These authors contributed equally to this work.

[1]Shusen Pu These authors contributed equally to this work.

[1]Department of Mathematics and Statistics, University of West Florida, 11000
University Pkwy, Pensacola, 32514, FL, USA

2]Department of Computer Science, University of West Florida, 11000 University
Pkwy, Pensacola, 32514, FL, USA

#  Advancing Continuous Distribution Generation: An Exponentiated Odds Ratio
Generator Approach

Xinyu Chen  Yuanqi Xie  Achraf Cohen<EMAIL_ADDRESS>*  [

###### Abstract

This paper presents a new methodology for generating continuous statistical
distributions, integrating the exponentiated odds ratio within the framework
of survival analysis. This new method enhances the flexibility and
adaptability of distribution models to effectively address the complexities
inherent in contemporary datasets. The core of this advancement is illustrated
by introducing a particular subfamily, the “Type-2 Gumbel Weibull-G Family of
Distributions.” We provide a comprehensive analysis of the mathematical
properties of these distributions, encompassing statistical properties such as
density functions, moments, hazard rate and quantile functions, Rényi entropy,
order statistics, and the concept of stochastic ordering. To establish the
robustness of our approach, we apply five distinct methods for parameter
estimation. The practical applicability of the Type-2 Gumbel Weibull-G
distributions is further supported through the analysis of three real-world
datasets. These empirical applications illustrate the exceptional statistical
precision of our distributions compared to existing models, thereby
reinforcing their significant value in both theoretical and practical
statistical applications.

###### keywords:

Continuous Statistical Distributions Generator; Exponentiated Odds Ratio;
Survival Analysis; Methods of Estimation; Statistical properties

###### pacs:

[

MSC Classification]62E99; 60E05

##  1 Introduction

With the ever-increasing complexity and volume of data across various
disciplines, developing and applying a diverse array of statistical
distributions has become a paramount area of research. These continuous
distributions are essential in modeling, forecasting, and interpreting complex
data, facilitating hidden patterns and relationships [1]. Nonetheless, the
rapid growth and evolving nature of contemporary data often pose challenges
that traditional distributions struggle to cope with, leading to a need for
novel statistical distribution generators [2].

Over the past several decades, numerous methods for creating new continuous
probability distributions have been explored. These include methods such as
transformations of random variables, the use of mixed models, and advanced
compounding methods [3, 4, 5, 6]. Among the recent contributions in this field
are the gamma-Topp-Leone-Type II-exponentiated half Logistic-G distribution
family [7], an enhanced version of the generalized Weibull distribution [8],
the innovative modified alpha power Weibull-X distribution set [9], the Topp-
Leone type II exponentiated half logistic-G distribution category [10], the
modified-half-Normal distribution by [11], an inventive extension of the power
Lindley distribution [11], the truncated inverse generalized Rayleigh
distribution [12], the Ristić–Balakrishnan–Topp–Leone–Gompertz-G family of
distributions [13], the odd Gompertz-G distribution family [14], the inverse
Burr-generalized distribution series [15], the gamma inverse paralogistic
distribution [16], and the shifted generalized truncated Nadarajah-Haghighi
distribution by [17].

This paper presents a new generator of continuous statistical distributions,
designed to offer greater flexibility and adaptability in addressing the
increasing complexity of contemporary datasets. The foundation of this
generator’s development lies in survival analysis, drawing inspiration from
the extreme values of odds ratio data, as elaborated in Section 3. The
generator proposed herein aims to provide a comprehensive framework for the
creation of a broad range of distributions, each characterized by unique
shapes and properties. This feature allows for more precise modeling and
analysis of an extensive variety of data. The adaptability of the proposed
generator is illustrated through its application in deriving new families of
distributions, demonstrating superior data fitting capabilities compared with
traditional methods.

The structure of the remaining sections of this paper is outlined as follows:
Section 2 provides a description of the methodology of the new generator.
Section 3 presents the new subfamily of distributions generated by the
proposed technique and investigates the mathematical properties of the
resulting distributions. Four estimation methods are discussed in Section 4 to
test the robustness of the model. Section 5 introduces several special cases
of the new family of distributions. In Section 6, we evaluate the performance
and flexibility of the generator through a set of experiments and comparative
analyses. Finally, we conclude the paper and discuss the future directions in
Section 7.

##  2 The New Generator Based on Odds Ratio

The field of survival analysis is a critical component in various research
domains, such as medical science, engineering, and social sciences. Central to
survival analysis is the odds ratio, which quantifies the probability that an
individual or component, defined by a specific lifespan following a continuous
distribution $H(x,\boldsymbol{\psi})$, will fail or expire at a particular
point in time, $x$ [18]. The odds ratio, expressed as
$\displaystyle{H(x,\boldsymbol{\psi})}/{\overline{H}(x,\boldsymbol{\psi})}$,
has become an important tool for understanding and assessing risk factors,
shedding light on the relative probabilities of event outcomes, namely, death
or failure [19, 20, 21].

This study presents a novel generator developed by integrating the
methodologies delineated in [4] and [22]. This new generator can be
efficiently applied to any baseline distribution with a cumulative
distribution function (cdf), represented as $R(t)$. It is mathematically
defined by:

$\displaystyle F_{{\text{RT--EOR-H}}}(x;\alpha,\beta)$
$\displaystyle=\int_{0}^{-\log(1-F_{WG}(x))}r(t)dt$ (1)
$\displaystyle=R\left\\{-\log(1-F_{WG}(x))\right\\}$ (2)
$\displaystyle=R\left\\{\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{\beta}\right\\}$
(3)

Here, $r(t)$ and $R(t)$ denote the probability density function (pdf) and cdf
of a continuous random variable, respectively. $f_{WG}(x)$, and $F_{WG}(x)$
represent the pdf and cdf of the Weibull-G family of distributions proposed by
Bourguignon et al. [4]. $H(x,\boldsymbol{\psi})$ and
$\overline{H}(x,\boldsymbol{\psi})$ represent the cdf and the survival
function of the baseline distribution, respectively. This newly designed
generator of continuous distributions is denoted as the R-transferred
exponentiated odds ratio generator (RT-EOR-H), denoted as $F_{{}_{\text{RT--
EOR-H}}}(x;\alpha,\beta)$.

This new approach provides a comprehensive framework, allowing researchers to
delve into the exponentiated odds ratio of any baseline function
$H(x,\boldsymbol{\psi})$, embedded within any conceivable distribution $R(t)$.
Crucially, this new generator, by permitting the investigation of any baseline
function within any distribution $R$, significantly enhances the flexibility
of survival analyses, yielding more accurate, adaptable, and precise
predictions. These advancements meet the need for continued evolution in
survival analysis, providing a path for enhanced comprehension of the
complexities of survival and risk.

Some example subfamilies are listed in Table 1.

Table 1: Families of distributions derived from different $R(t)$ Distribution | $R(t)$ | RT-EOR-H
---|---|---
Uniform | $\frac{t}{\theta}$ | $\frac{\alpha}{\theta}\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{\beta}$
Normal | $\Phi\left(\frac{x-\mu}{\sigma}\right)$ | $\Phi\left(\frac{\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{\beta}-\mu}{\sigma}\right)$
Gamma | $\frac{1}{\Gamma(k)}\gamma(k,\frac{t}{\theta})$ | $\frac{1}{\Gamma(k)}\gamma\left(k,\frac{\alpha}{\theta}\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{\beta}\right)$
Log-logistic | $\frac{1}{1+(t/c)^{-k}}$ | $\frac{1}{1+\left(\frac{\alpha}{c}\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{\beta}\right)^{-k}}$
Rayleigh | $1-e^{-\frac{t^{2}}{2\sigma^{2}}}$ | $1-\exp\left\\{-\frac{\alpha^{2}}{2\sigma^{2}}\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{2\beta}\right\\}$
Weibull | $1-e^{-(t/\lambda)^{k}}$ | $1-\exp\left\\{-\left(\frac{\alpha}{\lambda}\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{\beta}\right)^{k}\right\\}$
Type-2 Gumbel | $e^{-\lambda t^{-\delta}}$ | $\exp\left\\{-\lambda\left(\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{\beta}\right)^{-\delta}\right\\}$
Lomax | $1-(1+\lambda t)^{-k}$ | $1-\left(1+\lambda\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{\beta}\right)^{-k}$
Burr XII | $1-(1+t^{c})^{-k}$ | $1-\left\\{1+\left(\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{\beta}\right)^{c}\right\\}^{-k}$
Pareto | $1-\left(\frac{\gamma}{t}\right)^{k}$ | $1-\left(\frac{\gamma}{\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{\beta}}\right)^{k}$
Lévy | $\text{erfc}\left(\sqrt{\frac{c}{2t}}\right)$ | $\text{erfc}\left(\sqrt{\frac{c}{2\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{\beta}}}\right)$
Fréchet | $e^{-\lambda t^{\gamma}}$ | $\exp\left\\{-\lambda\left(\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{\beta}\right)^{\gamma}\right\\}$
Kumaraswamy | $(1-x^{k})^{-\lambda}-1$ | $\left\\{1-\left(\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{\beta}\right)^{k}\right\\}^{-\lambda}-1$

##  3 The Type-2 Gumbel Weibull-G Family of Distributions

To illustrate the applicability of the new generator, we will focus on one of
its subfamilies in this paper. Here, we combine the generator with the cdf of
Type-2 Gumbel distribution as follows

$\displaystyle R(t)=e^{-\lambda t^{-\delta}}$ (4)

Then we obtain the cdf and pdf of a new Type-2 Gumbel Weibll-G (T2GWG)family
of distributions as

$\displaystyle
F_{{}_{T2GWG}}(x)=\exp\left\\{-\lambda\alpha^{-\delta}\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta\delta}\right\\}$
(5)

To avoid over-parameterization, we let $\lambda=1$, $\delta=1$, and substitite
$\alpha=\alpha^{-1}$, now the cdf reduces to

$\displaystyle F_{{}_{T2GWG}}(x)$
$\displaystyle=\exp\left\\{-\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta}\right\\}$
(6)

and

$\displaystyle f_{{}_{T2GWG}}(x)=\alpha\beta
h(x,\boldsymbol{\psi})\frac{H(x,\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x,\boldsymbol{\psi})^{-\beta+1}}\exp\left\\{-\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta}\right\\}$
(7)

An interpretation of the T2GWG family of distributions can be given as
follows. Let $Z$ be a lifetime random variable with a baseline cdf
$H(x,\boldsymbol{\psi})$. The odds ratio that an individual following the
lifetime $Z$ will die (failure) at time $x$ is
$\displaystyle\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}$.
For a sequence of such independent and identical odds ratios
$Z_{1},Z_{2},\cdots,Z_{n}$, the maximum
$M_{n}=\text{max}(Z_{1},Z_{2},\cdots,Z_{n})$, then $M_{n}$ follows a
distribution which converges to the T2GWG distribution as provided in
equation(6) [3, 4].

