Cumulative Residual Tsallis Entropy-Based Test of Uniformity and Some New Findings

Mohamed S Mohamed,Mohamed A Abd Elgawad,Salem A Alyami,Haroon M Barakat

doi:10.3390/math10050771

Mohamed S Mohamed, Mohamed A Abd Elgawad + Show 2 more

Open Access

https://doi.org/10.3390/math10050771

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Abstract

The Tsallis entropy is an extension of the Shannon entropy and is used extensively in physics. The cumulative residual Tsallis entropy, which is a generalization of the Tsallis entropy, plays an important role in the measurement uncertainty of random variables and has simple relationships with other important information and reliability measures. In this paper, some novel properties of the cumulative residual Tsallis entropy are disclosed. Moreover, this entropy measure is applied to testing the uniformity, where the limit distribution and an approximation of the distribution of the test statistic are derived. In addition, the property of stability is discussed. Furthermore, the percentage points and power against seven alternative distributions of this test statistic are presented. Finally, to compare the power of the suggested test with that of other tests of uniformity, a simulation study is conducted.

Highlights

Received: 27 January 2022The classical measure of uncertainty in a discrete distribution (Shannon [1]) has been used in many areas, such as computer science [2], communication theory [2], the physical and chemical fields [3], fuzzy sets [4], and finance [5,6]
A straightforward extension of the discrete case to continuous distributions based on a probability density function (PDF)
Some novel properties of the cumulative residual Tsallis entropy (CRTE) quantity were presented such as sufficient conditions for the CRTE to be finite, the weak convergence of the CRTE, the connections between the CRTE, the cumulative residual entropy (CRE), and classic differential entropy, and the stability of the empirical CRTE

Summary

Introduction

The classical measure of uncertainty in a discrete distribution (Shannon [1]) has been used in many areas, such as computer science [2], communication theory [2], the physical and chemical fields [3], fuzzy sets [4], and finance [5,6]. Rajesh and Sunoj [12] introduced an alternate measure of CRTE of order θ, which possesses certain interesting properties with CTθ∗ ( X ), as CTθ ( X ) = CTθ ( F ) =. Noughabi [29] developed a test for uniformity based on the CRE and studied some of its features He compared the percentage points and power of seven alternative distributions. Sati and Gupta [11] introduced a cumulative residual Tsallis entropy of order θ and studied its various properties in the context of reliability modeling. Rajesh and Sunoj [12] introduced an alternate measure of CRTE (defined by (2)) and studied its properties.

Further Theoretical Aspects and Test Statistic

Percentage Points of the Test Statistic

Power Analysis

Conclusions