# A Systematic Comparison of Encrypted Machine Learning Solutions for Image
Classification

Veneta Haralampieva
Imperial College London
<EMAIL_ADDRESS>Daniel Rueckert
Imperial College London
Technical University of Munich
<EMAIL_ADDRESS>
Jonathan Passerat-Palmbach
Imperial College London
ConsenSys Health
<EMAIL_ADDRESS>
###### Abstract

This work provides a comprehensive review of existing frameworks based on
secure computing techniques in the context of private image classification.
The in-depth analysis of these approaches is followed by careful examination
of their performance costs, in particular runtime and communication overhead.

To further illustrate the practical considerations when using different
privacy-preserving technologies, experiments were conducted using four state-
of-the-art libraries implementing secure computing at the heart of the data
science stack: PySyft and CrypTen supporting private inference via Secure
Multi-Party Computation, TF-Trusted utilising Trusted Execution Environments
and HE- Transformer relying on Homomorphic encryption.

Our work aims to evaluate the suitability of these frameworks from a
usability, runtime requirements and accuracy point of view. In order to better
understand the gap between state-of-the-art protocols and what is currently
available in practice for a data scientist, we designed three neural network
architecture to obtain secure predictions via each of the four aforementioned
frameworks. Two networks were evaluated on the MNIST dataset and one on the
Malaria Cell image dataset. We observed satisfying performances for TF-Trusted
and CrypTen and noted that all frameworks perfectly preserved the accuracy of
the corresponding plaintext model.

##  1 Introduction

In recent years the use of Deep Learning models has become much more
prevalent, with these models achieving outstanding results in a variety of
fields such as Computer Vision [1] and Natural Language Processing [2]. At the
same time, Machine Learning-as-a-service (MLaaS) solutions hosted on cloud
platforms are a common strategy for deploying Deep Learning models. They
however rely on clients submitting raw data to the server which is
unacceptable when utilising sensitive data such as patient records or
financial statements.

In certain scenarios, such as medical applications, there are strict rules and
regulations governing the storage and usage of personal data such as General
Data Protection Regulation (GDPR) in the European Union and Health Insurance
Portability and Accountability Act (HIPAA) in the United States. Similarly,
the model owners consider the trained model and its parameters to be their own
intellectual property and do not want to reveal it to customers or
collaborators for inference.

These mutual privacy constraints have prompted the emergence of a body of work
leveraging Privacy-Preserving (PP) methods such as Homomorphic Encryption,
Secure Multi-Party Computation and Trusted Execution Environments in the
context of PP inference. The results exposed in the literature and summarised
in Section 2 encouraged us to produce a more practical analysis of the current
possibilities of secure inference for image classification. This work fills
the following gaps in existing studies: 1) the homogeneity of tested model
architectures and datasets, 2) the evaluation of these secure inference
techniques in the more realistic context of the most popular deep learning and
data science software stack: Python, PyTorch and TensorFlow.

Our study consists in experiments run over two datasets: MNIST [3]: A standard
Machine Learning dataset consisting of 60,000 training and 10,000 testing
greyscale images of handwritten digits from 0-9 of size 28x28, and the Malaria
Cell Images [4]: A balanced medical imaging dataset containing 27,558 images
with 2 classes: infected and uninfected individuals. The data was shuffled and
split randomly into $90\%$ training data and $10\%$ testing data as done
previously [5].

Section 3.1 introduces the four Python libraries that were used to compare
Private Inference solutions based on Secure Multi-Party Computation,
Homomorphic encryption and Trusted Execution Environments. All experiments
described in Section 3.2 were conducted by first training a plaintext model
and then performing secure inference.

##  2 Privacy-preserving inference in the literature

###  2.1 Homomorphic Encryption

One of the key seminal works in private inference, proposed by [6] was
CryptoNets which relied on Leveled Homomorphic Encryption (LHE) and supported
both fully connected and convolutional layers since they can be performed
using matrix multiplications. One characteristic disadvantage of the framework
was its inability to support commonly used non-linear activations, as well as
Max Pooling layers, due to the fact they cannot be computed using polynomials.
The authors instead proposed replacing these with HE friendly variants such as
using the square activation and a variation of average pooling instead of max
pooling. Later, [7] extended their work in CaRENets, where they proposed a new
image packing scheme to more efficiently store and process the encrypted data.
A radically different approach was explored by [8] who suggested using the
Matrix Operation for Randomization or Encryption (MORE) scheme [9]. Unlike the
ring based HE schemes used in previous solutions [6, 7] which represented the
data as polynomials and relied on noise additions, the MORE algorithm encodes
data as a matrix in a noise-free non-deterministic manner. While this approach
offered weaker security, it provided an innovative way of handling non-linear
activations using the properties of the encrypted matrix.

###  2.2 Secure Multi-Party Computation

Numerous frameworks based on Secure Multi-Party Computation (SMPC) techniques
such as Garbled Circuits (GC) [10], Secret Sharing (SS) [11] and Oblivious
Transfer (OT) [12] have been proposed in the context on private inference. All
operate in a two party honest-but-curious setting where one party is
considered the data owner and the other the model owner.

SecureML [13] was the first framework to provide both secure training and
inference. It used SS and Beaver triples [14], generated during an offline
phase, for matrix multiplications in the fully connected layers and
represented non-linear activations as GC, with the sigmoid and softmax
activations being approximated. Next, [15] proposed MiniONN, a private
inference framework based on SS and GC. Unlike SecureML [13], which
implemented only fully connected layers, MiniONN provided support for
convolutional ones and implemented them in a privacy-preserving manner using
SS. Additive SS was further used for average pooling layers, while max pooling
and non-linear activations were done by GC. Similarly, the Chameleon [16]
framework also relied on Additive SS for matrix multiplications, and used
either GC or Goldreich-Micali-Wigderson (GMW) circuits [17], depending on
their efficiency, for performing non-linear activations and max pooling. The
Gazelle framework [18] instead used an efficient HE scheme for matrix
multiplications, and switched to GC for activation functions and pooling
layers.

Further work by [19] explored relying only on GC for the computation of ReLU
activations, fully connected and convolutional layers in the DeepSecure
framework. The authors also suggested applying data pre-processing, which
projected the data to a low dimensional space and used both during training
and inference, and network post-processing, which removed neurons whose
activations were below a certain threshold and did not contribute
significantly to the accuracy [19]. Another approach based on GC was the XONN
framework [5], which was able to achieve a constant communication cost with
respect to the depth of the network, making it several orders of magnitude
more efficient than prior work [6, 15, 19]. Since it operated on Binarized
Neural Networks (BNNs) [20], where the weights and activations take only
binary values (either -1 or 1) [5], the costly multiplication operations were
replaced with XNOR gates, which are considered free in GC, and Max Pooling
layers efficiently computed using OR gates. Similarly to DeepSecure [19], pre
and post-processing phases could be done to further improve the runtime. The
SecureNN [21] and the recent Falcon [22] frameworks relied exclusively on
efficient SS protocols for performing not only matrix multiplications, but
also for computing ReLU activations, average and max pooling layers, thus
reducing the runtime and communication cost of private training and inference
considerably.

###  2.3 Trusted Execution Environments

Trusted Execution Environments (TEEs) such as Intel’s SGX offer a promising
avenue for performing private inference due to their isolation from the main
operating system, one can securely perform operations inside them. They could
be used to store an already pre-trained model which is never visible to the
main system and perform private inference. While they offer considerable speed
improvements in comparison to HE and SMPC approaches, they have limited memory
available and are still considerably slower than standard untrusted CPUs and
GPUs. To address these limitations, [23] proposed Slalom, a framework which
efficiently utilises TEEs, where non-linear activations and pooling operations
are executed, and a co-located untrusted hardware which can be used to
securely perform the costly matrix computation of fully connected and
convolutional layers.

###  2.4 Comparison of different approaches

Most of the works from the literature provide performance benchmarks against
four main neural networks evaluated on the MNIST dataset. Both Network A [13]
and Network B [6] used the square activation, however, the former included 3
fully connected layers, while the latter had a Convolutional and two linear
layers. Network C consisted of a single convolutional and two fully connected
layers with ReLUs. Finally, Network D had two sets of convolutions and max
pooling layers, followed by two linear layers and used ReLU throughout. It
should be noted that the XONN framework [5] had only binary activations.

Framework | Model | Parties | Accuracy | Runtime (s) | Comm (MB)
---|---|---|---|---|---
Offline | Online | Total | Offline | Online | Total
SecureML | A | 2PC | $93.1\%$ | 4.7 | 0.18 | 4.88 | - | - | -
MiniONN | 2PC | $97.6\%$ | 0.9 | 0.14 | 1.04 | 3.8 | 12 | 15.8
Gazelle | 2PC | $97.6\%$ | - | 0.03 | 0.03 | - | 0.5 | 0.5
SecureNN | 3PC | $93.4\%$ | - | 0.043 | 0.043 | - | 2.1 | 2.1
Falcon | 3PC | $93.4\%$ | - | 0.011 | 0.011 | - | 0.012 | 0.012
XONN | 2PC | $97.6\%$ | - | 0.13 | 0.13 | - | 4.29 | 4.29
CryptoNets | B | 2PC | $98.95\%$ | - | 297.5 | 297.5 | - | 372.2 | 372.2
MiniONN | 2PC | $98.95\%$ | 0.88 | 0.4 | 1.28 | 3.6 | 44 | 47.6
Gazelle | 2PC | $97.6\%$ | - | 0.03 | 0.03 | - | 0.5 | 0.5
XONN |  | 2PC | $98.64\%$ | - | 0.16 | 0.16 | - | 38.28 | 38.28
DeepSecure | C | 2PC | $98.95\%$ | - | 9.67 | 9.67 | - | 791 | 791
Chameleon | 3PC | $99.0\%$ | 1.25 | 0.99 | 2.24 | 5.4 | 5.1 | 10.5
Gazelle | 2PC | $99.0\%$ | 0.15 | 0.05 | 0.20 | 5.9 | 2.1 | 8.0
SecureNN | 3PC | $99.0\%$ | - | 0.076 | 0.076 | - | 4.05 | 4.05
Falcon | 3PC | $98.77\%$ | - | 0.009 | 0.009 | - | 0.049 | 0.049
XONN | 2PC | $98.64\%$ | - | 0.16 | 0.16 | - | 38.28 | 38.28
MiniONN | D | 2PC | $99.31\%$ | 3.58 | 5.74 | 9.32 | 20.9 | 636.6 | 657.5
Gazelle | 2PC | $99.0\%$ | 0.481 | 0.33 | 0.811 | 47.5 | 22.5 | 70.0
SecureNN | 3PC | $99.15\%$ | - | 0.13 | 0.13 | - | 8.86 | 8.86
Falcon | 3PC | $99.15\%$ | - | 0.042 | 0.042 | - | 0.51 | 0.51
XONN | 2PC | $99.0\%$ | - | 0.15 | 0.15 | - | 32.13 | 32.13
Table 1: Performance benchmarks obtained on the MNIST dataset of different
privacy-preserving frameworks when executing private inference on a single
image (LAN setting).

Table 1 demonstrates that the research in privacy-preserving frameworks has
lead to considerable reduction in communication and improvements in
performance. When examining a deeper model like Network D, Falcon [22] is over
200X faster than MiniONN [15] and requires over 1000X less memory. While
solutions based entirely on HE generally perform worse, a mixed approach like
Gazelle [18] showcases an efficient combination of HE and SMPC. The difference
in the accuracy reported by different frameworks can be attributed to varying
hyper-parameters used during training and the application of specific
techniques, such as batch normalisation in Network A by MiniONN [15].

Several frameworks did not provide benchmarks against these existing
architectures and are not present in Table 1, however, they discussed
improvements with respect to other baselines. CaRENets [7] reported a 5x
memory usage improvement on the MNIST dataset in comparison to CryptoNets,
while the Slalom framework [23] achieved significant performance improvements
on larger models such as VGG16 [24] and ResNet [25] when compared to a
solution where all computations are performed inside the secure enclave [23].
The approach proposed by [26], based on the MORE scheme, was successfully
applied to several deep models in the medical imaging domain.

##  3 Privacy-Preserving Inference in practice

###  3.1 Frameworks

####  3.1.1 PySyft [27]

A prominent private learning library which implements the SPDZ [28] and
SecureNN protocols [21] for private inference. It supports two-party SMPC and
relies on a trusted third party, the crypto provider, for generating
correlated randomness using the Beaver triplets technique [14]. As discussed
by [21], floating point numbers cannot be used directly, which is why PySyft
uses fixed precision encoding [27]. Each party is represented as either a
virtual or a network worker. The former is instantiated on the same device and
simulates network communications, while the latter can be run on the same
device or different ones and communicates over Web Sockets [27]. Both worker
types apply the same serialization and deserialization steps. In this work,
Virtual Workers were used for simplicity as they still allow the number of
communication bytes sent between parties to be measured.