In the subsequent subsections, we delve into the statistical properties of
this novel distribution. Our discussion will cover a broad range of topics,
including the expansion of the density function, hazard rate and quantile
functions, moments, incomplete moments, the generation function, Rényi
Entropy, order statistics, and stochastic ordering.

###  3.1 Expansion of the pdf

Consider the following expansion

$\displaystyle\exp\left\\{-\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta}\right\\}=\sum_{j=0}^{\infty}\frac{(-1)^{j}\alpha^{j}}{j!}\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-j\beta},$
(8)

then the pdf of T2GWG can be expanded as

$\displaystyle f_{{}_{T2GWG}}(x)$ $\displaystyle=\alpha\beta
h(x,\boldsymbol{\psi})\frac{H(x,\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x,\boldsymbol{\psi})^{-\beta+1}}\sum_{j=0}^{\infty}\frac{(-1)^{j}\alpha^{j}}{j!}\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-j\beta}$
$\displaystyle=\alpha\beta
h(x,\boldsymbol{\psi})\sum_{j=0}^{\infty}\frac{(-1)^{j}\alpha^{j}}{j!}\frac{\overline{H}(x,\boldsymbol{\psi})^{\beta+j\beta-1}}{H(x,\boldsymbol{\psi})^{\beta+j\beta+1}}$
(9)

Moreover, note that

$\displaystyle\overline{H}(x,\boldsymbol{\psi})^{\beta+j\beta-1}$
$\displaystyle=[1-H(x,\boldsymbol{\psi})]^{\beta+j\beta-1}$
$\displaystyle=\sum_{k=0}^{\infty}\binom{\beta(j+1)-1}{k}(-1)^{k}H(x,\boldsymbol{\psi})^{k}$
(10)

Then

$\displaystyle f_{{}_{T2GWG}}(x)$ $\displaystyle=\alpha\beta
h(x,\boldsymbol{\psi})\sum_{j=0}^{\infty}\frac{(-1)^{j}\alpha^{j}}{j!}\sum_{k=0}^{\infty}\binom{\beta(j+1)-1}{k}\frac{(-1)^{k}H(x,\boldsymbol{\psi})^{k}}{H(x,\boldsymbol{\psi})^{\beta+j\beta+1}}$
$\displaystyle=\alpha\beta
h(x,\boldsymbol{\psi})\sum_{j,k=0}^{\infty}\frac{(-1)^{j+k}\alpha^{j}}{j!}\binom{\beta(j+1)-1}{k}\left[H(x,\boldsymbol{\psi})\right]^{k-\beta-j\beta-1}$
$\displaystyle=\sum_{j,k=0}^{\infty}c_{j,k}r_{k-\beta(j+1)-1}(x,\boldsymbol{\psi})$
(11)

where

$\displaystyle
c_{j,k}=\frac{\alpha\beta(-1)^{j+k}\alpha^{j}}{j!(k-\beta(j+1))}\binom{\beta(j+1)-1}{k}$
(12)

and

$r_{k-\beta-j\beta-1}(x,\boldsymbol{\psi})=(k-\beta(j+1))h(x,\boldsymbol{\psi})H(x,\boldsymbol{\psi})^{k-\beta(j+1)-1}$

which is the pdf of exponentiated generalized (EG) distribution with parameter
$\beta^{*}=k-\beta(j+1)$.

###  3.2 Hazard rate and quantile functions

Building upon our earlier discussion on the odds ratio and survival analysis,
we now delve into the key mathematical structures: the hazard rate function
and the quantile function. These are crucial to the deeper understanding and
application of survival analysis as they allow the computation of survival
probabilities and survival times.

####  3.2.1 Hazard rate and quantile functions

The hazard rate function (hrf) plays an important role in survival analysis as
it defines the instantaneous potential per unit time for the occurrence of an
event given survival up to that time. On the other hand, the quantile function
is essential in determining the time at which a certain proportion of survival
is expected. In the newly proposed family of distributions, these functions
take a particularly interesting form as follows.

$\displaystyle h_{{}_{T2GWG}}(x)=\frac{\alpha\beta
h(x,\boldsymbol{\psi})\frac{H(x,\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x,\boldsymbol{\psi})^{-\beta+1}}\exp\left\\{-\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{1-\exp\left\\{-\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta}\right\\}}$
(13)

In addition to the hrf, it is also useful to consider its reciprocal, termed
the reverse hazard rate function. This function essentially reflects the
hazard function’s properties but is viewed from the perspective of the event
not occurring.

$\displaystyle\tau_{{}_{T2GWG}}(x)$ $\displaystyle=\frac{\alpha\beta
h(x,\boldsymbol{\psi})\frac{H(x,\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x,\boldsymbol{\psi})^{-\beta+1}}\exp\left\\{-\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{\exp\left\\{-\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta}\right\\}}$
$\displaystyle=\alpha\beta
h(x,\boldsymbol{\psi})\frac{H(x,\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x,\boldsymbol{\psi})^{-\beta+1}}$
(14)

####  3.2.2 Quantile function

Next, we will focus on the quantile function, which is particularly useful
when determining the survival time corresponding to a specific survival
probability.

$F_{{}_{T2GWG}}(x)=\exp\left\\{-\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta}\right\\}=p$

for $0\leq p\leq 1$, then, it is sufficient to solve

$\displaystyle H(x,\boldsymbol{\psi})=\displaystyle\frac{1}{\left(\frac{\log
p}{-\alpha}\right)^{\frac{1}{\beta}}+1}:=q$ (15)

Thus, the quantile $x_{p}$ of the distribution reduces to the quantile $x_{q}$
of the baseline distribution with cdf $H(x,\boldsymbol{\psi})$ and is given by

$\displaystyle x_{q}=H^{-1}(q)$ (16)

###  3.3 Moments, incomplete moments and generating functions

####  3.3.1 Moments

In the field of statistics, moments play a vital role in characterizing the
properties of a probability distribution. Moments provide important summary
measures of the characteristics of data sets. The first moment about the
origin, also known as the mean, measures the location of the distribution. The
second central moment is known as the variance, which quantifies the spread or
dispersion of the distribution. The third and fourth moments, skewness and
kurtosis, respectively, describe the shape of the distribution, capturing
aspects of its asymmetry and tailedness. We can present the $r^{th}$ moment of
the distributions as

$\displaystyle E(Y^{r})=\sum_{j,k=0}^{\infty}c_{j,k}E(Z_{j,k}^{r})$ (17)

where $Z_{j,k}$ is the exponentiated generalized distribution with the
parameter $\beta^{*}=k-\beta(j+1)$ and $c_{j,k}$ is defined by Eq. 12.

####  3.3.2 Incomplete Moments, Conditional Moments and Moment Generating
Function

While moments give us an understanding of the general characteristics of a
distribution, they do not always provide sufficient detail about specific
intervals or subsets within the data. This is where incomplete moments come
into play. Incomplete moments, also known as truncated or restricted moments,
are defined similarly to regular moments but are integrated over a subset of
the possible range of the variable. They offer a more granular insight into
the characteristics of the distribution within specific ranges. This makes
them particularly useful when analyzing left or right-skewed data or when
assessing the impact of outlier observations. The incomplete moment is
provided as

$\displaystyle
I_{Y}(z)=\int_{0}^{z}y^{s}f_{{}_{T2GWG}}(y)dy=\sum_{j,k=0}^{\infty}c_{j,k}I_{j,k}(y)$
(18)

where $I_{j,k}(y)=\int_{0}^{z}y^{s}r_{k-\beta(j+1)}(x,\boldsymbol{\psi})$.
The $rth$ conditional moments of the Type-2 Gumble Weibull-G family of
distributions is given by

$\displaystyle E(Y^{r}|Y\geq a)$
$\displaystyle=\frac{1}{\overline{F}_{T2GWG}(a;\alpha,\beta,\boldsymbol{\psi})}\int_{t}^{\infty}y^{r}f_{{}_{T2GWG}}(y;\alpha,\beta,\boldsymbol{\psi})dy$
$\displaystyle=\frac{1}{\overline{F}_{T2GWG}(a;\alpha,\beta,\boldsymbol{\psi})}\sum_{j,k=0}^{\infty}c_{j,k}I_{j,k}(y)$
(19)

where $I_{j,k}(y)$ is defined above.
The moment generating function is given by

$\displaystyle
M_{Y}(t)=E(e^{tY})=\sum_{j,k=0}^{\infty}c_{j,k}E(e^{tZ_{j,k}})=\sum_{j,k=0}^{\infty}c_{j,k}M_{Z_{j,k}}(t)$
(20)

and the characteristic function is defined as

$\displaystyle\phi(t)=E(e^{itY})=\sum_{j,k=0}^{\infty}c_{j,k}E(e^{itZ_{j,k}})=\sum_{j,k=0}^{\infty}c_{j,k}\phi_{k-\beta(j+1)}(t),$
(21)

where $\phi_{k-\beta(j+1)}(t)$ is the characteristic function of EG
distribution with parameter $\beta^{*}=k-\beta(j+1)$.

