestant revivalist preacher. Edwards was a major figure of his day, and an incredibly powerful author who, according to novelist William Dean Howells, “first gave our poor American provinciality world standing.” Edwards is best known for his renowned sermon Sinners in the Hands of an Angry God (seen right). Sermons aside, Edwards had an impact that rings through the years, as his philosophical theology has a strong presence in current theological discussions.
He was born on about 100 years after the Mayflower landed at Plymouth, on October 5, 1703 and died on March 22, 1758. He studied at Yale University from 1716-1722. He was born the son of a minster, and grew up filling his time with religious activities. Although, even though he was doing all the right things, Edwards’s faith in God was superficial. The first reading is an excerpt from Edwards’s personal narrative, and tells of his journey from a superficial believer to an incredibly devout follower of Christ.
The second reading is an excerpt from Edward’s famous sermon Sinners in the Hands of an Angry God.  He emphatically urges his congregation to repent and turn to God, and therefore be spared from damnation to hell.
The third reading is a text written by Edwards’s wife, Sarah Pierrepont Edwards, in which she reflect upon her relationship with God, and how beautiful and majestic He is to her. She writes about how she could spend time lost in the beauty of God in a trance like state, unsure if she was awake or asleep. Her writing style is similar to her husbands; it flows calmly but with power.
Discussion Questions:
- What was the bottom line of the message Edwards is sharing with his congregation?
- What tactics did Edwards use to try to persuade his audience?
Work Cited:
Edwards, Jonathan. “ Personal Narrative (Excerpt) ” The Open Anthology of Earlier American Literature. Ed. Robin DeRosa. Plymouth, NH: Public Commons Publishing, 2015. Digital File.
Wikipedia contributors. “Jonathan Edwards (theologian).” Wikipedia, The Free Encyclopedia . Wikipedia, The Free Encyclopedia, 12 Oct. 2015. Web. 13 Oct. 2015.