####  3.1.2 CrypTen [29]

A recent private inference library with support for PyTorch which can be used
for both Private Training and Inference. While there is no official white
paper published yet, the presentation given at NeurIPS 2019 outlines that
CrypTen utilises both secret sharing (for matrix multiplications) and garbled
circuits (for evaluating non-linearities) and includes protocols for
converting between these two secret sharing schemes [30]. The communication
between parties is implemented using the Gloo backend, a PyTorch distributed
backend where each party runs the same code in a different process. Similar to
PySyft it also uses a trusted third party for computing Beaver triplets and
operates in an honest-but-curious setting [30].

####  3.1.3 TF-Trusted [31]

A Tensorflow framework which allows models to be executed within an Intel SGX
secure enclave and is based on the Asylo library, a generic framework for
executing computations on secure enclaves [32]. A pre-trained Tensorlow model,
saved using the protobuf format, can be used for private inference. For the
purpose of this benchmark, we have added support for Tensorflow 1.14.0 in
place of 1.13.1 to compare identical TF versions.

####  3.1.4 HE-Transformer

Also known as nGraph-HE2 [33], it is a library for private inference using
Homomorphic Encryption (HE) built using Tensorflow 1.14.0, Intel nGraph’s Deep
Learning model compiler [34] and Microsoft SEAL version 3.3.0 which implements
the CKKS scheme [35]. It operates in a two party setting: a server who is the
model owner and the client who holds data. Additionally, the client also
performs non-linear activations such as ReLU and Max Pooling layers. After
computing a linear or a convolutional layer the server sends the encrypted
data to the client, who decrypts it, applies the function, re-encrypts it and
returns the result [33]. Similarly to Gazelle [18], that process helps manage
the noise growth by refreshing the ciphertext after every layer.

###  3.2 Conducted experiments

####  3.2.1 Model architectures

Three main architectures were evaluated, inspired by the ones reviewed in
section 2. Model X and Y were evaluated on the MNIST dataset, while Model Z
was applied to the Malaria dataset which is more complex and required a deeper
model. Model X consists of three fully connected layers, while Model Y inserts
both a convolution and a pooling operation before two fully connected ones.
Finally, Model Z, based on the network proposed by [5], includes two sets of
convolution and pooling layers followed by two fully connected ones.

####  3.2.2 Experimental setting

Both datasets were normalised to have zero mean and unit standard deviation
prior to using them for training or inference. The Malaria dataset was resized
to 32x32 similar to [5] which enables faster inference time. All experiments
were performed on a machine with a 2.6 GHz 8-Core Intel Xeon E5-2650 v2 and
256 GB RAM. The tf-trusted experiments that require an Intel SGX chip were run
on a 2.80GHz Intel Core i7-7600U CPU and 16GB RAM. Each runtime benchmark was
executed 20 times and the average time was reported in seconds. All three
networks were optimised using Adam [36] with a learning rate of $0.001$. The
MNIST models were trained with a batch size of $128$ for $15$ epochs, while
the Malaria one using batch size $52$ and $20$ epochs. The associated code is
available here 111https://github.com/venetahar/iso-privacy-
smpc222https://github.com/venetahar/iso-privacy-tee-he.

####  3.2.3 Results

Library | Model | Accuracy | Runtime (s) | Comm (MB)
---|---|---|---|---
PyTorch | X | $97.66\%$ | 0.0002 | -
PySyft | $97.66\%$ | 0.5 | 5.71
CrypTen | $97.66\%$ | 0.043 | 1.01
Tensorflow | $97.63\%$ | 0.0003 | -
TF-Trusted | $97.63\%$ | 0.14 | -
HE-Transformer | $97.63\%$ | 15.4 | 1171.94
PyTorch | Y | $98.47\%$ | 0.0003 | -
PySyft | $98.47\%$ | 0.66 | 1.28
CrypTen | $98.47\%$ | 0.035 | 0.35
Tensorflow | $98.05\%$ | 0.0003 | -
TF-Trusted | $98.05\%$ | 0.12 | -
HE-Transformer | $98.05\%$ | 12.6 | 2780.66
PyTorch | Z | $94.85\%$ | 0.0011 | -
PySyft | $94.85\%$ | 8.26 | 38.56
CrypTen | $94.85\%$ | 0.25 | 8.29
Tensorflow | $91.5\%$ | 0.0008 | -
TF-Trusted | $91.5\%$ | 0.14 | -
HE-Transformer | $91.5\%$ | 548.94 | 56703.68
Table 2: Performance benchmarks during inference using a single data instance
on MNIST [3] (Models X and Y) and the Malaria dataset [4] (Model Z).

As it can be observed in Table 2 all frameworks preserve the accuracy of the
plain text model when performing private inference. It can be seen that
CrypTen is significantly faster than PySyft which is clearer in Model Z. This
is an interesting find as CrypTen requires share conversions between Additive
SS and GCs [30] which has been previously reported as inefficient [21]. We
attribute the discrepancy in runtimes to the use of different communication
backends and the serialization and deserialization applied. Additionally, the
results show that CrypTen requires less bandwidth which could be explained by
the use of Garbled Circuits to represent ReLU activations while the PySyft
implementation of ReLU requires more communication between parties for the
additive shares to be exchanged. While there are inherent performance
differences between Tensorflow and PyTorch, it can be seen that TF-Trusted is
able to evaluate larger models such as Model Z much faster. Intuitively this
makes sense as the inference is performed inside the enclave using a plain
text version of the data. While HE-Transformer also preserves the accuracy,
this comes at a much higher computational and memory cost mainly due to the
communication with the client after every fully connected or convolutional
layer where the non-linearity is applied.

##  4 Conclusion

Overall, the experiments conducted demonstrate that it is possible to leverage
PPML methods purely from the traditional data science stack. Our experiments
have produced a unified codebase only diverging at the underlying deep-
learning framework level: PyTorch or TensorFlow, and could go even further by
manipulating ONNX models in the future. Most importantly, model accuracy was
preserved compared to the plaintext version, regardless of the PPML method
employed, demonstrating that the studied frameworks abstract the low-level
implementation constraints, such as fixed-precision arithmetic, in a
satisfying manner.

With regards to run times, Crypten is worth noting for displaying performances
very close to the hardware-based solution of TF-Trusted and as such might
appeal to a broader range of users by lifting the hardware requirement and
memory space constraints imposed by Intel SGX. On the other hand, HE-
Transformer suffers from the genericity of its approach, since a bespoke
protocol would have fared better than the automatic toolchain. Communication
and memory requirements currently make the HE option impractical. As part of
our future iterations on this work, we will thus integrate other secure
inference scenarios evaluating batched predictions, without real-time
constraints. This setting and further manual optimisations might prove a
better fit for HE-based solutions. It will also be interesting to evaluate how
other optimisations on the model side, such as quantisation, impact the
results for each framework.

## References

  * [1] Joseph Redmon, Santosh Divvala, Ross Girshick, and Ali Farhadi.  You only look once: Unified, real-time object detection.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 779–788, 2016.
  * [2] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.  Bert: Pre-training of deep bidirectional transformers for language understanding.  arXiv preprint arXiv:1810.04805, 2018.
  * [3] Yann LeCun, Corinna Cortes, and CJ Burges.  Mnist handwritten digit database.  ATT Labs [Online]. Available: http://yann. lecun. com/exdb/mnist, 2, 2010.
  * [4] Sivaramakrishnan Rajaraman, Sameer K Antani, Mahdieh Poostchi, Kamolrat Silamut, Md A Hossain, Richard J Maude, Stefan Jaeger, and George R Thoma.  Pre-trained convolutional neural networks as feature extractors toward improved malaria parasite detection in thin blood smear images.  PeerJ, 6:e4568, 2018.
  * [5] M Sadegh Riazi, Mohammad Samragh, Hao Chen, Kim Laine, Kristin Lauter, and Farinaz Koushanfar.  Xonn: Xnor-based oblivious deep neural network inference.  In 28th USENIX Security Symposium (USENIX Security 19), pages 1501–1518, 2019.
  * [6] Ran Gilad-Bachrach, Nathan Dowlin, Kim Laine, Kristin Lauter, Michael Naehrig, and John Wernsing.  Cryptonets: Applying neural networks to encrypted data with high throughput and accuracy.  In International Conference on Machine Learning, pages 201–210, 2016.
  * [7] Jin Chao, Ahmad Al Badawi, Balagopal Unnikrishnan, Jie Lin, Chan Fook Mun, James M Brown, J Peter Campbell, Michael Chiang, Jayashree Kalpathy-Cramer, Vijay Ramaseshan Chandrasekhar, et al.  Carenets: compact and resource-efficient cnn for homomorphic inference on encrypted medical images.  arXiv preprint arXiv:1901.10074, 2019.
  * [8] Anamaria Vizitiu, Cosmin Ioan Nitǎ, Andrei Puiu, Constantin Suciu, and Lucian Mihai Itu.  Applying deep neural networks over homomorphic encrypted medical data.  Computational and Mathematical Methods in Medicine, 2020, 2020.
  * [9] Aviad Kipnis and Eliphaz Hibshoosh.  Efficient methods for practical fully homomorphic symmetric-key encrypton, randomization and verification.  IACR Cryptology ePrint Archive, 2012:637, 2012.
  * [10] Andrew Chi-Chih Yao.  How to generate and exchange secrets.  In 27th Annual Symposium on Foundations of Computer Science (sfcs 1986), pages 162–167. IEEE, 1986.
  * [11] Adi Shamir.  How to share a secret.  Communications of the ACM, 22(11):612–613, 1979.
  * [12] Shimon Even, Oded Goldreich, and Abraham Lempel.  A randomized protocol for signing contracts.  Communications of the ACM, 28(6):637–647, 1985.
  * [13] Payman Mohassel and Yupeng Zhang.  Secureml: A system for scalable privacy-preserving machine learning.  In 2017 IEEE Symposium on Security and Privacy (SP), pages 19–38. IEEE, 2017.
  * [14] Donald Beaver.  Efficient multiparty protocols using circuit randomization.  In Annual International Cryptology Conference, pages 420–432. Springer, 1991.
  * [15] Jian Liu, Mika Juuti, Yao Lu, and Nadarajah Asokan.  Oblivious neural network predictions via minionn transformations.  In Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, pages 619–631, 2017.
  * [16] M Sadegh Riazi, Christian Weinert, Oleksandr Tkachenko, Ebrahim M Songhori, Thomas Schneider, and Farinaz Koushanfar.  Chameleon: A hybrid secure computation framework for machine learning applications.  In Proceedings of the 2018 on Asia Conference on Computer and Communications Security, pages 707–721, 2018.
  * [17] Oded Goldreich, S. Micali, and Avi Wigderson.  How to play any mental game.  pages 218–229, 01 1987.
  * [18] Chiraag Juvekar, Vinod Vaikuntanathan, and Anantha Chandrakasan.  Gazelle: A low latency framework for secure neural network inference.  In 27th USENIX Security Symposium (USENIX Security 18), pages 1651–1669, 2018.
  * [19] Bita Darvish Rouhani, M Sadegh Riazi, and Farinaz Koushanfar.  Deepsecure: Scalable provably-secure deep learning.  In Proceedings of the 55th Annual Design Automation Conference, pages 1–6, 2018.
  * [20] Matthieu Courbariaux, Itay Hubara, Daniel Soudry, Ran El-Yaniv, and Yoshua Bengio.  Binarized neural networks: Training deep neural networks with weights and activations constrained to+ 1 or-1.  arXiv preprint arXiv:1602.02830, 2016.
  * [21] Sameer Wagh, Divya Gupta, and Nishanth Chandran.  Securenn: Efficient and private neural network training.  IACR Cryptology ePrint Archive, 2018:442, 2018.
  * [22] Sameer Wagh, Shruti Tople, Fabrice Benhamouda, Eyal Kushilevitz, Prateek Mittal, and Tal Rabin.  Falcon: Honest-majority maliciously secure framework for private deep learning.  arXiv preprint arXiv:2004.02229, 2020.
  * [23] Florian Tramer and Dan Boneh.  Slalom: Fast, verifiable and private execution of neural networks in trusted hardware.  arXiv preprint arXiv:1806.03287, 2018.
  * [24] Karen Simonyan and Andrew Zisserman.  Very deep convolutional networks for large-scale image recognition.  arXiv preprint arXiv:1409.1556, 2014.
  * [25] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.
  * [26] Anamaria Vizitiu, Cosmin Ioan Nitǎ, Andrei Puiu, Constantin Suciu, and Lucian Mihai Itu.  Towards privacy-preserving deep learning based medical imaging applications.  In 2019 IEEE International Symposium on Medical Measurements and Applications (MeMeA), pages 1–6. IEEE, 2019.
  * [27] Theo Ryffel, Andrew Trask, Morten Dahl, Bobby Wagner, Jason Mancuso, Daniel Rueckert, and Jonathan Passerat-Palmbach.  A generic framework for privacy preserving deep learning.  arXiv preprint arXiv:1811.04017, 2018.
  * [28] Ivan Damgård, Valerio Pastro, Nigel Smart, and Sarah Zakarias.  Multiparty computation from somewhat homomorphic encryption.  In Annual Cryptology Conference, pages 643–662. Springer, 2012\.
  * [29] Facebook AI Research Revision.  Crypten.  https://crypten.readthedocs.io/en/latest/, 2019.  Accessed: 2020-03-28.
  * [30] Shubho Sengupta.  Private ai.  https://research.fb.com/neurips-2019-expo-workshops/, 2019.  Accessed: 2020-03-28.
  * [31] Cape Privacy (Formerly Dropout Labs).  Tf trusted.  https://github.com/capeprivacy/tf-trusted, 2019.  Accessed: 2020-03-28.
  * [32] Google).  Asylo.  https://asylo.dev, 2018.  Accessed: 2020-03-28.
  * [33] Fabian Boemer, Anamaria Costache, Rosario Cammarota, and Casimir Wierzynski.  ngraph-he2: A high-throughput framework for neural network inference on encrypted data.  In Proceedings of the 7th ACM Workshop on Encrypted Computing & Applied Homomorphic Cryptography, pages 45–56, 2019.
  * [34] Scott Cyphers, Arjun K Bansal, Anahita Bhiwandiwalla, Jayaram Bobba, Matthew Brookhart, Avijit Chakraborty, Will Constable, Christian Convey, Leona Cook, Omar Kanawi, et al.  Intel ngraph: An intermediate representation, compiler, and executor for deep learning.  arXiv preprint arXiv:1801.08058, 2018.
  * [35] Microsoft SEAL (release 3.3).  https://github.com/Microsoft/SEAL, 2019.  Microsoft Research, Redmond, WA.
  * [36] Diederik P. Kingma and Jimmy Ba.  Adam: A method for stochastic optimization, 2014.