###  3.4 Rényi Entropy

Rényi entropy is named after the Hungarian mathematician Alfréd Rényi [23].
While Shannon entropy is perhaps the most commonly referenced form of entropy
in the field of information theory, characterizing the average uncertainty or
unpredictability of a source of information, Rényi entropy provides a more
generalized measure. Rényi entropy finds applications in various domains
including physics, computer science, statistics, and quantum information
theory. For example, in the context of machine learning, it can be used to
measure the diversity or complexity of learned models. In statistical physics,
it is useful in understanding the thermodynamics of complex systems. The Rényi
Entropy of this new distribution is

$\displaystyle I_{R}(\omega)=$
$\displaystyle(1-\omega)^{-1}\log\left[\int_{-\infty}^{\infty}f^{\omega}(x)dx\right]$
$\displaystyle=$
$\displaystyle(1-\omega)^{-1}\left\\{\omega(\log\alpha+\log\beta)\right.$
$\displaystyle+\left.\log\left[\int_{-\infty}^{\infty}h^{\omega}(x,\boldsymbol{\psi})\frac{H(x,\boldsymbol{\psi})^{\omega(-\beta-1)}}{\overline{H}(x,\boldsymbol{\psi})^{\omega(-\beta+1)}}\exp\left\\{-\omega\alpha\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta}\right\\}\right]\right\\}$

where $\omega>0$ and $\omega\neq 1$. Apply the same expansion technique for
the pdf, we obtain

$\displaystyle I_{R}(\omega)$
$\displaystyle=(1-\omega)^{-1}\left\\{\omega(\log\alpha+\log\beta)\right.$
$\displaystyle+\left.\log\left[\sum_{i=0}^{\infty}\frac{(-1)^{i}(\omega\alpha)^{i}}{i!}\int_{-\infty}^{\infty}h^{\omega}(x,\boldsymbol{\psi})\frac{[\overline{H}(x,\boldsymbol{\psi})]^{\omega(\beta-1)+i\beta}}{[H(x,\boldsymbol{\psi})]^{\omega(\beta+1)+i\beta}}dx\right]\right\\}$
(22)

Consider that

$\displaystyle\overline{H}(x,\boldsymbol{\psi})^{\omega(\beta-1)+i\beta}$
$\displaystyle=[1-H(x,\boldsymbol{\psi})]^{\omega(\beta-1)+i\beta}$
$\displaystyle=\sum_{j=0}^{\infty}\binom{\omega(\beta-1)+i\beta}{j}(-1)^{j}H(x,\boldsymbol{\psi})^{j}$
(23)

Thus we can write the Reńyi Entropy as

$\displaystyle I_{R}(\omega)=$
$\displaystyle(1-\omega)^{-1}\left\\{\omega(\log\alpha+\log\beta)+\log\left[\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\frac{(-1)^{i+j}(\omega\alpha)^{i}}{i!}\right.\right.$
$\displaystyle\times\left.\left.\binom{\omega(\beta-1)+i\beta}{j}\int_{-\infty}^{\infty}h^{\omega}(x,\boldsymbol{\psi})(H(x,\boldsymbol{\psi}))^{j-\omega(\beta+1)-i\beta}dx\right]\right\\}$
$\displaystyle=$
$\displaystyle(1-\omega)^{-1}\left\\{\omega(\log\alpha+\log\beta)+\log\left[\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\frac{(-1)^{i+j}(\omega\alpha)^{i}}{i!}\right.\right.$
$\displaystyle\times\left.\binom{\omega(\beta-1)+i\beta}{j}\frac{\omega^{\omega}}{[j-\omega(\beta+1)-i\beta+\omega]^{\omega}}\right.$
$\displaystyle\times\left.\left.\int_{-\infty}^{\infty}[\frac{j-\omega(\beta+1)-i\beta+\omega}{\omega}h(x,\boldsymbol{\psi})(H(x,\boldsymbol{\psi}))^{\frac{j-\omega(\beta+1)-i\beta}{\omega}}]^{\omega}\right]\right\\}$
$\displaystyle=$
$\displaystyle(1-\omega)^{-1}\left\\{\omega(\log\alpha+\log\beta)+\log\left[\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\frac{(-1)^{i+j}(\omega\alpha)^{i}}{i!}\right.\right.$
$\displaystyle\times\binom{\omega(\beta-1)+i\beta}{j}\frac{\omega^{\omega}}{[j-\omega(\beta+1)-i\beta+\omega]^{\omega}}\times
e^{(1-\omega)I_{REG}}$ (24)

where $I_{REG}$ is the Rényi entropy of the exponentiated generalized
distribution with parameter
$\displaystyle\beta^{*}=\frac{j-\omega(\beta+1)-i\beta+\omega}{\omega}$.

###  3.5 Order statistics

In the field of statistics, order statistics are a fundamental concept that
allows for deeper analysis and understanding of sampled data. Specifically,
order statistics are the values from a random sample sorted in ascending or
descending order. This sorting process provides a powerful perspective on the
sample’s overall distribution and associated characteristics. Order statistics
are used in a variety of applications, including non-parametric statistics
(which does not rely on parameters defined in terms of a theoretical or
assumed population), reliability engineering, and statistical quality control.
They also play a central role in the construction of quantile-quantile plots,
which are used to assess if a data set follows a particular theoretical
distribution.

Let $X_{1},X_{2},...,X_{n}$ be independent identically distributed random
variables distributed by Eqn. (7). The pdf of the $i^{th}$ order statistic
$f_{i:n}(x)$ is given by

$\displaystyle f_{i:n}(x)$
$\displaystyle=\frac{n!f_{{}_{T2GWG}}(x)}{(i-1)!(n-i)!}[F_{{}_{T2GWG}}(x)]^{i-1}[1-F_{{}_{T2GWG}}(x)]^{n-i}$
$\displaystyle=\frac{n!f_{{}_{T2GWG}}(x)}{(i-1)!(n-i)!}\sum_{m=0}^{n-i}\binom{n-i}{m}(-1)^{m}[F_{{}_{T2GWG}}(x)]^{i-1+m}$
$\displaystyle=\frac{n!f_{{}_{T2GWG}}(x)}{(i-1)!(n-i)!}\sum_{m=0}^{n-i}\binom{n-i}{m}(-1)^{m}\exp\left\\{(i-1+m)(-\alpha)\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta}\right\\}$
$\displaystyle=\frac{n!}{(i-1)!(n-i)!}\sum_{m=0}^{n-i}\binom{n-i}{m}\frac{(-1)^{m}}{i+m}f_{{}_{T2GWG}}(x;(i+m)\alpha,\beta)$
(25)

Therefore, we can present $f_{i:n}(x)$ as a linear combination of the T2GWG
with parameter $(\alpha^{*},\beta)$, where $\alpha^{*}=(i+m)\alpha$.

###  3.6 Stochastic ordering

Stochastic ordering is a mathematical concept frequently applied in the realm
of statistics, probability theory, decision theory, and economics [24]. The
most basic form of stochastic ordering is the usual order of real numbers,
which extends naturally to random variables: a random variable $X$ is said to
be stochastically smaller than another random variable $Y$ if, for every real
number $x$, the probability that $X$ is less than or equal to $x$ is higher
than or equal to the probability that $Y$ is less than or equal to $x$. This
gives rise to the concept of one distribution being ”stochastically larger”
than another, which can be a valuable tool in comparing different probability
models or assessing risk. There are several types of stochastic orderings,
such as increasing convex order, likelihood ratio order, and hazard rate
order, each imposing a different structure on the sets of random variables or
distributions. Stochastic ordering is a significant concept because it enables
us to make statements about the relative behavior of different random
variables or distributions without specifying them precisely. It has been
widely used in various fields, such as reliability, insurance, finance,
operations research, and queueing theory.

Let $X_{1}\sim T2GWG(x;\alpha_{1},\beta,\boldsymbol{\psi})$ and $X_{2}\sim
T2GWG(x;\alpha_{2},\beta,\boldsymbol{\psi})$. The likelihood ratio is

$\displaystyle\frac{f_{X_{1}}(x)}{f_{X_{2}}(x)}$
$\displaystyle=\frac{\alpha_{1}\beta
h(x,\boldsymbol{\psi})\frac{H(x,\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x,\boldsymbol{\psi})^{-\beta+1}}\exp\left\\{-\alpha_{1}\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{\alpha_{2}\beta
h(x,\boldsymbol{\psi})\frac{H(x,\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x,\boldsymbol{\psi})^{-\beta+1}}\exp\left\\{-\alpha_{2}\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta}\right\\}}$
$\displaystyle=\frac{\alpha_{1}}{\alpha_{2}}\exp\left\\{(\alpha_{2}-\alpha_{1})\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta}\right\\}$
(26)

Then we differentiate the Eqn.26 and obtain

$\displaystyle\frac{d}{dx}\left(\frac{f_{X_{1}}(x)}{f_{X_{2}}(x)}\right)$
$\displaystyle=\frac{\alpha_{1}}{\alpha_{2}}\exp\left\\{(\alpha_{2}-\alpha_{1})\left[\frac{H(x,\boldsymbol{\psi})}{\overline{H}(x,\boldsymbol{\psi})}\right]^{-\beta}\right\\}$
$\displaystyle\times\left[(\alpha_{2}-\alpha_{1})h(x,\boldsymbol{\psi})\frac{H(x,\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x,\boldsymbol{\psi})^{-\beta+1}}\right]$
(27)

If $\alpha_{1}<\alpha_{2}$,
$\frac{d}{dx}\left(\frac{f_{X_{1}}(x)}{f_{X_{2}}(x)}\right)<0$. Thus it
indicates that $X\preceq_{lr}Y$. According to the theorem proposed by [24],
both $X\preceq_{hr}Y$ and $X\preceq Y$ hold.

##  4 Methods of Estimation

###  4.1 Maximum Likelihood Estimation

We can estimate the unknown parameters of the Type-2 Gumble Eibull-G family
distributions by using the widely used Maximum Likelihood Estimation (MLE).
Let $\mathbf{\Delta}=(\alpha,\beta,\boldsymbol{\psi})^{T}$. Then the log-
likelihood for $\mathbf{\Delta}$ is defined by

$\displaystyle\ell(\mathbf{\Delta})$
$\displaystyle=n\log(\alpha)+n\log(\beta)+\sum_{i=1}^{n}\log
h(x_{i};\boldsymbol{\psi})-(\beta+1)\sum_{i=1}^{n}\log
H(x_{i},\boldsymbol{\psi})$
$\displaystyle+(\beta-1)\sum_{i=1}^{n}\log[1-H(x_{i},\boldsymbol{\psi})]-\alpha\sum_{i=1}^{n}\left[\frac{H(x_{i},\boldsymbol{\psi})}{1-H(x_{i},\boldsymbol{\psi})}\right]^{-\beta}$
(28)

The first derivative of $\ell(\mathbf{\Delta})$ with respect to
$\mathbf{\Delta}$ are shown as following

$\displaystyle\frac{\partial\ell}{\partial\alpha}=\frac{n}{\alpha}-\sum_{i=1}^{n}\left[\frac{H(x_{i},\boldsymbol{\psi})}{1-H(x_{i},\boldsymbol{\psi})}\right]^{-\beta}$
(29) $\displaystyle\frac{\partial\ell}{\partial\beta}=$
$\displaystyle\frac{n}{\beta}-\sum_{i=1}^{n}\log
H(x_{i},\boldsymbol{\psi})+\sum_{i=1}^{n}\log[1-H(x_{i},\boldsymbol{\psi})]$
$\displaystyle+\alpha\sum_{i=1}^{n}\left[\frac{H(x_{i},\boldsymbol{\psi})}{1-H(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\log\left(\frac{H(x_{i},\boldsymbol{\psi})}{1-H(x_{i},\boldsymbol{\psi})}\right)$
(30)

and

$\displaystyle\frac{\partial\ell}{\partial\psi_{k}}=$
$\displaystyle\sum_{i=1}^{n}\frac{1}{h(x_{i},\boldsymbol{\psi})}\frac{\partial
h(x_{i},\boldsymbol{\psi})}{\partial\psi_{k}}-(\beta+1)\sum_{i=1}^{n}\frac{1}{H(x_{i},\boldsymbol{\psi})}\frac{\partial
H(x_{i},\boldsymbol{\psi})}{\partial\psi_{k}}$
$\displaystyle-(\beta-1)\sum_{i=1}^{n}\frac{1}{1-H(x_{i},\boldsymbol{\psi})}\frac{\partial
H(x_{i},\boldsymbol{\psi})}{\partial\psi_{k}}$
$\displaystyle+\alpha\beta\sum_{i=1}^{n}\frac{H(x_{i},\boldsymbol{\psi})^{-\beta-1}}{[1-H(x_{i},\boldsymbol{\psi})]^{-\beta+1}}\frac{\partial
H(x_{i},\boldsymbol{\psi})}{\partial\psi_{k}}$ (31)

where $\psi_{k}$ is the $k^{th}$ element of the vector $\boldsymbol{\psi}$.
We can maximize the log-likehood function $\ell(\mathbf{\Delta})$ by solving
the nonlinear equations
$\left(\frac{\partial\ell}{\partial\alpha},\frac{\partial\ell}{\partial\beta},\frac{\partial\ell}{\partial\psi_{k}}\right)=0$
with numerical methods such as Newton–Raphson approach.

###  4.2 Least Square and Weighted Least Square Estimation

The least squares (LS) method is a commonly used technique in regression
analysis for approximating the solution of overdetermined systems. The method
provides the best linear unbiased estimates of the unknown parameters if the
errors are homoscedastic and uncorrelated. On the other hand, the weighted
least squares (WLSE) approach extends the least squares technique by
incorporating the different variances of the observations. This method assigns
a weight to each data point based on the variance of its error term, placing
less emphasis on the observations with higher variances to make the overall
model more reliable.

The LSE and WLSE techniques can also provide estimators in the model. The LS
estimation is given by

$\displaystyle LS(\mathbf{\Delta})$
$\displaystyle=\sum_{i=1}^{n}\left(F(x_{i},\mathbf{\Delta})-\frac{i}{n+1}\right)^{2}$
$\displaystyle=\sum_{i=1}^{n}\left(\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\frac{i}{n+1}\right)^{2}$
(32)

By differentiating Eqn.32, we have the followings

$\displaystyle\frac{\partial LS}{\partial\alpha}=$ $\displaystyle
2\sum_{i=1}^{n}\left(\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\frac{i}{n+1}\right)$
$\displaystyle\times(-1)\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}$
(33) $\displaystyle\frac{\partial LS}{\partial\beta}=$ $\displaystyle
2\sum_{i=1}^{n}\left(\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\frac{i}{n+1}\right)$
$\displaystyle\times\alpha\log\left(\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right)\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}$
(34)

and

$\displaystyle\frac{\partial LS}{\partial\psi_{k}}=$ $\displaystyle
2\sum_{i=1}^{n}\left(\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\frac{i}{n+1}\right)$
$\displaystyle\times\alpha\beta\frac{\partial
H(x_{i},\boldsymbol{\psi})}{\partial\psi_{k}}\frac{H(x_{i},\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x_{i},\boldsymbol{\psi})^{-\beta+1}}\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}$
(35)

We can also apply Newton–Raphson procedure to minimize the least square
estimation $LS(\mathbf{\Delta})$ by solving the equations
$\left(\frac{\partial LS}{\partial\alpha},\frac{\partial
LS}{\partial\beta},\frac{\partial LS}{\partial\theta_{k}}\right)=0$
Similarly, the WLS estimation can be obtained by minimizing

$\displaystyle WLS(\mathbf{\Delta})$
$\displaystyle=\sum_{i=1}^{n}\frac{(n+1)^{2}(n+2)}{i(n-i+1)}\left(F(x_{i},\mathbf{\Delta})-\frac{i}{n+1}\right)^{2}$
$\displaystyle=\sum_{i=1}^{n}\frac{(n+1)^{2}(n+2)}{i(n-i+1)}\left(\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\frac{i}{n+1}\right)^{2}$
(36)

###  4.3 Maximum Product Spacing Approach of Estimation

The Maximum Product Spacing (MPS) approach is particularly useful when dealing
with unknown or complex distributions [25]. Unlike MLE, the MPS method does
not require the explicit formulation of a likelihood function, making it a
versatile and robust approach for different types of distributions. The
geometric mean of the MPS spacings is given by

$\displaystyle
G(\mathbf{\Delta})=\left\\{\prod_{i=1}^{n+1}D_{i}(x_{i},\mathbf{\Delta})\right\\}^{\frac{1}{n+1}}$
(37)

where

$\displaystyle D_{i}=\left\\{\begin{aligned} &F(x_{1},\mathbf{\Delta}),i=0\\\
&F(x_{i},\mathbf{\Delta})-F(x_{i-1},\mathbf{\Delta}),i=2,3,...,n\\\
&1-F(x_{n},\mathbf{\Delta}),i=n+1\end{aligned}\right.$

Thus, We can maximize

$\displaystyle G(\mathbf{\Delta})=$
$\displaystyle\left[\exp\left\\{-\alpha\left[\frac{H(x_{1},\boldsymbol{\psi})}{\overline{H}(x_{1},\boldsymbol{\psi})}\right]^{\beta}\right\\}\left(1-\exp\left\\{-\alpha\left[\frac{H(x_{n},\boldsymbol{\psi})}{\overline{H}(x_{n},\boldsymbol{\psi})}\right]^{-\beta}\right\\}\right)\right.$
$\displaystyle\left.\times\prod_{i=2}^{n}\left(\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\exp\left\\{-\alpha\left[\frac{H(x_{i-1},\boldsymbol{\psi})}{\overline{H}(x_{i-1},\boldsymbol{\psi})}\right]^{-\beta}\right\\}\right)\right]^{\frac{1}{n+1}}$
(38)

Equivalently, we can also maximize $H=\log G$

$\displaystyle H(\mathbf{\Delta})=$
$\displaystyle\frac{1}{n+1}\sum_{i=1}^{n+1}\log D_{i}(x_{i},\mathbf{\Delta})$
$\displaystyle=$
$\displaystyle\frac{1}{n+1}\left\\{-\alpha\left[\frac{H(x_{1},\boldsymbol{\psi})}{\overline{H}(x_{1},\boldsymbol{\psi})}\right]^{-\beta}+\log\left(1-\exp\left\\{-\alpha\left[\frac{H(x_{n},\boldsymbol{\psi})}{\overline{H}(x_{n},\boldsymbol{\psi})}\right]^{-\beta}\right\\}\right)\right.$
$\displaystyle\left.+\sum_{i=2}^{n}\log\left(\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\exp\left\\{-\alpha\left[\frac{H(x_{i-1},\boldsymbol{\psi})}{\overline{H}(x_{i-1},\boldsymbol{\psi})}\right]^{-\beta}\right\\}\right)\right\\}$
(39)

By solving $\left(\frac{\partial H}{\partial\alpha},\frac{\partial
H}{\partial\beta},\frac{\partial H}{\partial\psi_{k}}\right)=0$, we can obtain
the estimators. The partial derivatives are provided in Appendix-A.

###  4.4 Cramér-von Mises Approach of Estimation

The Cramér–von Mises method is another approach to estimate the parameters of
a distribution [26]. The Cramér-von-Mises statistic measures the difference
between the empirical distribution function of the data and the cumulative
distribution function of the proposed model. This technique has an advantage
over methods such as maximum likelihood estimation in that it considers the
whole data set, not just the location and dispersion, resulting in a more
comprehensive estimation. We can apply the Cramér–von Mises criterion to
obtain the estimators by minimizing the function $S(x;\mathbf{\Delta})$ with
respect to $\mathbf{\Delta}$, where

$\displaystyle S(x;\mathbf{\Delta})$
$\displaystyle=\frac{1}{12n^{2}}+\frac{1}{n}\sum_{i=1}^{n}\left(F(x_{i};\mathbf{\Delta})-\frac{2i-1}{2n}\right)^{2}$
$\displaystyle=\frac{1}{12n^{2}}+\frac{1}{n}\sum_{i=1}^{n}\left(\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\frac{2i-1}{2n}\right)^{2}$
(40)

Take the first partial derivatives of S, and we can have the following

$\displaystyle\frac{\partial S}{\partial\alpha}=$
$\displaystyle\frac{2}{n}\sum_{i=1}^{n}\left(\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\frac{2i-1}{2n}\right)$
$\displaystyle\times(-1)\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}$
(41) $\displaystyle\frac{\partial S}{\partial\beta}=$
$\displaystyle\frac{2}{n}\sum_{i=1}^{n}\left(\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\frac{2i-1}{2n}\right)$
$\displaystyle\times\alpha\log\left(\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right)\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}$
(42) $\displaystyle\frac{\partial S}{\partial\psi_{k}}=$
$\displaystyle\frac{2}{n}\sum_{i=1}^{n}\left(\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\frac{2i-1}{2n}\right)$
$\displaystyle\times\alpha\beta\frac{\partial
H(x_{i},\boldsymbol{\psi})}{\partial\psi_{k}}\frac{H(x_{i},\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x_{i},\boldsymbol{\psi})^{-\beta+1}}\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}$
(43)