#  $U$-Statistics Based Jackknife Empirical Likelihood Tests for the
Generalized Lorenz Curves

Suthakaran Ratnasingam111Corresponding author. Email:
<EMAIL_ADDRESS>Anton Butenko
Department of Mathematics
California State University, San Bernardino, San Bernardino, CA 92407, USA

###### Abstract

A Lorenz curve is a graphical representation of the distribution of income or
wealth within a population. The generalized Lorenz curve can be created by
scaling the values on the vertical axis of a Lorenz curve by the average
output of the distribution. In this paper, we propose two non-parametric
methods for testing the equality of two generalized Lorenz curves. Both
methods are based on empirical likelihood and utilize a $U$-statistic. We
derive the limiting distribution of the likelihood ratio, which is shown to
follow a chi-squared distribution with one degree of freedom. We performed
simulations to evaluate how well the proposed methods perform compared to an
existing method, by examining their Type I error rates and power across
different sample sizes and distribution assumptions. Our results show that the
proposed methods exhibit superior performance in finite samples, particularly
in small sample sizes, and are robust across various scenarios. Finally, we
use real-world data to illustrate the methods of testing two generalized
Lorenz curves.

##  1 Introduction

A Lorenz curve is a visual representation of an income or wealth distribution
within a population. It is named after American economist Max Lorenz (Lorenz
(1905)) who developed it in 1905. The Lorenz curve is constructed by plotting
the cumulative percentage of the population on the $x-$axis against the
cumulative percentage of the variable (such as income or wealth) on the
$y-$axis. The resulting curve represents the distribution of the variable in
the population. A Lorenz curve that is close to the line of equality (which is
a straight line that represents perfect equality) indicates that the
distribution of the variable is relatively equal across the population. On the
other hand, a Lorenz curve that is farther away from the line of equality
indicates a higher degree of inequality in the distribution of the variable.
Following Gastwirth (1971), a general definition of the Lorenz curve is given
as

$\displaystyle\xi(t)=\dfrac{1}{\mu}\int_{0}^{\psi_{t}}xdF(x),\quad t\in[0,1]$
(1)

where $\mu$ denotes the mean of $F$, and $\psi_{t}=F^{-1}(t)=\inf\\{x:F(x)\geq
t\\}$ is the $t-$th quantile of $F$. For a fixed $t\in[0,1]$, the Lorenz
ordinate $\xi(t)$ is the proportion of the cumulative income of the lowest
$t$-th quantile of households. The generalized Lorenz curve can be constructed
from a Lorenz curve by scaling the values on the vertical axis by the average
output of the distribution. Similarly, the generalized Lorenz curve is defined
by

$\displaystyle\eta(t)=\int_{0}^{\psi_{t}}xdF(x),\quad t\in[0,1]$ (2)

where $\psi_{t}$ is the $t-$th quantile of $F$ as defined above. For a fixed
$t\in[0,1]$, the generalized Lorenz ordinate $\eta(t)$ is the average income
of the lowest $t$-th quantile of households. While Lorenz curves are
frequently used in economics to represent financial inequality, they also can
be used in other fields of study to visualize the inequality of the
distribution within any system. For example, the Lorenz curve has been used by
several researchers to analyze physician distributions. Chang Halfon (1997)
examined variations in the distribution of pediatricians among the states
between 1982 and 1992 using Lorenz curves and Gini indices. Kobayashi Takaki
(1992) used the Lorenz curve and the Gini coefficient to study the disparity
in physician distribution in Japan.

The empirical likelihood (EL) method, introduced by Owen (1988), is a powerful
nonparametric approach that offers numerous advantages over traditional
methods. Unlike traditional methods, EL does not require strong assumptions to
utilize the likelihood ratio approach, yet still preserves many of its
desirable features, including Wilk’s theorem, asymmetric confidence intervals,
and better coverage for small sample sizes. However, the EL method has some
computational difficulties when using nonlinear statistics, as demonstrated by
Jing . (2009), and when constraints’ solutions do not exist, as shown by Chen
. (2008). Specifically, Jing . (2009) showed that the EL-based approach loses
its appeal when using nonlinear $U$-statistics with a degree of $m\geq{2}$
because of the computational difficulty of solving a system of nonlinear
equations simultaneously using Lagrange multipliers. To address this issue,
Jing . (2009) proposed the jackknife empirical likelihood (JEL) approach. The
JEL method turns the statistic of interest into a sample mean based on
jackknife pseudo-values (Quenouille (1956)), which are asymptotically
independent under mild conditions (Shi (1984)). Then, Owen’s EL method can be
applied consecutively, resulting in a simpler system of equations. On the
other hand, Chen . (2008) pointed out that under certain conditions, it can be
challenging to determine the parameter region over which the likelihood ratio
function is well-defined. This makes it difficult to identify the maximum
likelihood ratio or find a proper initial value. To tackle this challenge,
Chen proposed the adjusted empirical likelihood (AEL) approach, which extends
the convex hull to include the origin by adding a pseudo-value. With this
adjustment, the empirical likelihood is well-defined for all parameter values,
making it easier to find the maximum.

Multiple studies have been conducted on EL for the Lorenz curve by various
researchers. For instance, Belinga-Hall (2007) and Yang . (2012) developed
plug-in empirical likelihood-based inferences to construct confidence
intervals for the generalized Lorenz curve. Most recently, Ratnasingam .
(2023) developed three nonparametric EL-based methods to construct confidence
intervals for the generalized Lorenz curve using adjusted empirical likelihood
(AEL), transformed empirical likelihood (TEL), and transformed adjusted
empirical likelihood (TAEL). Moreover, several studies have focused on
comparing two Lorenz curves. For example, Arora Jain (2006) investigated the
generalized Lorenz dominance and proposed tests for the equality of two
generalized Lorenz curves over a specified interval. Li Wei (2018) noted that
normal approximation-based methods may have poor performance, especially for
the skewed income data, or the limiting distributions are nonstandard and
bootstrap calibrations are needed hence more effective inferences for Lorenz
curves are desirable. All of these tests were parametric and they involve
making assumptions about the underlying distribution of the data. Xu (1997)
proposed an asymptotically distribution-free statistical (ADF) test to
evaluate the equality of two generalized Lorenz curves and showed that the
test statistic follows the weighted sum of $\chi^{2}$ with different degrees
of freedom.

As far as we know, there have been no previous studies examining the testing
of the equality of two generalized Lorenz curves using EL methods. Therefore,
this is the first study to investigate the equality of two generalized Lorenz
curves using a nonparametric approach. We propose two novel nonparametric
methods that employ a $U$-statistic based on the jackknife empirical
likelihood (JEL) method and its extension to the adjusted jackknife empirical
likelihood (AJEL). These methods combine two EL-based approaches, namely the
JEL and AEL, previously discussed in Jing . (2009) and Chen . (2008),
respectively.

The remainder of the paper is structured as follows. In Section 2, we present
two novel nonparametric techniques for testing the similarity between two
generalized Lorenz curves. Section 4 describes the simulation studies carried
out to evaluate the effectiveness of the proposed methods in different
scenarios and to compare their performance with an existing method. In Section
5, we demonstrate the application of these methods to real datasets. Our
findings are discussed in Section 6. All proofs are provided in the appendix.

##  2 Main Results

In this section we develop two new testing procedures using jackknife EL
methods. Let $X_{1},X_{2},\cdots,X_{n_{1}}$ and $Y_{1},Y_{2},\cdots,Y_{n_{2}}$
be two random samples from two independent populations. The generalized Lorenz
curve for these two samples are

$\displaystyle\eta_{1}(t)=\int_{0}^{\psi_{t}}xdF(x),\quad t\in[0,1]$ (3)

and

$\displaystyle\eta_{2}(t)=\int_{0}^{\psi_{t}}ydF(y),\quad t\in[0,1]$ (4)

where $\psi_{t}=F^{-1}(t)=\inf\\{x:F(x)\geq t\\}$ is the $t-$th quantile of
$F$. We are interested in testing the following hypotheses.

$\displaystyle H_{0}:\eta_{1}(t)=\eta_{2}(t)\quad vs\quad
H_{1}:\eta_{1}(t)\neq\eta_{2}(t)$ (5)

From the definition of the generalized Lorenz curve, it can be clearly seen
that

$\displaystyle E[X\,I(X\leq\psi_{t})]-\eta_{1}(t)=0.$

and

$\displaystyle E[Y\,I(Y\leq\psi_{t})]-\eta_{2}(t)=0.$

As a result, the generalized Lorenz ordinates $\eta_{1}(t)$ and $\eta_{2}(t)$
are the means of the random variable $X$ and $Y$ truncated at $\psi_{t}$
respectively. Let’s consider the kernel function,

$\displaystyle
h(X,Y)=X\,I(X\leq\psi_{t})I(Y\leq\psi_{t})-Y\,I(X\leq\psi_{t})I(Y\leq\psi_{t})$
(6)

We can easily show that $\theta(t)\equiv
E\big{[}h(X_{i},Y_{j}\big{]}=\big{(}\eta_{1}(t)-\eta_{2}(t)\big{)}P\big{(}X\leq\psi_{t}\big{)}P\big{(}Y\leq\psi_{t}\big{)}$.
Thus, we are interested in testing

$\displaystyle H_{0}:\theta(t)=0\quad vs\quad H_{1}:\theta(t)\neq 0.$ (7)

Now consider, the two-sample $U$-statistics of degree (1,1) with the kernel
$h$ is given by,

$\displaystyle U_{n_{1},n_{2}}$
$\displaystyle=\dfrac{1}{n_{1}}\dfrac{1}{n_{2}}\sum_{1\leq i\leq
n_{1}}\sum_{1\leq j\leq n_{2}}h(X_{i},Y_{j})$ (8)
$\displaystyle=\dfrac{1}{n_{1}}\dfrac{1}{n_{2}}\sum_{1\leq i\leq
n_{1}}\sum_{1\leq j\leq
n_{2}}X_{i}\,I(X_{i}\leq\psi_{t})I(Y_{j}\leq\psi_{t})-Y_{j}\,I(X_{i}\leq\psi_{t})I(Y_{j}\leq\psi_{t})$

Let $n=n_{1}+n_{2}$. We can write the $U$-statistics

$\displaystyle
U_{n_{1},n_{2}}(X_{1},\dots,X_{n_{1}},Y_{1},\dots,Y_{n_{2}})=U_{n}(Z_{1},Z_{2},\dots,Z_{n})$
(9)

where

$Z_{k}=\begin{cases}X_{k}&k=1,2,\dots,n_{1}\\\
Y_{k-n_{1}}&k=n_{1}+1,\dots,n\end{cases}$