###  4.5 Anderson and Darling Approach of Estimation

The Anderson-Darling approach was proposed by [27] to test if a data set
follows a specific distribution, and can also be used to estimate parameters.
This method gives greater weight to the tails of the distribution compared to
other methods, like the Kolmogorov-Smirnov test. The Anderson-Darling
statistic is minimized to find the parameters of the best-fitting
distribution. This estimation method is highly sensitive to deviations in the
tails and thus can be more powerful for identifying whether a particular
distribution fits the data. The Anderson-Darling estimators can be obtained by
minimizing

$\displaystyle AD(\mathbf{\Delta})=$
$\displaystyle-n-\frac{1}{n}\sum_{i=1}^{n}(2i-1)[\log
F(x_{i},\mathbf{\Delta})-\log\overline{F}(x_{n+1-i},\mathbf{\Delta})]$
$\displaystyle=$
$\displaystyle-n-\frac{1}{n}\sum_{i=1}^{n}(2i-1)\left[-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right.$
$\displaystyle-\left.\log\left(1-\exp\left\\{-\alpha\left[\frac{H(x_{n+1-i},\boldsymbol{\psi})}{\overline{H}(x_{n+1-i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}\right)\right]$
(44)

Similarly, we take the first derivatives of $AD(\mathbf{\Delta})$ and obtain

$\displaystyle\frac{\partial AD}{\partial\alpha}=$
$\displaystyle-\frac{1}{n}\sum_{i=1}^{n}(2i-1)\left(-\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right.$
$\displaystyle-\left.\left[\frac{H(x_{n+1-i},\boldsymbol{\psi})}{\overline{H}(x_{n+1-i},\boldsymbol{\psi})}\right]^{-\beta}\frac{\exp\left\\{-\alpha\left[\frac{H(x_{n+1-i},\boldsymbol{\psi})}{\overline{H}(x_{n+1-i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{1-\exp\left\\{-\alpha\left[\frac{H(x_{n+1-i},\boldsymbol{\psi})}{\overline{H}(x_{n+1-i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}\right)$
(45) $\displaystyle\frac{\partial AD}{\partial\beta}=$
$\displaystyle-\frac{1}{n}\sum_{i=1}^{n}(2i-1)\left[\alpha\log\left(\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right)\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right.$
$\displaystyle+\alpha\log\left(\frac{H(x_{n+1-i},\boldsymbol{\psi})}{\overline{H}(x_{n+1-i},\boldsymbol{\psi})}\right)\left[\frac{H(x_{n+1-i},\boldsymbol{\psi})}{\overline{H}(x_{n+1-i},\boldsymbol{\psi})}\right]^{-\beta}\frac{\exp\left\\{-\alpha\left[\frac{H(x_{n+1-i},\boldsymbol{\psi})}{\overline{H}(x_{n+1-i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{1-\exp\left\\{-\alpha\left[\frac{H(x_{n+1-i},\boldsymbol{\psi})}{\overline{H}(x_{n+1-i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}$
(46) $\displaystyle\frac{\partial AD}{\partial\psi_{k}}=$
$\displaystyle-\frac{1}{n}\sum_{i=1}^{n}(2i-1)\left[\alpha\beta\frac{H(x_{i},\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x_{i},\boldsymbol{\psi})^{-\beta+1}}\frac{\partial
H(x_{i},\boldsymbol{\psi})}{\partial\psi_{k}}\right.$
$\displaystyle\left.+\alpha\beta\frac{H(x_{n+1-i},\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x_{n+1-i},\boldsymbol{\psi})^{-\beta+1}}\frac{\partial
H(x_{n+1-i},\boldsymbol{\psi})}{\partial\psi_{k}}\frac{\exp\left\\{-\alpha\left[\frac{H(x_{n+1-i},\boldsymbol{\psi})}{\overline{H}(x_{n+1-i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{1-\exp\left\\{-\alpha\left[\frac{H(x_{n+1-i},\boldsymbol{\psi})}{\overline{H}(x_{n+1-i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}\right]$
(47)

###  4.6 Simulation and Estimation

We combined Monte Carlo simulation with the above techniques to estimate the
parameters. The parameters are set as $\alpha=2.5,\beta=0.8$, and
$\gamma=1.3$. The sample sizes $N=50,100,250,500$, and $1000$ were used to
generate random samples. For each sample size, the experiment was replicated
for $N=1000$ times. Then the bias and mean squared error (MSE) were
calculated. Table 2 and Fig. 1 show the estimation results. The MSE converges
to $0$ when $N$ increases, confirming the estimations’ stability and
reliability.

Figure 1: MSE of parameters in Table 3

Table 2: Different estimations for $\alpha=2.5,\beta=0.8,\gamma=1.3$ | MLE | LS | WLS | MPS | CVM | AD
---|---|---|---|---|---|---
$N$ |  | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE
50 | $\alpha$ | 0.0909 | 2.0761 | -0.0418 | 2.9292 | 0.0307 | 2.6645 | 0.8349 | 2.8174 | 0.1095 | 3.2913 | 0.1459 | 2.5743
$\beta$ | 0.0512 | 0.0778 | 0.0107 | 0.1906 | 0.0066 | 0.1520 | -0.1492 | 0.1068 | 0.0262 | 0.1929 | 0.0010 | 0.1192
$\gamma$ | 0.1512 | 0.8690 | 0.9373 | 8.7264 | 0.8477 | 9.4008 | 2.4409 | 63.8413 | 0.8518 | 6.9832 | 0.6359 | 5.4760
100 | $\alpha$ | 0.0662 | 1.1061 | 0.0595 | 1.6917 | 0.0878 | 1.4001 | 0.5914 | 1.5128 | 0.1357 | 1.7947 | 0.1327 | 1.3275
$\beta$ | 0.0182 | 0.0379 | -0.0229 | 0.0897 | -0.0158 | 0.0586 | -0.1047 | 0.0511 | -0.0161 | 0.0895 | -0.0163 | 0.0520
$\gamma$ | 0.0963 | 0.3519 | 0.4901 | 3.1145 | 0.2684 | 1.1972 | 0.8138 | 14.1750 | 0.4755 | 2.7094 | 0.2518 | 0.9787
250 | $\alpha$ | 0.0203 | 0.4578 | 0.0211 | 0.8317 | 0.0336 | 0.5983 | 0.3204 | 0.5872 | 0.0510 | 0.8502 | 0.0533 | 0.5754
$\beta$ | 0.0097 | 0.0148 | -0.0027 | 0.0349 | -0.0010 | 0.0211 | -0.0544 | 0.0177 | -0.0002 | 0.0350 | -0.0027 | 0.0197
$\gamma$ | 0.0265 | 0.1085 | 0.1088 | 0.3554 | 0.0614 | 0.1707 | 0.2180 | 0.2020 | 0.1141 | 0.3529 | 0.0660 | 0.1620
500 | $\alpha$ | 0.0226 | 0.2394 | 0.0168 | 0.4176 | 0.0220 | 0.2933 | 0.2066 | 0.2890 | 0.0318 | 0.4223 | 0.0321 | 0.2900
$\beta$ | 0.0049 | 0.0075 | -0.0015 | 0.0168 | 0.0002 | 0.0102 | -0.0335 | 0.0086 | -0.0003 | 0.0168 | -0.0008 | 0.0100
$\gamma$ | 0.0189 | 0.0543 | 0.0498 | 0.1280 | 0.0307 | 0.0742 | 0.1269 | 0.0809 | 0.0529 | 0.1282 | 0.0347 | 0.0744
1000 | $\alpha$ | 0.0072 | 0.1146 | 0.0071 | 0.2278 | 0.0113 | 0.1497 | 0.1163 | 0.1310 | 0.0146 | 0.2290 | 0.0167 | 0.1504
$\beta$ | 0.0027 | 0.0036 | -0.0003 | 0.0086 | 0.0001 | 0.0050 | -0.0198 | 0.0039 | 0.0003 | 0.0086 | -0.0007 | 0.0050
$\gamma$ | 0.0079 | 0.0254 | 0.0258 | 0.0673 | 0.0160 | 0.0373 | 0.0691 | 0.0331 | 0.0274 | 0.0673 | 0.0186 | 0.0380

##  5 Special Cases

In this section, we will explore a variety of special cases that emerge from
our novel distribution model. By closely examining these unique instances, we
aim to illustrate the multifaceted aspects and potential applications of the
T2GWG family of distributions.

###  5.1 Type-2 Gumbel Weibull-Exponential (T2GWE) distribution

Suppose the baseline distribution $H(x,\boldsymbol{\psi})$ is an exponential
distribution with parameter $\gamma>0$. Then $h(x;\gamma)=\gamma e^{-\gamma
x}$ and $H(x;\gamma)=1-e^{-\gamma x}$.

####  5.1.1 cdf and pdf of the T2GWE distribution

The cdf of the Type-2 Gumbel Weibull-Exponential distribution is presented as

$\displaystyle F_{T2GWE}(x)=\exp\left\\{-\alpha(e^{\gamma
x}-1)^{-\beta}\right\\}$ (48)

and the pdf is given by

$\displaystyle f_{T2GWE}(x)=\beta\alpha(e^{\gamma
x}-1)^{-\beta-1}\exp\left\\{\gamma x-\alpha(e^{\gamma x}-1)^{-\beta}\right\\}$
(49)

where $x\geq 0$ and $\alpha,\beta>0$

####  5.1.2 Hazard rate and quantile functions

The hrf is shown as

$\displaystyle h_{T2GWE}=\frac{\beta\alpha(e^{\gamma
x}-1)^{-\beta-1}\exp\left\\{\gamma x-\alpha(e^{\gamma
x}-1)^{-\beta}\right\\}}{1-\exp\left\\{-\alpha(e^{\gamma
x}-1)^{-\beta}\right\\}}$ (50)

and the reverse hrf is given by

$\displaystyle\tau_{T2GWE}=\beta\alpha(e^{\gamma x}-1)^{-\beta-1}$ (51)

Moreover, the quantile function is obtained as

$\displaystyle x_{p}=-\frac{1}{\gamma}\log\left(\left[\frac{\log
p}{-\alpha}\right]^{-\frac{1}{\beta}}+1\right)$ (52)

Fig. 2 displays several typical configurations of the pdf and hrf for the
T2GWE distribution. The pdf of the T2GWE distribution shows various
configurations, including almost symmetric, right-skewed, decreasing, and
increasing. Additionally, the hrf of the T2GWE distribution can exhibit a
range of shapes, such as decreasing, increasing, and right-skewed.

Figure 2: Left: The pdf of T2GWE distribution for different parameters. Right:
The hrf of T2GWE for different parameter values.

###  5.2 Type-2 Gumbel Weibull-Uniform (T2GWU) distribution

Let the baseline distribution $H(x,\boldsymbol{\psi})$ be a uniform
distribution with parameter $\gamma>0$. Then $h(x,\gamma)=\frac{1}{\gamma}$
and $H(x,\gamma)=\frac{x}{\gamma}$.

####  5.2.1 cdf and pdf of the T2GWU distribution

The cdf of the T2GWU distribution is

$\displaystyle
F_{T2GWU}=\exp\left\\{-\alpha\left(\frac{x}{\gamma-x}\right)^{-\beta}\right\\},$
(53)

and the corresponding pdf is

$\displaystyle f_{T2GWU}(x)=\alpha\beta\frac{\gamma
x^{-\beta-1}}{(\gamma-x)^{-\beta+1}}\exp\left\\{-\alpha\left(\frac{x}{\gamma-x}\right)^{-\beta}\right\\}.$
(54)

####  5.2.2 Hazard rate and quantile functions

The hrf of T2GWU is displayed by

$\displaystyle h_{T2GWU}=\frac{\alpha\beta\frac{\gamma
x^{-\beta-1}}{(\gamma-x)^{-\beta+1}}\exp\left\\{-\alpha\left(\frac{x}{\gamma-x}\right)^{-\beta}\right\\}}{1-\exp\left\\{-\alpha\left(\frac{x}{\gamma-x}\right)^{-\beta}\right\\}},$
(55)

and the reverse hrf is given by

$\displaystyle\tau_{T2GWU}=\alpha\beta\frac{\gamma
x^{-\beta-1}}{(\gamma-x)^{-\beta+1}}.$ (56)

Moreover, the quantile function can be obtained as

$\displaystyle x_{p}=\frac{1}{1+\left[\frac{\log
p}{-\alpha}\right]^{\frac{1}{\beta}}}.$ (57)

Fig. 3 plots of the pdf and hrf for the T2GWU distribution with several
combinations of parameter values. The pdf plots show different shapes,
including right-skewed, decreasing, and increasing. In addition, the hrf plots
capture various possibilities such as increasing, decreasing, bathtub, and
shallow bathtub.

Figure 3:  Left: pdf of T2GWU distribution for different values of parameters
$\alpha$, $\beta$, and $\gamma$. Right: hrf of T2GWU for selected parameters
$\alpha$, $\beta$, and $\gamma$.

###  5.3 Type-2 Gumbel Weibull-Pareto (T2GWP) distribution

If we set $H(x,\boldsymbol{\psi})$ as a Pareto distribution with parameter
$\theta,k>0$, then $\displaystyle h(x,\theta,k)=\frac{k\theta^{k}}{x^{k+1}}$
and $\displaystyle H(x,\theta,k)=1-\left(\frac{\theta}{x}\right)^{k}$.

####  5.3.1 cdf and pdf of the T2GWP distribution

Thus the cdf of the T2GWP distribution is given by

$\displaystyle
F_{T2GWP}=\exp\left\\{-\alpha\left(\left[\frac{x}{\theta}\right]^{k}-1\right)^{-\beta}\right\\},$
(58)

with a pdf

$\displaystyle f_{T2GWP}(x)=\beta\alpha
kx^{k-1}\theta^{-k}\left(\left[\frac{x}{\theta}\right]^{k}-1\right)^{-\beta-1}\exp\left\\{-\alpha\left(\left[\frac{x}{\theta}\right]^{k}-1\right)^{-\beta}\right\\}.$
(59)

####  5.3.2 Hazard rate and quantile functions

The hrf of T2GWP is displayed by

$\displaystyle h_{T2GWP}=\frac{\beta\alpha
kx^{k-1}\theta^{-k}\left(\left[\frac{x}{\theta}\right]^{k}-1\right)^{-\beta-1}\exp\left\\{-\alpha\left(\left[\frac{x}{\theta}\right]^{k}-1\right)^{-\beta}\right\\}}{1-\exp\left\\{-\alpha\left(\left[\frac{x}{\theta}\right]^{k}-1\right)^{-\beta}\right\\}},$
(60)

and the reverse hrf is

$\displaystyle\tau_{T2GWP}=\beta\delta\alpha^{-\delta}kx^{k-1}\theta^{-k}\left(\left[\frac{x}{\theta}\right]^{k}-1\right)^{-\beta\delta-1}.$
(61)

Moreover, the quantile function is obtained as

$\displaystyle x_{p}=\theta\left(\left[\frac{\log
p}{-\alpha}\right]^{-\frac{1}{\beta}}+1\right)^{\frac{1}{k}}$ (62)

Shapes of the pdf and hrf for the T2GWP distribution with selected parameters
are shown in Fig. 4. The pdfs exhibit a variety of shapes including right-
skewed, decreasing, and increasing. Moreover, hrf plots for the T2GWP
distribution display growing, decreasing, and right-skewed forms.

Figure 4:  Left: The pdf of T2GWP distribution for selected values of
$\alpha$, $\beta$, $\theta$ and $k$. Right: The hrf of T2GWP for various
$\alpha$, $\beta$, $\theta$ and $k$.

##  6 Applications

In this section, we will investigate beyond theoretical constructs and delve
into the practical implications of our model, demonstrating its applicability
using real-world data sets. This will validate the practical utility of our
newly devised model and shed light on how it can be effectively employed in
handling concrete data-driven scenarios. The objective is to ensure that our
theoretical advancements resonate with tangible applications, thereby
significantly contributing to both the academic discourse and the operational
applications of statistical distributions.

In this section, we present three applications of the Type-2 Gumbel Weibull-
exponential distribution. We compared it with the Exponentiated Gumbel Type-2
(EGT) [28], Weibull Generalized Exponential (WGE) [29], Lomax Gumbel Type-2
(LGT) [30], Type-2 Gumbel (T2G), and with Exponentiated Weibull-Logistic
distributions (EWL) [31]. The pdf and cdf of those distributions are provided
in Appendix B. The goodness-of-fit statistics including -2log-likelihood
statistic, Cramér-von Mises statistic ($W^{*}$), Anderson-Darling statistic
($A^{*}$), Akaike Information Criterion (AIC), Bayesian Information Criterion
(BIC), Consistent Akaike Information Criterion (CAIC), Hannan-Quinn criterion
(HQIC), Kolmogorov-Smirnov test statistic (K-S) and its corresponding p-value
are reported.

To evaluate and compare the performance of different models, we can examine
their goodness-of-fit statistics. Generally, a model with smaller values in
these statistics fits the data better. However, it’s important to note that
for the p-values, which are a measure of expectation, a larger value indicates
a better fit.

###  6.1 Chemotherapy data

This dataset is a subset of data reported by Bekker et al. [32], which
represents the survival times (in years) of a group of patients who received
chemotherapy treatment alone. Please refer to the “Declarations” section for
guidance on how to access the data.

The estimates of the parameters and the goodness-of-fit statistics are
summarized in Table 3. Fig. 5 plots of the fitted densities alongside the
histogram and the expected probability. From Table 3, we can conclude that the
Type-2 Gumbel Weibull-exponential distribution has better performance than
other distributions because of its lowest values among all measures of
goodness-of-fit and the highest p-value in the K-S test.

In Fig. 6, the Kaplan-Meier (K–M) survival curve, as well as the theoretical
and empirical cumulative distribution functions (ECDF), and total time on test
(TTT) scaled are displayed. The closely matched empirical and theoretical
plots suggest that our model is an excellent fit for the given data.
Additionally, the TTT scaled plot shows that the model is suitable for a
hazard rate structure that is not monotonic.

Figure 5: left) Fitted density superposed on the histogram and observed
probability for the Chemotherapy data. right) Expected probability plots for
the Chemotherapy data.

Figure 6: Fitted K-M survival curve, theoretical and ECDF, the TTT statistics,
and the hrf for the Chemotherapy data. Table 3: MLEs and Goodness-of-Fit
Statistics for Chemotherapy Data

|  Estimates (SE) | Statistics
---|---|---
Model | $\alpha$ | $\beta$ | $\gamma$ |  | $-2\log\,L$ | $AIC$ | $CAIC$ | $BIC$ | $HQIC$ | $W^{*}$ | $A^{*}$ | K-S | p-value
T2GWE | 1.1328 | 0.5416 | 1.4015 | - | 113.3334 | 119.3334 | 119.9188 | 124.7534 | 121.3539 | 0.0415 | 0.3113 | 0.0756 | 0.9421
| (0.4388) | (0.1170) | (0.5530) |  |  |  |  |  |  |  |  |  |
| $\alpha$ | $\phi$ | $\theta$ |  |  |  |  |  |  |  |  |  |
EGT | 1000.1282 | 0.1452 | 7.1554 | - | 115.9096 | 121.9096 | 122.4949 | 127.3295 | 123.9301 | 0.0608 | 0.4231 | 0.0927 | 0.0926
| ($3.6913\times 10^{3}$) | ($7.4598\times 10^{-2}$) | (3.7373) |  |  |  |  |  |  |  |  |  |
| $\alpha$ | $\theta$ | $\gamma$ |  |  |  |  |  |  |  |  |  |
WGE | 3.9393 | 0.9508 | 0.1484 | - | 115.9251 | 121.9251 | 122.5105 | 127.3451 | 123.9457 | 0.0917 | 0.6079 | 0.1120 | 0.5864
| (8.8328) | (0.2076) | (0.2675) |  |  |  |  |  |  |  |  |  |
| $\alpha$ | $\beta$ | $\theta$ | $k$ |  |  |  |  |  |  |  |  |
LGT | 17.9903 | 0.0196 | 7.0332 | 0.1503 | 116.3564 | 124.3564 | 125.3564 | 131.5831 | 127.0504 | 0.0610 | 0.4270 | 0.0892 | 0.835
| (24.8990) | (0.0298) | (1.9777) | (0.0459) |  |  |  |  |  |  |  |  |
| $\alpha$ | $\nu$ |  |  |  |  |  |  |  |  |  |  |
T2G | 0.4987 | 0.8672 | - | - | 127.6381 | 131.6381 | 131.9238 | 135.2515 | 132.9851 | 0.1430 | 0.9790 | 0.1382 | 0.3253
| (0.0979) | (0.0928) |  |  |  |  |  |  |  |  |  |  |
| $\alpha$ | $\beta$ | $\lambda$ | $\theta$ |  |  |  |  |  |  |  |  |
EWL | 4.6147 | 0.3981 | 0.4506 | 221.7210 | 140.0032 | 148.0032 | 149.0032 | 155.2298 | 150.6972 | 0.3318 | 2.0776 | 0.1853 | 0.0796
| (1.1065) | (0.6466) | (0.7320) | (250.5359) |  |  |  |  |  |  |  |  |

###  6.2 Depressive data

This dataset consists of scores from the “General Rating of Affective Symptoms
for Preschoolers” (GRASP) scale, which is used to assess behavioral and
emotional problems in children. The data set is studied by [33]. Please refer
to the “Declarations” section for detailed instructions on how to access the
data.