We define the corresponding jackknife pseudo-values by

$\displaystyle\widehat{V}_{k}=nU_{n}-(n-1)U_{n-1}^{-k},\quad k=1,2,\cdots n,$
(10)

where
$U_{n-1}^{-k}=U_{n}\big{(}Z_{1},Z_{2},\dots,Z_{k-1},Z_{k+1},\dots,Z_{n}\big{)}$.
Further, the jackknife estimator of $\theta$ is
$n^{-1}\sum_{i=1}^{n}\widehat{V}_{i}$. In particular, under mild conditions,
the $\widehat{V}_{k}$’s are asymptotically independent. For more details,
readers are referred to Shi (1984). Thus, we can use the EL approach to the
$\widehat{V}_{k}$’s. It should be noted that $\widehat{V}_{k}(t)$ is the
function of $t$ and can be calculated at a fixed value $t_{0}$ such that
$t_{0}\in[0,1]$. For the simplicity of notations, we use $\widehat{V}_{k}$
instead of $\widehat{V}_{k}(t)$. The JEL for $\theta(t)$ is defined as
follows:

$\displaystyle
L(\theta(t))=\sup_{\mathbf{p}}\bigg{\\{}\prod_{k=1}^{n}p_{k}:\sum_{k=1}^{n}p_{k}=1,\sum_{k=1}^{n}p_{k}\big{(}\widehat{V}_{k}-\mathbf{E}\widehat{V}_{k}\big{)}=0\bigg{\\}},$
(11)

where $\mathbf{p}=(p_{1},p_{2},\dots,p_{n})$ is a probability vector
satisfying $\sum_{k=1}^{n}p_{k}=1$ and $p\geq 0$ for all $k$, and
$\mathbf{E}\widehat{V}_{k}$ can be determined using the equation (14) in Jing
. (2017). Note that $\prod_{k=1}^{n}p_{k}$, subject to
$\sum_{k=1}^{n}p_{k}=1$, attains its maximum $n^{-n}$ at $p_{k}=n^{-1}$. Thus,
the JEL ratio for $\theta(t)$ is given as

$\displaystyle\mathcal{R}(\theta(t))=\sup\bigg{\\{}\prod_{k=1}^{n}np_{k}:\sum_{k=1}^{n}p_{k}=1,\sum_{k=1}^{n}p_{k}\big{(}\widehat{V}_{k}-\mathbf{E}\widehat{V}_{k}\big{)}=0\bigg{\\}}$
(12)

Further, under null hypothesis $H_{0}:\theta(t)=0$, the JEL ratio becomes,

$\displaystyle\mathcal{R}(0)=\sup\bigg{\\{}\prod_{k=1}^{n}np_{k}:\sum_{k=1}^{n}p_{k}=1,\sum_{k=1}^{n}p_{k}\widehat{V}_{k}=0\bigg{\\}}.$
(13)

Using the Lagrange multiplier method, we have

$\displaystyle
p_{k}=\dfrac{1}{n}\bigg{\\{}1+\lambda\widehat{V}_{k}\bigg{\\}}^{-1},\quad
k=1,\dots,n.$

where $\lambda$ is the solution to

$\displaystyle\dfrac{1}{n}\sum_{k=1}^{n}\dfrac{\widehat{V}_{k}}{1+\lambda\widehat{V}_{k}}=0.$

Hence, the profile jackknife empirical log-likelihood ratio for $\theta(t)$
becomes

$\displaystyle\ell(\theta(t))=-2\log\mathcal{R}(\theta(t))=2\sum_{k=1}^{n}\log\big{\\{}1+\lambda\widehat{V}_{k}\big{\\}}.$
(14)

Let
$h_{1,0}(x)=\mathbf{E}h(x,Y_{1}),~{}~{}\sigma_{1,0}^{2}=Var\big{(}h_{1,0}(X_{1})\big{)},~{}~{}h_{0,1}(y)=\mathbf{E}h(X_{1},y),$
and $\sigma_{0,1}^{2}=Var\big{(}h_{0,1}(Y_{1})\big{)}$. We have the following
theorem for the JEL.

######  Theorem 2.1.

Assume that

  1. 1.

$E(X^{2})<\infty$, and $E(Y^{2})<\infty$

  2. 2.

$\mathbf{E}h^{2}(X_{1},Y_{1})<\infty$, $\sigma_{1,0}^{2}>0$, and
$\sigma_{0,1}^{2}>0$

  3. 3.

$n_{1}/n_{2}\longrightarrow r$, where $0<r<\infty$

For any given $t=t_{0}\in(0,1)$, the limiting distribution of
$\ell(\theta(t_{0}))$ defined by (14) is a chi-square distribution with one
degree of freedom,

$\displaystyle\ell(\theta(t_{0}))\longrightarrow\chi^{2}_{1},\quad\text{as}\,\,\min(n_{1},n_{2})\longrightarrow\infty.$
(15)

###### Proof.

Proof of Theorem 2.1 is given in Appendix. ∎

Further, Chen . (2008) proposed the AEL method by adding a pseudo-observation
to the data set. This method bypasses the convex hull constraint and ensures a
solution at any parameter point. By adopting Chen . (2008)’s idea, we extend
the proposed JEL method by employing the adjusted jackknife empirical
likelihood (AJEL) to examine the equality of two Lorenz curves. The AJEL for
$\theta(t)$ is defined as follows:

$\displaystyle
L^{\textnormal{Adj}}(\theta(t))=\sup_{\mathbf{p}}\bigg{\\{}\prod_{k=1}^{n+1}p^{\textnormal{Adj}}_{k}:\sum_{k=1}^{n+1}p^{\textnormal{Adj}}_{k}=1,\sum_{k=1}^{n+1}p^{\textnormal{Adj}}_{k}g^{\textnormal{Adj}}_{k}(t)=0\bigg{\\}},$
(16)

where
$g^{\textnormal{Adj}}_{k}(t)=\widehat{V}_{k}-\mathbf{E}\widehat{V}_{k},\,\,k=1,\dots,n$,
and
$g^{\textnormal{Adj}}_{n+1}=-a_{n}\bar{g}_{n}(t)=-\frac{a_{n}}{n}\sum_{i=1}^{n}g_{i}(t)$.
As recommended by Chen . (2008), $a_{n}=\max\\{1,\log(n)/2\\}$. Using the
Lagrange multiplier method, we can determine $L^{\textnormal{Adj}}(\theta(t))$
as follows.

$\displaystyle
p^{\textnormal{Adj}}_{k}=\dfrac{1}{n+1}\bigg{\\{}1+\lambda^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t)\bigg{\\}}^{-1},\quad
k=1,\dots,n+1.$

where $\lambda^{\textnormal{Adj}}$ is the solution to

$\displaystyle\dfrac{1}{n+1}\sum_{k=1}^{n+1}\dfrac{g^{\textnormal{Adj}}_{k}(t)}{1+\lambda^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t)}=0.$

Note that $\prod_{k=1}^{n+1}p^{\textnormal{Adj}}_{k}$, subject to
$\sum_{k=1}^{n+1}p^{\textnormal{Adj}}_{k}=1$, attains its maximum
$(n+1)^{-n-1}$ at $p_{k}=(n+1)^{-1}$. Thus, the AJEL ratio for $\theta(t)$ is
given as

$\displaystyle\mathcal{R}^{\textnormal{Adj}}(\theta(t))=\prod_{k=1}^{n+1}(n+1)p^{\textnormal{Adj}}_{k}=\prod_{k=1}^{n+1}\big{\\{}1+\lambda^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t)\big{\\}}^{-1}$
(17)

Hence, the profile-adjusted jackknife empirical log-likelihood ratio for
$\theta(t)$ is

$\displaystyle\ell^{\textnormal{Adj}}(\theta(t))=-2\log\mathcal{R}^{\textnormal{Adj}}(\theta(t))=2\sum_{k=1}^{n+1}\log\big{\\{}1+\lambda^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t)\big{\\}}.$
(18)

######  Theorem 2.2.

Under the same conditions of Theorem 2.1 and for any given $t=t_{0}\in(0,1)$,
the limiting distribution of $\ell^{\textnormal{Adj}}(\theta(t_{0}))$ defined
by (18) is a chi-square distribution with one degree of freedom,

$\displaystyle\ell^{\textnormal{Adj}}(\theta(t_{0}))\longrightarrow\chi^{2}_{1},\quad\text{as}\,\,n\longrightarrow\infty.$
(19)

###### Proof.

Proof of Theorem 2.2 is given in Appendix. ∎

##  3 Simulation Study

In this section, we conduct a simulation study to evaluate the performance of
the proposed testing methods. In our simulation analysis, we take into account
Chi-Square, Exponential, and Half-Normal distributions as the overall
distribution function $F(x)$ because the majority of income distributions are
positively skewed. Under the null hypothesis, we examine the distributions of
$\chi^{2}_{4},~{}Exp(4)$, and $HN(1)$ using different sample sizes
$(n_{1},n_{2})$ such as $(20,30),(40,50),(75,75),$ and $(100,100)$. We first
assess the Type I error probabilities of the ADF, JEL and AJEL methods with a
nominal level of $\alpha=0.05$. Tables 1-3 provide a summary of the findings,
including the probabilities of Type I errors (TE) and their corresponding
standard errors (SE), whereas Figure 1 depicts the outcomes graphically. The
JEL method appears to perform slightly better or similar to the AJEL method.
For instance, when using the $\chi^{2}_{4}$ distribution with sample sizes of
(20, 30) at $t=0.1$, the Type I error probability for the ADF method is 0.007
with a standard error of 0.0027, the JEL method is 0.027 with a standard error
of 0.0051, and the AJEL method has a probability of 0.075 and a standard error
of 0.0083. When using the $\chi^{2}_{4}$ test with sample sizes of 20 and 30,
the AJEL method produces a Type I error rate that is slightly higher than the
expected level. The ADF method comes next, with the JEL method following.
However, when testing for the $Exp(1)$ distribution, the ADF method results in
a Type I error rate that is much lower than the expected level, and the test
becomes more conservative for $t>0.2$. When using the $HN(1)$ distribution,
the JEL method performs the best among the three methods, while the ADF method
performs the worst for all sample sizes. The Type I error probabilities are
slightly above the nominal level for small sample sizes, but improve for
larger sample sizes and remain within an acceptable range.

Table 1: Type I error (TE) and standard error (SE) comparison of ADF, JEL, and AJEL tests with nominal level $\alpha=0.05$ when $X,Y\sim\chi_{4}^{2}$ |  | ADF | JEL | AJEL
---|---|---|---|---
$(n_{1},n_{2})$ | $t$ | TE | SE | TE | SE | TE | SE
(20, 30) | 0.1 | 0.007 | 0.0027 | 0.027 | 0.0051 | 0.075 | 0.0083
| 0.2 | 0.011 | 0.0033 | 0.019 | 0.0043 | 0.058 | 0.0074
| 0.3 | 0.027 | 0.0051 | 0.018 | 0.0042 | 0.062 | 0.0076
| 0.4 | 0.037 | 0.0060 | 0.017 | 0.0041 | 0.065 | 0.0078
| 0.5 | 0.069 | 0.0080 | 0.017 | 0.0041 | 0.065 | 0.0078
| 0.6 | 0.062 | 0.0076 | 0.059 | 0.0075 | 0.086 | 0.0089
| 0.7 | 0.069 | 0.0080 | 0.052 | 0.0070 | 0.070 | 0.0081
| 0.8 | 0.071 | 0.0081 | 0.046 | 0.0066 | 0.065 | 0.0078
| 0.9 | 0.070 | 0.0081 | 0.038 | 0.0060 | 0.052 | 0.0075
(40,50) | 0.1 | 0.005 | 0.0023 | 0.016 | 0.0040 | 0.055 | 0.0072
| 0.2 | 0.009 | 0.0030 | 0.019 | 0.0042 | 0.039 | 0.0061
| 0.3 | 0.014 | 0.0037 | 0.010 | 0.0031 | 0.044 | 0.0065
| 0.4 | 0.027 | 0.0051 | 0.013 | 0.0036 | 0.054 | 0.0071
| 0.5 | 0.047 | 0.0067 | 0.022 | 0.0046 | 0.061 | 0.0076
| 0.6 | 0.044 | 0.0065 | 0.049 | 0.0068 | 0.072 | 0.0082
| 0.7 | 0.051 | 0.0070 | 0.050 | 0.0069 | 0.065 | 0.0078
| 0.8 | 0.059 | 0.0075 | 0.042 | 0.0063 | 0.065 | 0.0078
| 0.9 | 0.060 | 0.0075 | 0.043 | 0.0063 | 0.061 | 0.0076
(75,75) | 0.1 | 0.001 | 0.0012 | 0.010 | 0.0029 | 0.012 | 0.0034
| 0.2 | 0.002 | 0.0014 | 0.011 | 0.0033 | 0.013 | 0.0039
| 0.3 | 0.015 | 0.0038 | 0.019 | 0.0042 | 0.016 | 0.0040
| 0.4 | 0.016 | 0.0040 | 0.012 | 0.0034 | 0.018 | 0.0048
| 0.5 | 0.022 | 0.0046 | 0.023 | 0.0047 | 0.024 | 0.0051
| 0.6 | 0.024 | 0.0048 | 0.037 | 0.0060 | 0.046 | 0.0073
| 0.7 | 0.025 | 0.0049 | 0.031 | 0.0056 | 0.031 | 0.0056
| 0.8 | 0.028 | 0.0052 | 0.030 | 0.0054 | 0.035 | 0.0071
| 0.9 | 0.036 | 0.0059 | 0.032 | 0.0056 | 0.031 | 0.0056
(100, 100) | 0.1 | 0.002 | 0.0016 | 0.011 | 0.0030 | 0.038 | 0.0060
| 0.2 | 0.006 | 0.0024 | 0.012 | 0.0034 | 0.040 | 0.0062
| 0.3 | 0.014 | 0.0037 | 0.020 | 0.0044 | 0.048 | 0.0068
| 0.4 | 0.024 | 0.0048 | 0.015 | 0.0038 | 0.046 | 0.0066
| 0.5 | 0.033 | 0.0056 | 0.010 | 0.0031 | 0.044 | 0.0065
| 0.6 | 0.036 | 0.0059 | 0.031 | 0.0055 | 0.046 | 0.0066
| 0.7 | 0.038 | 0.0060 | 0.035 | 0.0058 | 0.046 | 0.0066
| 0.8 | 0.047 | 0.0067 | 0.035 | 0.0058 | 0.046 | 0.0066
| 0.9 | 0.045 | 0.0066 | 0.033 | 0.0057 | 0.045 | 0.0065