From Table 4 and Fig. 7, we can conclude that the Type-2 Gumbel Weibull-
exponential distribution has better performance than other distributions as it
possesses the smallest values for all goodness-of-fit statistics and the
highest p-value in the K-S test. As shown in Fig. 8, the close resemblance
between the fitted empirical and theoretical plots indicates a strong fit of
our model to the provided data. Furthermore, the TTT scaled plot provides
clear evidence that the model is appropriate for a non-monotonic hazard rate
structure.

Figure 7: left) Fitted density superposed on the histogram and observed
probability for the Depressive data. right) Expected probability plots for the
Depressive data.

Figure 8: Fitted K-M survival curve, theoretical and ECDF, the TTT statistics,
and the hrf for the Depressive data. Table 4: MLEs and Goodness-of-Fit
Statistics for Depressive Data

|  Estimates (SE) | Statistics
---|---|---
Model | $\alpha$ | $\beta$ | $\gamma$ |  | $-2\log\,L$ | $AIC$ | $CAIC$ | $BIC$ | $HQIC$ | $W^{*}$ | $A^{*}$ | K-S | p-value
T2GWE | 0.1127 | 5.0122 | 0.0223 | - | 785.7463 | 791.7463 | 791.9309 | 800.4398 | 795.2791 | 0.1814 | 1.3478 | 0.1092 | 0.0821
| (0.0818) | (0.3542) | (0.0024) |  |  |  |  |  |  |  |  |  |
| $\alpha$ | $\phi$ | $\theta$ |  |  |  |  |  |  |  |  |  |
EGT | 9.7621 | 2.0192 | 1691.0601 | - | 803.2078 | 809.2078 | 809.3924 | 817.9013 | 812.7406 | 0.3185 | 2.1927 | 0.1108 | 0.0746
| (3.7163) | (0.2403) | (1088.5591) |  |  |  |  |  |  |  |  |  |
| $\alpha$ | $\theta$ | $\gamma$ |  |  |  |  |  |  |  |  |  |
WGE | 44.2733 | 3.7778 | 0.0115 | - | 852.253 | 858.253 | 858.4376 | 866.9465 | 861.7858 | 0.8203 | 5.0577 | 0.1913 | 0.0001
| (163.1340) | (0.5147) | (0.0008) |  |  |  |  |  |  |  |  |  |
| $\alpha$ | $\beta$ | $\theta$ | $k$ |  |  |  |  |  |  |  |  |
LGT | 2.7374 | 0.0006 | 161.2461 | 1.0193 | 807.7372 | 815.7372 | 816.0473 | 827.3286 | 820.4476 | 0.3392 | 2.3111 | 0.1252 | 0.0301
| (0.9395) | (0.0028) | (46.6043) | (0.1060) |  |  |  |  |  |  |  |  |
| $\alpha$ | $\nu$ |  |  |  |  |  |  |  |  |  |  |
T2G | 1684.3935 | 2.3918 | - | - | 919.7708 | 923.7708 | 923.8624 | 929.5665 | 926.126 | 0.2340 | 1.6765 | 0.2480 | $1.392e-07$
| (404.6525) | (0.0807) |  |  |  |  |  |  |  |  |  |  |
| $\alpha$ | $\beta$ | $\lambda$ | $\theta$ |  |  |  |  |  |  |  |  |
EWL | 1.2679 | 0.0770 | 0.6636 | 61.1849 | 816.1935 | 824.1935 | 824.5036 | 835.7849 | 828.9039 | 0.4525 | 2.9886 | 0.1310 | 0.0202
| (0.3130) | (0.0271) | (0.2349) | (31.6238) |  |  |  |  |  |  |  |  |

###  6.3 Covid-Mexico data

This dataset recorded the mortality rates of the patients infected by the
COVID-19 pandemic in Mexico and was studied by [34]. This dataset consists of
106 observations from March 31, 2020, to July 20, 2020. To access the data,
please follow the instructions in the “Declarations” section.

Table 5 and Fig. 9 provide the estimates of the parameters and the goodness-
of-fit. We can conclude that the Type-2 Gumbel Weibull-exponential
distribution has the best performance among other distributions given its
smallest values in all goodness-of-fit statistics and the highest p-value in
the K-S test. Displayed in Fig. 10 are the K-M survival curve, as well as the
theoretical and ECDF, and TTT scaled. The convergence of the fitted empirical
and theoretical plots suggests that our model accurately represents the given
data. In addition, the TTT scaled plot clearly indicates that the model is
well-suited for a hazard rate structure that is not strictly monotonic.

Figure 9: left) Fitted density superposed on the histogram and observed
probability for the Covid-Mexico data. right) Expected probability plots for
the Covid-Mexico data.

Figure 10: Fitted K-M survival curve, theoretical and ECDF functions, the TTT
statistics, and the hrf for the Covid-Mexico data. Table 5: MLEs and Goodness-
of-Fit Statistics for Covid-Mexico Data

|  Estimates (SE) | Statistics
---|---|---
Model | $\alpha$ | $\beta$ | $\gamma$ |  | $-2\log\,L$ | $AIC$ | $CAIC$ | $BIC$ | $HQIC$ | $W^{*}$ | $A^{*}$ | K-S | p-value
T2GWE | 3.9220 | 0.7209 | 0.9767 | - | 375.7089 | 381.7089 | 381.9442 | 389.6993 | 384.9475 | 0.0454 | 0.2525 | 0.0577 | 0.8721
| (1.4023) | (0.2117) | (0.3942) |  |  |  |  |  |  |  |  |  |
| $\alpha$ | $\phi$ | $\theta$ |  |  |  |  |  |  |  |  |  |
EGT | 56.8310 | 0.4537 | 6.7577 | - | 376.3527 | 382.3527 | 382.5879 | 390.343 | 385.5912 | 0.0521 | 0.2877 | 0.0646 | 0.7686
| (78.7004) | (0.1409) | (1.3261) |  |  |  |  |  |  |  |  |  |
| $\alpha$ | $\theta$ | $\gamma$ |  |  |  |  |  |  |  |  |  |
WGE | 49.8868 | 1.8037 | 0.0326 | - | 382.8119 | 388.8119 | 389.0472 | 396.8022 | 392.0504 | 0.1128 | 0.7313 | 0.0664 | 0.7387
| (72.3253) | (0.1413) | (0.0222) |  |  |  |  |  |  |  |  |  |
| $\alpha$ | $\beta$ | $\theta$ | $k$ |  |  |  |  |  |  |  |  |
LGT | 11.9707 | 0.0343 | 8.5300 | 0.3374 | 376.8091 | 384.8091 | 385.2052 | 395.4629 | 389.1271 | 0.0557 | 0.3106 | 0.0688 | 0.6975
| (11.5930) | (0.0615) | (1.6317) | (0.0982) |  |  |  |  |  |  |  |  |
| $\alpha$ | $\nu$ |  |  |  |  |  |  |  |  |  |  |
T2G | 2.8185 | 1.6830 | - | - | 400.2692 | 404.2692 | 404.3857 | 409.5961 | 406.4282 | 0.2734 | 1.7490 | 0.0928 | 0.3213
| (0.2928) | (0.1163) |  |  |  |  |  |  |  |  |  |  |
| $\alpha$ | $\beta$ | $\lambda$ | $\theta$ |  |  |  |  |  |  |  |  |
EWL | 3.6179 | 0.3384 | 0.4196 | 150.8827 | 390.9746 | 398.9747 | 399.3707 | 409.6284 | 403.2927 | 0.1807 | 1.1892 | 0.0957 | 0.2857
| (0.6784) | (2.5979) | (3.2208) | (109.2607) |  |  |  |  |  |  |  |  |

##  7 Conclusion

This paper introduces a novel methodology for generating continuous
statistical distributions by utilizing the exponentiated odds ratio, which is
based on the concepts of survival analysis. The approach described herein
constitutes a substantial advancement in the field of statistical modeling,
effectively augmenting the flexibility and accuracy of distribution models to
effectively address the requirements posed by intricate contemporary data
architectures. The major point of this progress lies in the formulation of the
“Type-2 Gumbel Weibull-G Family of Distributions,” which has undergone a
comprehensive mathematical analysis. The scope of this investigation covered
various statistical properties, such as expansions of density functions,
moments, hazard rate and quantile functions, Rényi entropy, order statistics,
and an examination of stochastic ordering.

In order to assess the robustness and reliability of the new generator, we
employed a set of five advanced parameter estimation techniques: Maximum
Likelihood, Least Square, Weighted Least Square, Maximum Product Spacing,
Cramér-von Mises, and Anderson and Darling. The efficacy and utility of the
Type-2 Gumbel Weibull-G distributions were further validated through a
comprehensive analysis of three datasets obtained from real-world scenarios.
These practical implementations demonstrated the superior statistical accuracy
of our proposed distributions over existing models, thereby emphasizing their
relevance and applicability in both theoretical and practical statistical
domains.

Our future research endeavors will entail a thorough investigation of several
sub-families within the new generator. This exploration will focus on their
distinct properties and potential applications in diverse scenarios, setting
them in comparison with other established distribution models. Furthermore, we
are now developing an R package with the objective of optimizing the parameter
estimate process through the utilization of diverse methodologies. This will
result in improved efficiency of data-fitting procedures, thus increasing the
accessibility and practicality of our study for wider applications.

## Abbreviations

The following abbreviations are used in this paper:

T2GWG | The Type-2 Gumbel Weibull-G
---|---
cdf | cumulative distribution function
pdf | probability density function
hrf | hazard rate function
Exp-G | exponentiated-G
EGT | Exponentiated Gumbel Type-2
WGE | Weibull Generalized Exponential
LGT | Lomax Gumbel Type-2
T2G | Type-2 Gumbel
EWL | Exponentiated Weibull-Logistic distributions
MLE | maximum likelihood estimates
MPS | maximum product spacing estimates
LS | least square estimates
WLS | weighted least square estimates
CVM | Cramér-von Mises estimates
AD | Anderson and Darling estimates
T2GWE | Type-2 Gumbel Weibull-Exponential
T2GWU | Type-2 Gumbel Weibull-Uniform
T2GWP | Type-2 Gumbel Weibull-Pareto
AIC | Akaike Information Criterion
CAIC | Consistent Akaike Information Criterion
BIC | Bayesian Information Criterion
HQIC | Hannan-Quinn Criterion
$W^{*}$ | Cramér-von Mises statistic
$A^{*}$ | Anderson-Darling statistic
K-S | Kolmogorov-Smirnov statistic
ECDF | empirical cumulative distribution function
TTT | total time on test
K-M | Kaplan-Meier

Supplementary information Not applicable

Acknowledgments

The authors wish to thank all members of the CSDA lab at the University of
West Florida for their helpful comments on the manuscript.

## Declarations

  * •

Funding: This research received no specific grant from any funding agency in
the public, commercial, or not-for-profit sectors.

  * •

Conflict of interest/Competing interests: On behalf of all authors, the
corresponding author states that there is no conflict of interest.

  * •

Ethics approval: Not applicable

  * •

Consent to participate: Not applicable

  * •

Consent for publication: Not applicable

  * •

Availability of data and materials: All data utilized in this study is openly
available on GitHub. The dataset can be accessed through the provided link:
https://github.com/shusenpu/Generator_Data. The data can also be accessed
using its Digital Object Identifier (DOI): 10.5281/zenodo.10215787. To
retrieve the data, visit the following link:
https://doi.org/10.5281/zenodo.10215787.

  * •

Code availability: available upon request.

  * •

Authors’ contributions: All authors have contributed significantly to writing
and editing the paper.

##  Appendix A The first derivatives of H

The first partial derivatives of $H$ in Section 4 with respect to
$\alpha,\beta,\delta,\psi_{k}$ are given by

$\displaystyle\frac{\partial H}{\partial\alpha}=$
$\displaystyle\frac{1}{n+1}\left\\{-\left[\frac{H(x_{1},\boldsymbol{\psi})}{\overline{H}(x_{1},\boldsymbol{\psi})}\right]^{-\beta}+\frac{\left[\frac{H(x_{n},\boldsymbol{\psi})}{\overline{H}(x_{n},\boldsymbol{\psi})}\right]^{-\beta}\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{1-\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}\right.$
$\displaystyle+\sum_{i=2}^{n}\left[\frac{\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\exp\left\\{-\alpha\left[\frac{H(x_{i-1},\boldsymbol{\psi})}{\overline{H}(x_{i-1},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}\right.$
$\displaystyle\left.\left.-\frac{\left[\frac{H(x_{i-1},\boldsymbol{\psi})}{\overline{H}(x_{i-1},\boldsymbol{\psi})}\right]^{-\beta}\exp\left\\{-\alpha\left[\frac{H(x_{i-1},\boldsymbol{\psi})}{\overline{H}(x_{i-1},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\exp\left\\{-\alpha\left[\frac{H(x_{i-1},\boldsymbol{\psi})}{\overline{H}(x_{i-1},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}\right]\right\\}$
(63) $\displaystyle\frac{\partial H}{\partial\beta}=$
$\displaystyle\frac{1}{n+1}\left\\{\alpha\log\left(\frac{H(x_{1},\boldsymbol{\psi})}{\overline{H}(x_{1},\boldsymbol{\psi})}\right)\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right.$
$\displaystyle+\frac{\log\left(\frac{H(x_{n},\boldsymbol{\psi})}{\overline{H}(x_{n},\boldsymbol{\psi})}\right)\left[\frac{H(x_{n},\boldsymbol{\psi})}{\overline{H}(x_{n},\boldsymbol{\psi})}\right]^{-\beta}\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{1-\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}$
$\displaystyle+\sum_{i=2}^{n}\left[\frac{\log\left(\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right)\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\exp\left\\{-\alpha\left[\frac{H(x_{i-1},\boldsymbol{\psi})}{\overline{H}(x_{i-1},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}\right.$
$\displaystyle\left.\left.\left.-\frac{\log\left(\frac{H(x_{i-1},\boldsymbol{\psi})}{\overline{H}(x_{i-1},\boldsymbol{\psi})}\right)\left[\frac{H(x_{i-1},\boldsymbol{\psi})}{\overline{H}(x_{i-1},\boldsymbol{\psi})}\right]^{-\beta}\exp\left\\{-\alpha\left[\frac{H(x_{i-1},\boldsymbol{\psi})}{\overline{H}(x_{i-1},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\exp\left\\{-\alpha\left[\frac{H(x_{i-1},\boldsymbol{\psi})}{\overline{H}(x_{i-1},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}\right]\right)\right\\}$
(64)

$\displaystyle\frac{\partial H}{\partial\psi_{k}}=$
$\displaystyle\frac{1}{n+1}\left\\{\alpha\beta\frac{H(x_{1},\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x_{1},\boldsymbol{\psi})^{-\beta+1}}\frac{\partial
H(x_{1},\boldsymbol{\psi})}{\partial\psi_{k}}+\frac{\frac{H(x_{n},\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x_{n},\boldsymbol{\psi})^{-\beta+1}}\frac{\partial
H(x_{n},\boldsymbol{\psi})}{\partial\psi_{k}}\exp\left\\{-\alpha\left[\frac{H(x_{n},\boldsymbol{\psi})}{\overline{H}(x_{n},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{1-\exp\left\\{-\alpha\left[\frac{H(x_{n},\boldsymbol{\psi})}{\overline{H}(x_{n},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}\right.$
$\displaystyle+\sum_{i=2}^{n}\left[\frac{\frac{H(x_{i},\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x_{i},\boldsymbol{\psi})^{-\beta+1}}\frac{\partial
H(x_{i},\boldsymbol{\psi})}{\partial\psi_{k}}\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\exp\left\\{-\alpha\left[\frac{H(x_{i-1},\boldsymbol{\psi})}{\overline{H}(x_{i-1},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}\right.$
$\displaystyle\left.\left.\left.-\frac{\frac{H(x_{i-1},\boldsymbol{\psi})^{-\beta-1}}{\overline{H}(x_{i-1},\boldsymbol{\psi})^{-\beta+1}}\frac{\partial
H(x_{i-1},\boldsymbol{\psi})}{\partial\psi_{k}}\exp\left\\{-\alpha\left[\frac{H(x_{i-1},\boldsymbol{\psi})}{\overline{H}(x_{i-1},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}{\exp\left\\{-\alpha\left[\frac{H(x_{i},\boldsymbol{\psi})}{\overline{H}(x_{i},\boldsymbol{\psi})}\right]^{-\beta}\right\\}-\exp\left\\{-\alpha\left[\frac{H(x_{i-1},\boldsymbol{\psi})}{\overline{H}(x_{i-1},\boldsymbol{\psi})}\right]^{-\beta}\right\\}}\right]\right)\right\\}$
(65)

##  Appendix B Distributions Used in the Application Section

  * •

Exponentiated Gumbel Type-2 Distribution:

$F_{{}_{EGT}}(x)=1-\left(1-e^{-\theta x^{-\phi}}\right)^{\alpha},$
$f_{{}_{EGT}}(x)=\alpha\phi\theta x^{-\phi-1}e^{-\theta
x^{-\phi}}\left(1-e^{-\theta x^{-\phi}}\right)^{\alpha-1},$
$\alpha,\phi,\theta>0$

  * •

Weibull Generalized Exponential Distribution:

$F_{{}_{WGE}}(x)=1-e^{-a\left[e^{\lambda x}-1\right]^{b}},$
$f_{{}_{WGE}}(x)=ab\lambda e^{\lambda x}\left[e^{\lambda
x}-1\right]^{b-1}e^{-a\left[e^{\lambda x}-1\right]^{b}},$ $a,b,\lambda>0$

  * •

Lomax Gumbel Type-2 Distribution:

$F_{{}_{LGT}}(x)=1-\beta^{\alpha}\left(\beta-\log\left[1-e^{-\theta
x^{-k}}\right]\right)^{-\alpha}$
$f_{{}_{LGT}}(x)=\frac{\alpha\beta^{\alpha}\theta kx^{-k-1}e^{-\theta
x^{-k}}}{\left[1-e^{-\theta x^{-k}}\right]}\left[\beta-\log\left[1-e^{-\theta
x^{-k}}\right]\right]^{-(\alpha+1)}$ $\alpha,\beta,\theta,k>0$

  * •

Type-2 Gumbel Distribution:

$F_{{}_{T2G}}(x)=e^{-\theta x^{-\phi}}$ $f_{{}_{T2G}}(x)=\phi\theta
x^{-\phi-1}e^{-\theta x^{-\phi}}$ $\phi,\theta>0$

  * •

Exponentiated Weibull-Logistic Distribution:

$F_{{}_{EWL}}(x)=\left(1-e^{-\alpha e^{\lambda\beta x}}\right)^{\theta}$
$f_{{}_{EWL}}(x)=\theta\left(1-e^{-\alpha e^{\lambda\beta
x}}\right)^{\theta-1}\left(\lambda\alpha\beta e^{\lambda\beta x-\alpha
e^{\lambda\beta x}}\right),$ $\lambda,\alpha,\beta,\theta>0.$

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