Table 2: Type I error and standard error comparison of ADF, JEL, and AJEL tests with nominal level $\alpha=0.05$ when $X,Y\sim Exp(4)$ |  | ADF | JEL | AJEL
---|---|---|---|---
$(n_{1},n_{2})$ | $t$ | TE | SE | TE | SE | TE | SE
(20,30) | 0.1 | 0.037 | 0.0060 | 0.081 | 0.0086 | 0.091 | 0.0091
| 0.2 | 0.022 | 0.0046 | 0.047 | 0.0067 | 0.060 | 0.0075
| 0.3 | 0.010 | 0.0031 | 0.074 | 0.0083 | 0.075 | 0.0083
| 0.4 | 0.006 | 0.0024 | 0.061 | 0.0076 | 0.074 | 0.0083
| 0.5 | 0.006 | 0.0024 | 0.060 | 0.0075 | 0.061 | 0.0076
| 0.6 | 0.001 | 0.0010 | 0.096 | 0.0093 | 0.101 | 0.0095
| 0.7 | 0.001 | 0.0010 | 0.088 | 0.0090 | 0.087 | 0.0089
| 0.8 | 0.001 | 0.0010 | 0.065 | 0.0078 | 0.075 | 0.0083
| 0.9 | 0.000 | 0.0000 | 0.062 | 0.0076 | 0.064 | 0.0078
(40,50) | 0.1 | 0.024 | 0.0049 | 0.060 | 0.0075 | 0.076 | 0.0084
| 0.2 | 0.010 | 0.0031 | 0.043 | 0.0064 | 0.052 | 0.0070
| 0.3 | 0.000 | 0.0000 | 0.048 | 0.0068 | 0.049 | 0.0068
| 0.4 | 0.000 | 0.0000 | 0.045 | 0.0066 | 0.052 | 0.0070
| 0.5 | 0.000 | 0.0000 | 0.057 | 0.0073 | 0.056 | 0.0073
| 0.6 | 0.000 | 0.0000 | 0.068 | 0.0080 | 0.070 | 0.0081
| 0.7 | 0.000 | 0.0000 | 0.065 | 0.0078 | 0.064 | 0.0077
| 0.8 | 0.000 | 0.0000 | 0.067 | 0.0079 | 0.072 | 0.0082
| 0.9 | 0.000 | 0.0000 | 0.059 | 0.0074 | 0.061 | 0.0075
(75,75) | 0.1 | 0.018 | 0.0043 | 0.062 | 0.0076 | 0.071 | 0.0081
| 0.2 | 0.005 | 0.0022 | 0.050 | 0.0069 | 0.055 | 0.0072
| 0.3 | 0.000 | 0.0000 | 0.049 | 0.0058 | 0.050 | 0.0059
| 0.4 | 0.000 | 0.0000 | 0.051 | 0.0070 | 0.056 | 0.0073
| 0.5 | 0.000 | 0.0000 | 0.056 | 0.0074 | 0.060 | 0.0076
| 0.6 | 0.000 | 0.0000 | 0.069 | 0.0080 | 0.069 | 0.0080
| 0.7 | 0.000 | 0.0000 | 0.052 | 0.0072 | 0.052 | 0.0072
| 0.8 | 0.000 | 0.0000 | 0.054 | 0.0071 | 0.056 | 0.0073
| 0.9 | 0.000 | 0.0000 | 0.052 | 0.0072 | 0.055 | 0.0074
(100, 100) | 0.1 | 0.012 | 0.0035 | 0.062 | 0.0076 | 0.071 | 0.0081
| 0.2 | 0.006 | 0.0024 | 0.049 | 0.0068 | 0.055 | 0.0072
| 0.3 | 0.001 | 0.0010 | 0.063 | 0.0077 | 0.061 | 0.0076
| 0.4 | 0.001 | 0.0010 | 0.054 | 0.0071 | 0.059 | 0.0075
| 0.5 | 0.000 | 0.0000 | 0.054 | 0.0071 | 0.051 | 0.0070
| 0.6 | 0.000 | 0.0000 | 0.051 | 0.0070 | 0.051 | 0.0070
| 0.7 | 0.000 | 0.0000 | 0.057 | 0.0073 | 0.057 | 0.0073
| 0.8 | 0.000 | 0.0000 | 0.057 | 0.0073 | 0.057 | 0.0073
| 0.9 | 0.000 | 0.0000 | 0.052 | 0.0072 | 0.052 | 0.0072

Table 3: Type I error and standard error comparison of ADF, JEL, and AJEL tests with nominal level $\alpha=0.05$ when $X,Y\sim HN(1)$ |  | ADF | JEL | AJEL
---|---|---|---|---
$(n_{1},n_{2})$ | $t$ | TE | SE | TE | SE | TE | SE
(20,30) | 0.1 | 0.068 | 0.0080 | 0.074 | 0.0083 | 0.094 | 0.0092
| 0.2 | 0.076 | 0.0084 | 0.063 | 0.0077 | 0.065 | 0.0078
| 0.3 | 0.084 | 0.0088 | 0.050 | 0.0069 | 0.047 | 0.0067
| 0.4 | 0.087 | 0.0089 | 0.051 | 0.0070 | 0.052 | 0.0070
| 0.5 | 0.129 | 0.0106 | 0.050 | 0.0069 | 0.056 | 0.0073
| 0.6 | 0.115 | 0.0101 | 0.077 | 0.0084 | 0.090 | 0.0090
| 0.7 | 0.099 | 0.0094 | 0.063 | 0.0077 | 0.069 | 0.0080
| 0.8 | 0.108 | 0.0098 | 0.062 | 0.0076 | 0.078 | 0.0085
| 0.9 | 0.103 | 0.0096 | 0.041 | 0.0063 | 0.071 | 0.0081
(40,50) | 0.1 | 0.062 | 0.0076 | 0.056 | 0.0073 | 0.066 | 0.0079
| 0.2 | 0.066 | 0.0079 | 0.037 | 0.0060 | 0.042 | 0.0063
| 0.3 | 0.068 | 0.0080 | 0.049 | 0.0068 | 0.054 | 0.0071
| 0.4 | 0.081 | 0.0086 | 0.065 | 0.0078 | 0.058 | 0.0074
| 0.5 | 0.101 | 0.0095 | 0.049 | 0.0068 | 0.059 | 0.0075
| 0.6 | 0.094 | 0.0092 | 0.080 | 0.0086 | 0.087 | 0.0089
| 0.7 | 0.096 | 0.0093 | 0.066 | 0.0079 | 0.064 | 0.0077
| 0.8 | 0.108 | 0.0098 | 0.053 | 0.0071 | 0.070 | 0.0081
| 0.9 | 0.098 | 0.0094 | 0.034 | 0.0057 | 0.055 | 0.0072
(75,75) | 0.1 | 0.038 | 0.0061 | 0.051 | 0.0070 | 0.053 | 0.0071
| 0.2 | 0.045 | 0.0066 | 0.055 | 0.0072 | 0.054 | 0.0071
| 0.3 | 0.059 | 0.0075 | 0.051 | 0.0058 | 0.058 | 0.0063
| 0.4 | 0.058 | 0.0074 | 0.053 | 0.0071 | 0.053 | 0.0071
| 0.5 | 0.077 | 0.0084 | 0.044 | 0.0048 | 0.048 | 0.0052
| 0.6 | 0.079 | 0.0085 | 0.057 | 0.0073 | 0.065 | 0.0078
| 0.7 | 0.081 | 0.0086 | 0.045 | 0.0050 | 0.048 | 0.0052
| 0.8 | 0.091 | 0.0091 | 0.055 | 0.0072 | 0.061 | 0.0076
| 0.9 | 0.098 | 0.0094 | 0.036 | 0.0059 | 0.062 | 0.0076
(100, 100) | 0.1 | 0.049 | 0.0069 | 0.046 | 0.0066 | 0.047 | 0.0067
| 0.2 | 0.055 | 0.0072 | 0.071 | 0.0081 | 0.064 | 0.0077
| 0.3 | 0.060 | 0.0075 | 0.054 | 0.0071 | 0.056 | 0.0073
| 0.4 | 0.067 | 0.0079 | 0.060 | 0.0075 | 0.059 | 0.0075
| 0.5 | 0.072 | 0.0082 | 0.048 | 0.0068 | 0.053 | 0.0071
| 0.6 | 0.078 | 0.0085 | 0.065 | 0.0078 | 0.071 | 0.0081
| 0.7 | 0.082 | 0.0087 | 0.066 | 0.0079 | 0.066 | 0.0079
| 0.8 | 0.087 | 0.0089 | 0.063 | 0.0077 | 0.072 | 0.0082
| 0.9 | 0.086 | 0.0089 | 0.039 | 0.0061 | 0.057 | 0.0073
Figure 1: Type I error comparison for ADF, JEL, and AJEL methods for different
distributions, sample sizes, and values of $t$

Next, we conduct a power analysis for the ADF, JEL, and AJEL methods. Figure 2
displays the generalized Lorenz curves for Chi-Square, Exponential, and Half-
Normal distribution under two sets of parameters. The difference between the
generalized Lorenz curves of $\chi^{2}(4)$ and $\chi^{2}(5.5)$ increases as
$t$ changes from 0 to 0.5, then the difference decreases as $t$ changes from
0.5 to 1. The difference between the generalized Lorenz curves for $Exp(2)$
and $Exp(4)$ increases significantly as $t$ changes from 0 to 1, while the
difference between the generalized Lorenz curves for $HN(1)$ and $HN(1.5)$
also increases but not as significant as in the case of the Exponential
distributions. Tables 4-6 present a summary of the outcomes, depicting the
power and standard errors, whereas Figure 3 illustrates the results
graphically. As expected, the power for Chi-Square distributions tends to
increase as the value of $t$ ranges from 0.1 to 0.5, followed by a slight drop
as $t$ ranges from 0.5 to 0.9. The AJEL method exhibits better power among the
three methods, while the ADF method is the weakest. Moreover, for Exponential
distributions, the power of the JEL and AJEL methods tends to increase as $t$
ranges from 0 to 0.9, while the power of the ADF method tends to decrease as
the value of $t$ changes from 0 to 0.9. When considering the Half-Normal
distributions, all three methods show a similar pattern, with the ADF method
being the most effective when $t\leq 0.5$, and the JEL and AJEL methods being
superior when $t>0.5$. The increase in power is more pronounced for
Exponential distributions than for Half-Normal distributions in the case of
the JEL and AJEL methods. In all three cases, the AJEL method outperforms the
JEL method slightly.

Figure 2: Generalized Lorenz Curves for Chi-Square, Exponential, and Half-Normal Distributions Table 4: Power comparison of ADF, JEL, and AJEL tests with nominal level $\alpha=0.05$ when $X\sim\chi^{2}_{4}$ and $Y\sim\chi^{2}_{5.5}$ |  | ADF | JEL | AJEL
---|---|---|---|---
$(n_{1},n_{2})$ | $t$ | Power | SE | Power | SE | Power | SE
(20,30) | 0.1 | 0.128 | 0.0106 | 0.357 | 0.0152 | 0.372 | 0.0153
| 0.2 | 0.190 | 0.0124 | 0.400 | 0.0155 | 0.415 | 0.0156
| 0.3 | 0.330 | 0.0149 | 0.423 | 0.0156 | 0.441 | 0.0157
| 0.4 | 0.360 | 0.0152 | 0.454 | 0.0157 | 0.471 | 0.0158
| 0.5 | 0.454 | 0.0157 | 0.443 | 0.0157 | 0.461 | 0.0158
| 0.6 | 0.475 | 0.0158 | 0.834 | 0.0118 | 0.840 | 0.0116
| 0.7 | 0.439 | 0.0157 | 0.771 | 0.0133 | 0.781 | 0.0131
| 0.8 | 0.450 | 0.0157 | 0.717 | 0.0142 | 0.725 | 0.0141
| 0.9 | 0.488 | 0.0158 | 0.594 | 0.0155 | 0.610 | 0.0154
(40,50) | 0.1 | 0.182 | 0.0122 | 0.483 | 0.0158 | 0.497 | 0.0158
| 0.2 | 0.369 | 0.0153 | 0.595 | 0.0155 | 0.607 | 0.0154
| 0.3 | 0.508 | 0.0158 | 0.660 | 0.0150 | 0.673 | 0.0148
| 0.4 | 0.578 | 0.0156 | 0.664 | 0.0149 | 0.676 | 0.0148
| 0.5 | 0.657 | 0.0150 | 0.644 | 0.0151 | 0.654 | 0.0150
| 0.6 | 0.646 | 0.0151 | 0.935 | 0.0078 | 0.936 | 0.0077
| 0.7 | 0.655 | 0.0150 | 0.914 | 0.0089 | 0.915 | 0.0088
| 0.8 | 0.670 | 0.0149 | 0.878 | 0.0103 | 0.883 | 0.0102
| 0.9 | 0.694 | 0.0146 | 0.783 | 0.0130 | 0.793 | 0.0128
(75,75) | 0.1 | 0.438 | 0.0157 | 0.688 | 0.0147 | 0.695 | 0.0146
| 0.2 | 0.595 | 0.0155 | 0.811 | 0.0124 | 0.820 | 0.0036
| 0.3 | 0.757 | 0.0136 | 0.986 | 0.0037 | 0.987 | 0.0036
| 0.4 | 0.790 | 0.0129 | 0.869 | 0.0107 | 0.872 | 0.0106
| 0.5 | 0.841 | 0.0116 | 0.983 | 0.0041 | 0.983 | 0.0041
| 0.6 | 0.852 | 0.0112 | 0.979 | 0.0045 | 0.980 | 0.0044
| 0.7 | 0.856 | 0.0111 | 0.791 | 0.0129 | 0.797 | 0.0127
| 0.8 | 0.872 | 0.0106 | 0.954 | 0.0066 | 0.957 | 0.0064
| 0.9 | 0.875 | 0.0105 | 0.927 | 0.0082 | 0.928 | 0.0082
(100,100) | 0.1 | 0.625 | 0.0153 | 0.809 | 0.0124 | 0.813 | 0.0123
| 0.2 | 0.786 | 0.0130 | 0.899 | 0.0095 | 0.901 | 0.0094
| 0.3 | 0.871 | 0.0106 | 0.931 | 0.0080 | 0.933 | 0.0079
| 0.4 | 0.923 | 0.0084 | 0.932 | 0.0080 | 0.933 | 0.0079
| 0.5 | 0.932 | 0.0080 | 0.936 | 0.0077 | 0.94 | 0.0075
| 0.6 | 0.933 | 0.0079 | 0.993 | 0.0026 | 0.993 | 0.0026
| 0.7 | 0.942 | 0.0074 | 0.991 | 0.0030 | 0.991 | 0.0030
| 0.8 | 0.951 | 0.0068 | 0.984 | 0.0040 | 0.986 | 0.0037
| 0.9 | 0.953 | 0.0067 | 0.972 | 0.0052 | 0.973 | 0.0051

Table 5: Power comparison of ADF, JEL, and AJEL tests with nominal level $\alpha=0.05$ when $X\sim Exp(4)$ and $Y\sim Exp(2)$ |  | ADF | JEL | AJEL
---|---|---|---|---
$(n_{1},n_{2})$ | $t$ | Power | SE | Power | SE | Power | SE
(20, 30) | 0.1 | 0.695 | 0.0146 | 0.363 | 0.0152 | 0.382 | 0.0154
| 0.2 | 0.678 | 0.0148 | 0.463 | 0.0158 | 0.478 | 0.0158
| 0.3 | 0.520 | 0.0158 | 0.551 | 0.0157 | 0.566 | 0.0157
| 0.4 | 0.417 | 0.0156 | 0.613 | 0.0154 | 0.626 | 0.0153
| 0.5 | 0.418 | 0.0156 | 0.664 | 0.0149 | 0.682 | 0.0147
| 0.6 | 0.367 | 0.0152 | 0.893 | 0.0098 | 0.901 | 0.0094
| 0.7 | 0.295 | 0.0144 | 0.884 | 0.0101 | 0.894 | 0.0097
| 0.8 | 0.284 | 0.0143 | 0.868 | 0.0107 | 0.875 | 0.0105
| 0.9 | 0.238 | 0.0135 | 0.834 | 0.0118 | 0.846 | 0.0114
(40, 50) | 0.1 | 0.835 | 0.0117 | 0.471 | 0.0158 | 0.485 | 0.0158
| 0.2 | 0.812 | 0.0124 | 0.606 | 0.0155 | 0.615 | 0.0154
| 0.3 | 0.582 | 0.0156 | 0.724 | 0.0141 | 0.730 | 0.0140
| 0.4 | 0.492 | 0.0158 | 0.792 | 0.0128 | 0.803 | 0.0126
| 0.5 | 0.470 | 0.0158 | 0.837 | 0.0117 | 0.851 | 0.0113
| 0.6 | 0.368 | 0.0153 | 0.969 | 0.0055 | 0.971 | 0.0053
| 0.7 | 0.324 | 0.0148 | 0.971 | 0.0053 | 0.972 | 0.0052
| 0.8 | 0.317 | 0.0147 | 0.961 | 0.0061 | 0.966 | 0.0057
| 0.9 | 0.290 | 0.0143 | 0.958 | 0.0063 | 0.960 | 0.0062
(75, 75) | 0.1 | 0.901 | 0.0094 | 0.616 | 0.0154 | 0.623 | 0.0153
| 0.2 | 0.886 | 0.0101 | 0.787 | 0.0129 | 0.791 | 0.0129
| 0.3 | 0.719 | 0.0142 | 0.975 | 0.0049 | 0.976 | 0.0048
| 0.4 | 0.586 | 0.0156 | 0.930 | 0.0081 | 0.930 | 0.0081
| 0.5 | 0.546 | 0.0157 | 0.993 | 0.0026 | 0.993 | 0.0026
| 0.6 | 0.440 | 0.0157 | 0.995 | 0.0022 | 0.995 | 0.0022
| 0.7 | 0.408 | 0.0155 | 0.987 | 0.0036 | 0.988 | 0.0034
| 0.8 | 0.371 | 0.0153 | 0.995 | 0.0022 | 0.995 | 0.0022
| 0.9 | 0.348 | 0.0151 | 0.994 | 0.0024 | 0.995 | 0.0022
(100, 100) | 0.1 | 0.946 | 0.0071 | 0.661 | 0.0150 | 0.667 | 0.0149
| 0.2 | 0.930 | 0.0081 | 0.851 | 0.0113 | 0.856 | 0.0111
| 0.3 | 0.717 | 0.0142 | 0.927 | 0.0082 | 0.928 | 0.0082
| 0.4 | 0.629 | 0.0153 | 0.965 | 0.0058 | 0.968 | 0.0056
| 0.5 | 0.551 | 0.0157 | 0.984 | 0.0040 | 0.985 | 0.0038
| 0.6 | 0.483 | 0.0158 | 1.000 | 0.0000 | 1.000 | 0.0000
| 0.7 | 0.429 | 0.0157 | 1.000 | 0.0000 | 1.000 | 0.0010
| 0.8 | 0.419 | 0.0156 | 0.999 | 0.0010 | 0.999 | 0.0010
| 0.9 | 0.355 | 0.0151 | 0.999 | 0.0010 | 0.999 | 0.0010

Table 6: Power comparison of ADF, JEL, and AJEL tests with nominal level $\alpha=0.05$ when $X\sim HN(1)$ and $Y\sim HN(1.5)$ |  | ADF | JEL | AJEL
---|---|---|---|---
$(n_{1},n_{2})$ | $t$ | Power | SE | Power | SE | Power | SE
(20, 30) | 0.1 | 0.415 | 0.0156 | 0.260 | 0.0139 | 0.274 | 0.0141
| 0.2 | 0.438 | 0.0157 | 0.292 | 0.0144 | 0.303 | 0.0145
| 0.3 | 0.479 | 0.0158 | 0.309 | 0.0146 | 0.327 | 0.0148
| 0.4 | 0.485 | 0.0158 | 0.373 | 0.0153 | 0.390 | 0.0154
| 0.5 | 0.541 | 0.0158 | 0.377 | 0.0153 | 0.395 | 0.0155
| 0.6 | 0.559 | 0.0157 | 0.721 | 0.0142 | 0.737 | 0.0139
| 0.7 | 0.520 | 0.0158 | 0.694 | 0.0146 | 0.716 | 0.0143
| 0.8 | 0.537 | 0.0158 | 0.670 | 0.0149 | 0.688 | 0.0147
| 0.9 | 0.548 | 0.0157 | 0.597 | 0.0155 | 0.616 | 0.0154
(40, 50) | 0.1 | 0.654 | 0.0150 | 0.312 | 0.0147 | 0.330 | 0.0149
| 0.2 | 0.661 | 0.0150 | 0.393 | 0.0154 | 0.400 | 0.0155
| 0.3 | 0.676 | 0.0148 | 0.455 | 0.0157 | 0.468 | 0.0158
| 0.4 | 0.690 | 0.0146 | 0.544 | 0.0158 | 0.561 | 0.0157
| 0.5 | 0.725 | 0.0141 | 0.577 | 0.0156 | 0.589 | 0.0156
| 0.6 | 0.713 | 0.0143 | 0.838 | 0.0117 | 0.845 | 0.0114
| 0.7 | 0.723 | 0.0142 | 0.840 | 0.0116 | 0.847 | 0.0114
| 0.8 | 0.733 | 0.0140 | 0.841 | 0.0116 | 0.847 | 0.0114
| 0.9 | 0.746 | 0.0138 | 0.809 | 0.0124 | 0.816 | 0.0123
(75, 75) | 0.1 | 0.839 | 0.0116 | 0.381 | 0.0154 | 0.390 | 0.0154
| 0.2 | 0.843 | 0.0115 | 0.508 | 0.0158 | 0.515 | 0.0158
| 0.3 | 0.853 | 0.0112 | 0.896 | 0.0097 | 0.897 | 0.0096
| 0.4 | 0.866 | 0.0108 | 0.689 | 0.0146 | 0.700 | 0.0145
| 0.5 | 0.877 | 0.0104 | 0.912 | 0.0090 | 0.913 | 0.0089
| 0.6 | 0.876 | 0.0104 | 0.933 | 0.0079 | 0.936 | 0.0077
| 0.7 | 0.882 | 0.0102 | 0.846 | 0.0114 | 0.851 | 0.0113
| 0.8 | 0.891 | 0.0099 | 0.932 | 0.0080 | 0.934 | 0.0079
| 0.9 | 0.892 | 0.0098 | 0.929 | 0.0081 | 0.931 | 0.0080
(100, 100) | 0.1 | 0.938 | 0.0076 | 0.390 | 0.0155 | 0.400 | 0.0155
| 0.2 | 0.940 | 0.0075 | 0.554 | 0.0157 | 0.561 | 0.0157
| 0.3 | 0.941 | 0.0075 | 0.691 | 0.0146 | 0.695 | 0.0146
| 0.4 | 0.944 | 0.0073 | 0.787 | 0.0129 | 0.792 | 0.0128
| 0.5 | 0.951 | 0.0068 | 0.837 | 0.0117 | 0.839 | 0.0116
| 0.6 | 0.953 | 0.0067 | 0.958 | 0.0063 | 0.958 | 0.0063
| 0.7 | 0.952 | 0.0068 | 0.969 | 0.0055 | 0.970 | 0.0054
| 0.8 | 0.957 | 0.0064 | 0.973 | 0.0051 | 0.973 | 0.0051
| 0.9 | 0.958 | 0.0063 | 0.972 | 0.0052 | 0.975 | 0.0049

Figure 3: Power comparison for ADF, JEL, and AJEL methods for different
distributions, sample sizes, and values of $t$

##  4 Applications

In this section, we use the proposed methods to evaluate the equality of the
generalized Lorenz curves for various subgroups of employees of California
State University (CSU) and University of California (UC) systems in
2021.222The most recent data was obtained and is available on the State
Controller’s Office website at https://publicpay.ca.gov/Reports/Explore.aspx.
The 2021 data comprises 105,414 records of salaries for CSU and 299,448
records of salaries for UC. The data is anonymous but grouped by employer name
and type of position.333To identify CSU instructional faculty, we filtered the
data set based on the following keywords: “Instructional Faculty”, “Teaching
Associate”, “Visiting Faculty”, “Lecturer”, ”Academic-Related”, “Department
Chair”. To identify UC instructional faculty, we filtered the data set based
on the following keywords: “Assoc Prof”, “Assoc Adj”, “Asst Adj”, “Asst Prof”,
“Prof In”, “VIS Prof”, “Adj Instr”, “Lect”, “Grad”, “Adj”. All other employees
that didn’t possess the listed keywords in the description of their position
were identified as non-teaching staff. Further, log-transformed data was used
to avoid the unnecessary computational burden. For the purpose of this
analysis, we considered testing the hypothesis defined in equation (7) at a
significance level of 5% for the following three scenarios. In each scenario,
we apply the proposed testing procedures for $t=0.0,0.2,0.4,0.6,0.8,1.0$, and
obtain the corresponding test statistics and p-values.

###  4.1 Using complete data to compare income distributions of faculty at CSU
Monterey Bay and CSU San Bernardino

In this scenario, we examine the salaries of all instructional faculty from
CSU Monterey Bay and CSU San Bernardino. Filtering data based on the employer
name and the type of position resulted in obtaining a total of 546 records for
CSU Monterey Bay faculty salaries and 1,265 records for CSU San Bernardino
faculty. These two institutions were chosen because their Lorenz and
generalized Lorenz curves appear to be very similar as shown in Figure 4.

Figure 4: (a) Lorenz curves and (b) generalized Lorenz curves for salaries of CSU Monterey Bay and CSU San Bernardino faculty Table 7: Test statistic and p-value |  | $t$
---|---|---
Method | Value | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0
ADF | Test statistic | - | 0.0119 | 0.0119 | 0.0119 | 0.0128 | 0.0135
| p-value | - | 0.9133 | 0.9133 | 0.9133 | 0.9100 | 0.9076
JEL | Test statistic | 0.2116 | 667.8884 | 417.0143 | 0.0380 | 0.0070 | 0.0138
| p-value | 0.6455 | 0.0000 | 0.0000 | 0.8454 | 0.9332 | 0.9065
AJEL | Test statistic | 0.2126 | 669.2835 | 417.9359 | 0.0382 | 0.0071 | 0.0139
| p-value | 0.6448 | 0.0000 | 0.0000 | 0.8450 | 0.9331 | 0.9063

Table 7 shows the computed test statistics and p-values for ADF, JEL, and AJEL
methods at values of $t=0.0,0.2,0.4,0.6,0.8,1.0$. It is observed that both
methods reject the null hypothesis at a 5% significance level for $t=0.2$ and
$t=0.4$, indicating sufficient evidence to conclude that the generalized
Lorenz curves considered are significantly different for the 20th and 40th
percentiles. However, for $t=0.0$, $t=0.6$, $t=0.8$, and $t=1.0$, the p-values
obtained by both methods are greater than 0.05, implying that we do not have
enough evidence to infer significant differences in the considered generalized
Lorenz curves for the 0th, 60th, 80th, and 100th percentiles. These mixed
results can be attributed to the multiple intersections of the considered
generalized Lorentz curves.

###  4.2 Using complete data to compare income distributions of faculty at CSU
San Bernardino and CSU San Francisco

In this scenario, we examine the salaries of all instructional faculty from
CSU San Bernardino and CSU San Francisco. Filtering data based on the employer
name and the type of position resulted in obtaining the total of 1,265 records
for CSU San Bernardino faculty salaries and 2,294 records for CSU San
Francisco faculty. These two institutions were chosen because their Lorenz and
generalized Lorenz curves appear to be very distinct as shown in Figure 5.

Figure 5: (a) Lorenz Curves and (b) Generalized Lorenz Curves for salaries of CSU San Bernardino and CSU San Francisco faculty Table 8: Test statistic and p-value |  | $t$
---|---|---
Method | Value | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0
ADF | Test statistic | - | 17.3485 | 20.1831 | 23.0600 | 24.9908 | 28.8444
| p-value | - | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
JEL | Test statistic | 1.5821 | 217.8804 | 942.6197 | 250.0000 | 30.2795 | 20.0987
| p-value | 0.2085 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
AJEL | Test statistic | 1.5860 | 218.2452 | 943.7490 | 250.0000 | 30.3477 | 20.1438
| p-value | 0.2079 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000

Table 8 presents the computed test statistics and p-values for ADF, JEL, and
AJEL methods at $t=0.0,0.2,0.4,0.6,0.8,1.0$. The results indicate that both
methods reject the null hypothesis for all values of $t$ except $t=0$,
implying sufficient evidence to conclude that the two generalized Lorenz
curves are significantly different for the 20th, 40th, 60th, 80th, and 100th
percentiles at a 5% significance level. However, for $t=0.0$, both methods
yield p-values greater than 0.05, suggesting insufficient evidence to infer
significant differences in the considered generalized Lorenz curves for the
0th percentile. These findings can be explained by the fact that the minimum
salaries for both institutions are nearly identical, but the difference
between the institutions increases as $t$ increases.

###  4.3 Using incomplete data to compare income distributions of all faculty
at CSU and UC

In this scenario, we examine the instructional faculty salaries across all CSU
and UC institutions. Filtering data based on the type of position resulted in
obtaining the total of 34,927 records for CSU faculty salaries and 18,104
records for UC faculty salaries. The Lorenz and generalized Lorenz curves are
shown in Figure 6.

Figure 6: (a) Lorenz curves and (b) generalized Lorenz curves for salaries of
CSU and UC faculty

Since both data sets are quite large, we decided to utilize the proposed
procedures using the samples. As the empirical likelihood approach is known to
be effective for relatively small samples, we selected sample sizes of
$n_{1}=n_{2}=100$. Given that the difference between the generalized Lorenz
curves is noticeable, we anticipate that both procedures will be capable of
detecting the difference even with such modest sample sizes.

Table 9: Test statistic and p-value |  | $t$
---|---|---
Method | Value | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0
ADF | Test statistic | - | 14.9673 | 17.3378 | 18.9842 | 19.8941 | 24.9073
| p-value | - | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000
JEL | Test statistic | 0.6586 | 48.6775 | 20.5069 | 66.4861 | 38.4036 | 25.2639
| p-value | 0.4171 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
AJEL | Test statistic | 0.6833 | 49.7646 | 20.9485 | 67.5089 | 39.0850 | 25.7541
| p-value | 0.4084 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000

Table 9 displays the test statistics and p-values computed for ADF, JEL, and
AJEL methods at $t=0.0,0.2,0.4,0.6,0.8,$ and $1.0$. The results indicate that
all three methods reject the null hypothesis for all values of $t$ except
$t=0$ at a 5% significance level, providing sufficient evidence to conclude
that the generalized Lorenz curves are significantly different for the 20th,
40th, 60th, 80th, and 100th percentiles. Conversely, for $t=0.0$, the p-values
exceed 0.05, indicating insufficient evidence to conclude that the curves are
significantly different at the 0th percentile. Similarly to the previous
scenario, these findings can be attributed to the fact that the minimum
salaries are nearly identical for both institutions, but the difference
between them increases as $t$ increases. However, these findings are
noteworthy because they were obtained using relatively small samples.

###  4.4 Using complete data to compare income distributions of all faculty at
CSUSB in Years 2009 and 2020

In this scenario, we examine the salaries of all instructional faculty at
CSUSB in the years 2009 and 2020. Filtering data based on the employer name
and the type of position resulted in obtaining a total of 1,111 records for
2009 and 1,331 records for 2020. The Lorenz and generalized Lorenz curves are
graphed in Figure 7.

Figure 7: (a) Lorenz curves and (b) generalized Lorenz curves for salaries of CSU faculty in years 2009 and 2020 Table 10: Test statistic and p-value |  | $t$
---|---|---
Method | Value | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0
ADF | Test statistic | - | 1.5990 | 2.2190 | 2.2614 | 2.5820 | 2.5776
| p-value | - | 0.2060 | 0.1363 | 0.1326 | 0.1081 | 0.1084
JEL | Test statistic | 0.1719 | 52.4042 | 29.2928 | 0.1161 | 0.0624 | 0.1486
| p-value | 0.6785 | 0.0000 | 0.0000 | 0.7333 | 0.8028 | 0.6999
AJEL | Test statistic | 0.1725 | 52.5651 | 29.3841 | 0.1165 | 0.0626 | 0.1490
| p-value | 0.6779 | 0.0000 | 0.0000 | 0.7329 | 0.8025 | 0.6994

##  5 Conclusions

In this paper, we developed two non-parametric JEL-based methods using a
$U$-statistic to test the equality of two generalized Lorenz curves. The
limiting distribution of the likelihood ratios is shown to follow a chi-
squared distribution with one degree of freedom. Simulations studies with
different distribution types and sample sizes illustrate that both methods
show improved Type I error probability and power as the sample size increases.
In general, the AJEL resulted in higher test powers in comparison to JEL
across all distributions, sample sizes, and values of $t$, except for a few
cases. However, the AJEL method has a higher Type I error rate than JEL,
though still within an acceptable range. Moreover, these methods exhibit
robustness across a range of scenarios, outperforming the existing ADF method.
The proposed testing methods are applied to three distinct scenarios using CSU
and UC salary data. The results are found to be acceptable for both complete
and incomplete data. This suggests that the proposed testing methods can be
applied to a variety of data types, providing satisfactory results regardless
of data completeness.

##  Appendix A Proofs of Theorems

###### Proof.

Theorem 2.1

Let $n_{1}\leq n_{2}$. As shown in Arvesen (1969), the jackknife procedure for
the two sample $U$-statistics, $U_{n_{1},n_{2}}$, we have

$\displaystyle V_{i,0}$
$\displaystyle=n_{1}U_{n_{1},n_{2}}-(n_{1}-1)U_{n_{1}-1,n_{2}}^{-i,0},\quad
i=1,\cdots,n_{1}$ (20) $\displaystyle V_{0,j}$
$\displaystyle=n_{2}U_{n_{1},n_{2}}-(n_{2}-1)U_{n_{1}-1,n_{2}}^{0,-j},\quad
j=1,\cdots,n_{2}$

Further, they proposed a consistent estimator of $Var(U_{n_{1},n_{2}})$ given
as

$\displaystyle\widehat{Var}_{\text{Jack}}(U_{n_{1},n_{2}})=\dfrac{1}{n_{1}(n_{1}-1)}\sum_{i=1}^{n_{1}}\bigg{(}V_{i,0}-\bar{V}_{\cdot,0}\bigg{)}^{2}+\dfrac{1}{n_{2}(n_{2}-1)}\sum_{j=1}^{n_{2}}\bigg{(}V_{0,j}-\bar{V}_{0,\cdot}\bigg{)}^{2},$

where $\bar{V}_{\cdot,0}$ and $\bar{V}_{0,\cdot}$ are the means of $V_{i,0}$
and $V_{0,j}$ respectively.

######  Lemma A.1.

(See Arvesen (1969))

  1. 1.

Assume that $E|h(X,Y)|<\infty$, then
$U_{n_{1},n_{2}}\xrightarrow[]{~{}~{}~{}\text{a.s.}~{}~{}~{}}\theta$ as
$n\longrightarrow\infty$.

  2. 2.

Assume that $Eh^{2}(X,Y)<\infty,\sigma_{1,0}^{2}>0$ and $\sigma_{0,1}^{2}>0$,
let
$S^{2}_{n_{1},n_{2}}=\dfrac{1}{n_{1}}\sigma_{1,0}^{2}+\dfrac{1}{n_{2}}\sigma_{0,1}^{2}$,
then

$\displaystyle\dfrac{U_{n_{1},n_{2}}-\theta}{S_{n_{1},n_{2}}}$
$\displaystyle\xrightarrow[]{~{}~{}~{}\text{d}~{}~{}~{}}N(0,1)\quad\text{and}\quad\widehat{Var}_{\text{Jack}}\big{(}U_{n_{1},n_{2}}\big{)}-S^{2}_{n_{1},n_{2}}=o_{p}(n_{1}^{-1})\quad\text{as}\quad
n_{1}\longrightarrow\infty.$

In order to apply JEL, using (20), we can determine

$\displaystyle
V_{i,0}=\dfrac{1}{n_{2}}I(X_{i}\leq\psi_{t})\sum_{r=1}^{n_{2}}\big{(}X_{i}-Y_{r}\big{)}\,I(Y_{r}\leq\psi_{t}),\quad
i=1,\dots,n_{1}$

and

$\displaystyle
V_{0,j}=\dfrac{1}{n_{1}}I(Y_{j}\leq\psi_{t})\sum_{s=1}^{n_{1}}\big{(}X_{s}-Y_{j}\big{)}\,I(X_{s}\leq\psi_{t}),\quad
j=1,\dots,n_{2}$

Let $n=n_{1}+n_{2}$. Consider

$\displaystyle U_{n}=\dfrac{1}{n_{1}n_{2}}\sum_{1\leq i\leq{n_{1}}<j\leq
n}\big{(}X_{i}-Y_{j-n_{1}}\big{)}I(X_{i}\leq t)I(Y_{j-n_{1}}\leq\psi_{t})$

and

$\displaystyle U_{n}^{-i}$
$\displaystyle=U\big{(}Z_{1},Z_{2},\dots,Z_{i-1},Z_{i+1},\dots,Z_{n}\big{)}$
$\displaystyle={n-1\choose
2}^{-1}\dfrac{1}{n_{1}n_{2}}\sum_{\begin{subarray}{c}1\leq r<s\leq n\\\
r,s\neq i\end{subarray}}\big{(}X_{r}-Y_{s-n_{1}}\big{)}I(X_{r}\leq
t)I(Y_{s-n_{1}}\leq\psi_{t})$
$\displaystyle=\begin{cases}\dfrac{n}{(n-2)}\bigg{[}U_{n}-\dfrac{1}{n_{1}n_{2}}{\sum}_{i\leq{n_{1}}<j}\big{(}X_{i}-Y_{j-n_{1}}\big{)}I(X_{i}\leq
t)I(Y_{j-n_{1}}\leq\psi_{t})\bigg{]}&,~{}~{}1\leq i\leq n_{1}\\\ \\\
\dfrac{n}{(n-2)}\bigg{[}U_{n}-\dfrac{1}{n_{1}n_{2}}{\sum}_{j\leq{n_{1}}<i}\big{(}X_{j}-Y_{i-n_{1}}\big{)}I(X_{j}\leq
t)I(Y_{i-n_{1}}\leq\psi_{t})\bigg{]}&,~{}~{}n_{1}<i\leq n\end{cases}$

It can be seen that

$\displaystyle\dfrac{1}{n_{1}n_{2}}\sum_{i\leq{n_{1}}<j}\big{(}X_{i}-Y_{j-n_{1}}\big{)}I(X_{i}\leq
t)I(Y_{j-n_{1}}\leq\psi_{t})=\dfrac{1}{n_{1}}V_{i,0}~{},\quad 1\leq i\leq
n_{1}$

and

$\displaystyle\dfrac{1}{n_{1}n_{2}}\sum_{j\leq{n_{1}}<i}\big{(}X_{j}-Y_{i-n_{1}}\big{)}I(X_{j}\leq
t)I(Y_{i-n_{1}}\leq\psi_{t})=\dfrac{1}{n_{2}}V_{0,i}~{},\quad n_{1}<i\leq n$

Now, consider JEL given in (10), for $1\leq k\leq n$, we have

$\displaystyle\widehat{V}_{k}$ $\displaystyle=nU_{n}-(n-1)U_{n-1}^{-k}$
$\displaystyle=\dfrac{n(n-1)}{n-2}\Bigg{[}\Bigg{(}\dfrac{V_{k,0}}{n_{1}}\Bigg{)}I(1\leq
k\leq n_{1})+\Bigg{(}\dfrac{V_{0,k-n_{1}}}{n_{2}}\Bigg{)}I(n_{1}<k\leq
n)\bigg{]}-\dfrac{n}{n-2}U_{n_{1},n_{2}}$

Thus,

$\displaystyle
E\widehat{V}_{k}=\dfrac{n\theta}{n-2}\Bigg{[}\Bigg{(}\dfrac{n_{2}-1}{n_{1}}\Bigg{)}I(1\leq
k\leq n_{1})+\Bigg{(}\dfrac{n_{1}-1}{n_{2}}\Bigg{)}I(n_{1}<k\leq n)\Bigg{]}$

Under $H_{0}$, $E\widehat{V}_{k}=0$. Next, following the similar arguments
given in Jing . (2009), for fixed $t=t_{0}\in[0,1]$, it can be proven that
$\ell(\theta(t_{0}))\longrightarrow\chi^{2}_{1}$, as
$n_{1}\longrightarrow\infty$ . Thus, details are omitted here.

∎

###### Proof.

Theorem 2.2 The proof of this theorem is similar to Theorem 1 given in Chen .
(2008). Let $\lambda^{\textnormal{Adj}}(t)$ be the solution to

$\displaystyle\sum_{k=1}^{n+1}\frac{g^{\textnormal{Adj}}_{k}(t)}{1+\lambda^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t)}=0.$
(21)

The first step is to show that
$\lambda^{\textnormal{Adj}}(t)=O_{p}(n^{-1/2})$. By using Lemma 3 of Owen
(1990) and the fact that $E(\hat{V}_{1}^{2}(t))<\infty$, we can establish that
$g^{*}=\max_{1\leq k\leq n}\left\|\hat{V}_{k}\right\|=o_{p}(n^{1/2})$ and
$\bar{g}_{n}(t)=O_{p}(n^{-1/2})$. Let
$\rho=\left\|\lambda^{\textnormal{Adj}}(t)\right\|$, $a_{n}=o_{p}(n)$ and
$\hat{\lambda}^{\textnormal{Adj}}(t)=\lambda^{\textnormal{Adj}}(t)/\rho$.
Multiplying $\hat{\lambda}^{\textnormal{Adj}}(t)/n$ to both sides gives

$\displaystyle 0$
$\displaystyle=\frac{\hat{\lambda}^{\textnormal{Adj}}(t)}{n}\sum_{k=1}^{n+1}\frac{g^{\textnormal{Adj}}_{k}(t)}{1+\lambda^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t)}$
(22)
$\displaystyle=\frac{\hat{\lambda}^{\textnormal{Adj}}(t)}{n}\sum_{k=1}^{n+1}g^{\textnormal{Adj}}_{k}(t)-\dfrac{\rho}{n}\sum_{k=1}^{n+1}\frac{(\hat{\lambda}^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t))^{2}}{1+\rho\hat{\lambda}^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t)}$
$\displaystyle\leq\hat{\lambda}^{\textnormal{Adj}}(t)\bar{g}_{n}(t)(1-a_{n}/n)-\frac{\rho}{n(1+\rho
g^{*}(t))}\sum_{k=1}^{n}(\hat{\lambda}^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t))^{2}$
$\displaystyle=\hat{\lambda}^{\textnormal{Adj}}(t)\bar{g}_{n}(t)-\frac{\rho}{n(1+\rho
g^{*}(t))}\sum_{k=1}^{n}\big{(}\hat{\lambda}^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t)\big{)}^{2}+O_{p}(n^{-3/2}a_{n}).$

The inequality stated above is valid due to the non-negativity of the
$(n+1)$th term in the second summation. According to Chen . (2008), for any
given $\epsilon>0$, we have

$\displaystyle\frac{1}{n}\sum_{k=1}^{n}\big{(}\lambda^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t)\big{)}^{2}\geq
1-\epsilon.$ (23)

Therefore, as long as $a_{n}=o_{p}(n)$, equation (22) implies that

$\displaystyle\frac{\rho}{(1+\rho
g^{*}(t))}\leq\hat{\lambda}^{\textnormal{Adj}}(t)\frac{\bar{g}_{n}(t)(t)}{(1-\epsilon)}=O_{p}(n^{-1/2}).$
(24)

Thus, we get $\rho=O_{p}(n^{-1/2})$ and hence
$\lambda^{\textnormal{Adj}}(t)=O_{p}(n^{-1/2})$. Now, consider

$\displaystyle 0$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n+1}\frac{g^{\textnormal{Adj}}_{k}(t)}{1+\lambda^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t)}$
(25)
$\displaystyle=\bar{g}_{n}(t)(t)-\lambda^{\textnormal{Adj}}(t)\hat{V}_{n}(t)+o_{p}(n^{-1/2}),$

where $\hat{V}_{n}=(1/n)\sum_{k=1}^{n}g^{\textnormal{Adj}}_{k}(t)^{2}$. Hence,
when
$n\longrightarrow\infty,\lambda^{\textnormal{Adj}}(t)=\hat{V}_{n}^{-1}\bar{g}_{n}(t)+o_{p}(n^{-1/2})$.
Now, we expand $l^{*}(\theta(t))$ as follows

$\displaystyle l^{*}(\theta(t))$
$\displaystyle=\sum_{k=1}^{n+1}\log\big{(}1+\lambda^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t)\big{)}$
(26)
$\displaystyle=\sum_{k=1}^{n+1}\bigg{\\{}\lambda^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t)-\frac{\big{(}\lambda^{\textnormal{Adj}}(t)g^{\textnormal{Adj}}_{k}(t)\big{)}^{2}}{2}\bigg{\\}}+o_{p}(1).$

Substituting the expansion of $\lambda^{\textnormal{Adj}}$, we get that

$\displaystyle-2l^{*}(\theta(t_{0}))$
$\displaystyle=n\hat{V}_{n}^{-1}\bar{g}_{n}(t)^{2}+o_{p}(1)$ (27)
$\displaystyle\xrightarrow[]{d}\chi^{2}_{1}.$

This completes the proof. ∎

## References

## References

  * Arora Jain ((2006)) arora2006Arora, S. Jain, K.  2006\.  Testing for generalized Lorenz dominance Testing for generalized lorenz dominance.  Statistical Methods and Applications1575–88.
  * Arvesen ((1969)) arv1969Arvesen, JN.  1969\.  Jackknifing U-statistics Jackknifing U-statistics.  Ann. Math. Stat402076-2100.
  * Belinga-Hall ((2007)) belinga2007Belinga-Hall, N.  2007\.  Empirical Likelihood Confidence Intervals for Generalized Lorenz Curve Empirical likelihood confidence intervals for generalized lorenz curve.  Thesis, Georgia State University.
  * Chang Halfon ((1997)) chang1997Chang, R. Halfon, N.  1997\.  Graphical distribution of pediatricians in the united states: An analysis of the fifty states and washington, dc. Graphical distribution of pediatricians in the united states: An analysis of the fifty states and washington, dc.  Pediatrics100172-179.
  * Chen . ((2008)) chen2008Chen, J., Variyath, A. Abraham, B.  2008\.  Adjusted empirical likelihood and its properties Adjusted empirical likelihood and its properties.  Journal of Computational and Graphical Statistics17(2)426-443.
  * Gastwirth ((1971)) gas1971Gastwirth, JL.  1971\.  A General Definition of Lorenz Curve A general definition of lorenz curve.  Econometrica391037-1039.
  * Jing . ((2017)) jing2017Jing, BY., Tsao, M. Zhou, W.  2017\.  Transforming the empirical likelihood towards better accuracy Transforming the empirical likelihood towards better accuracy.  The Canadian Journal of Statistics45(3)340-352.
  * Jing . ((2009)) jing2009Jing, BY., Yuan, J. Zhou, W.  2009\.  Jackknife Empirical Likelihood Jackknife empirical likelihood.  Journal of the American Statistical Association104(487)1224-1232.
  * Kobayashi Takaki ((1992)) kob1992Kobayashi, Y. Takaki, H.  1992\.  Geographic distribution of physicians in Japan Geographic distribution of physicians in japan.  Lancet.3401391–1393.
  * Li Wei ((2018)) li2018Li, Z. Wei, Y.  2018\.  Statistical inference for the difference of two Lorenz curves Statistical inference for the difference of two lorenz curves.  Statistics52(5)1128-1155.
  * Lorenz ((1905)) lor1905Lorenz, MC.  1905\.  Method of Measuring the Concentration of Wealth Method of measuring the concentration of wealth.  J. Ame. Statist.9209-219.
  * Owen ((1988)) owen1988Owen, AB.  1988\.  Empirical likelihood ratio confidence intervals for a single functional Empirical likelihood ratio confidence intervals for a single functional.  Biometrika75237-249.
  * Owen ((1990)) owen1990Owen, AB.  1990\.  Empirical likelihood confidence regions Empirical likelihood confidence regions.  The Annals of Statistics1890-120.
  * Quenouille ((1956)) que1956Quenouille, MH.  1956\.  Notes on Bias in Estimation Notes on bias in estimation.  Biometrika43(3/4)353–360.
  * Ratnasingam . ((2023)) ratlorenz2023Ratnasingam, S., Wallace, S., Amani, I. Romero, J.  2023\.  Non-Parametric Confidence Intervals for Generalized Lorenz Curve using Modified Empirical Likelihood Non-parametric confidence intervals for generalized lorenz curve using modified empirical likelihood.  (submitted).
  * Shi ((1984)) shi1984Shi, X.  1984\.  The approximate independence of jackknife pseudo-values and the bootstrap methods The approximate independence of jackknife pseudo-values and the bootstrap methods.  J. Wuhan Inst. Hydra-Elect. Eng.283-90.
  * Xu ((1997)) xu1997Xu, K.  1997\.  Asymptotically distribution-free statistical test for generalized lorenz curves: An alternative approach Asymptotically distribution-free statistical test for generalized lorenz curves: An alternative approach.  Journal of income distribution45 - 62.
  * Yang . ((2012)) yang2012Yang, BY., Qin, GS. Belinga-Hill, NE.  2012\.  Non-parametric inferences for the generalized Lorenz curve Non-parametric inferences for the generalized lorenz curve.  Sci Sin Math42(3)235-250.