Estimation of Entropy for Inverse Lomax Distribution under Multiple Censored Data

The inverse Lomax distribution has been widely used in many applied fields such as reliability, geophysics, economics and engineering sciences. In this paper, an unexplored practical problem involving the inverse Lomax distribution is investigated: the estimation of its entropy when multiple censored data are observed. To reach this goal, the entropy is defined through the Rényi and q-entropies, and we estimate them by combining the maximum likelihood and plugin methods. Then, numerical results are provided to show the behavior of the estimates at various sample sizes, with the determination of the mean squared errors, two-sided approximate confidence intervals and the corresponding average lengths. Our numerical investigations show that, when the sample size increases, the values of the mean squared errors and average lengths decrease. Also, when the censoring level decreases, the considered of Rényi and q-entropies estimates approach the true value. The obtained results validate the usefulness and efficiency of the method. An application to two real life data sets is given.


Introduction
Entropy is one of the most popular measure of uncertainty. As former mathematical work, Reference [1] proposed a theory on the concept of entropy, with numerical indicators as well. This theory was enhanced by numerous other entropy-like measures, arising from various applied fields. In this regard, a complete survey can be found in Reference [2]. Here, we focus our attention on two of the most famous entropy measures-the Rényi entopy by Reference [3] and the q-entropy by Reference [4] (also called Tsallis entropy). The Rényi entropy finds its source in the information theory and the q-entropy comes from the statistical physics, with a plethora of applications in their respective fields. For a random variable X having the probability density function (pdf) f (x; ϕ), where ϕ represents the corresponding parameters, these two entropy measures are, respectively, defined by , x, α, θ > 0, respectively. Among the advantages of the IL distribution, the corresponding probability functions are tractable, it is parsimonious in parameters and possesses a non-monotonic hazard rate function; it possesses decreasing and upside-down bathtub shapes. The practical usefulness of the related model is illustrated in Reference [19] for its application to the analyses of geophysical data and in Reference [20] for its application in economics and actuarial sciences. Also, we refer to Reference [21] for the estimation of the reliability parameter via Type II censoring samples, in Reference [22] for the estimation of the parameters based on hybrid censored samples, and in Reference [23] for the Bayesian estimation of the two-component mixture of the IL distribution under the Type I censoring scheme. Also, recent studies have proposed extensions of the IL distributions for further purposes. In this regard, let us cite Reference [24] for the inverse power Lomax (or power IL) distribution, Reference [25] for the Weibull IL distribution, Reference [26] for the alpha power transformed IL distribution, Reference [27] for the Marshall-Olkin IL distribution and Reference [28] for the odd generalized exponentiated IL distribution. However, as far as we know, despite its interest, the estimation of entropy measures for the IL distribution, such as the Rényi and q-entropies, remains an unexplored aspect. This study fills this gap by considering this problem under the realistic scenario of multiple censored data. This scenario commonly occurs where several censoring levels logically exist, which is often the case for many applications in life testing and survival analysis. We refer the reader to References [29][30][31], as well as the recent estimation studies of Reference [32] and Reference [15]. In our statistical framework, after investigating the maximum likelihood estimates of α and θ, estimates for the Rényi and q-entropies are derived. Then, two-sided approximate confidence intervals of the Rényi and q-entropies are discussed. A complete numerical study is performed, showing the favorable behavior of the obtained estimates at various sample sizes. In particular, the mean squared errors, approximate confidence intervals along with the corresponding average lengths are used as benchmarks. Our numerical investigations show that, when the sample size increases, the values of the mean squared errors and average lengths decrease. Also, when the censoring level decreases, the considered of Rényi and q-entropies estimates approach the true value. Two real life data sets, one physiological data set and one economic data set, are used to illustrate the findings.
The rest of the article is arranged as follows. The Rényi and q-entropies for the IL distribution are expressed in Section 2. Section 3 studies their estimation under multiple censored data. Simulation and numerical results are given in Section 4. An application to real data sets is presented in Section 5. The article ends with some concluding remarks in Section 6.

Mathematical Basics on Multiple Censored Data Setting
For our estimation study, we consider the situation of multiple censored data (including the type I and type II censoring), following the setting of Reference [30] (Section 1.3.2). We may also refer to Reference [25] in the context of the inverse Weibull distribution. The general framework can be summarized as follows. Let X be a random variable having the cdf and pdf given by f (x; ϕ) and F(x; ϕ), respectively. Based on n units under a certain test, we get n values x 1 , . . . , x n of which with n m + n f = n. Then, the likelihood function for ϕ can be expressed as where ε i, f = 1 if the ith unit failed, and 0 otherwise (so and 0 otherwise (so n ∑ i=1 ε i,m = n m ), and K denotes a secondary constant (independent of ϕ). Then, the maximum likelihood estimates (MLEs) of ϕ are obtained by maximizing L(ϕ) with respect to ϕ. All the details in this regard can be found in Reference [30] (Section 1.3.2).

Confidence Intervals
Owing to the invariance property,Î δ (X) andĤ q (X) are also the MLEs of I δ (X; α, θ) and H q (X; α, θ), respectively. Therefore, the well-known theory of the maximum likelihood method can be applied toÎ δ (X) andĤ q (X). In particular, invoking the so-called Delta theorem, under some technical regularity conditions, the subjacent asymptotic distribution ofÎ δ (X) can be approximated by the normal distribution with mean I δ (X; α, θ) and variance DJ −1 D T , where J −1 denotes the inverse of the observed information matrix and D = (∂I δ (X; α, θ)/∂α, ∂I δ (X; α, θ)/∂θ) | α=α,θ=θ , both can be determined from (5). Therefore, the two-sided approximate confidence interval for the Rényi entropy at the confidence level 100 A similar result holds forĤ q (X). We also refer to References [15,31] for more detail.

Simulation Study
Here, a simulation study is assessed to investigate the performance of the Rényi and q-entropies estimates given by (7) and (8), respectively. In this regard, we use the mean squared errors (MSEs), two-sided approximate confidence intervals along with their corresponding average lengths (ALs) (i.e., defined AL is the average value of U-L, where L and U denotes the lower and upper bounds of the corresponding interval, respectively) based on multiple censored data (or samples). We adopt the methodology of Reference [15]. Thus, the following procedure is conducted: • 3000 random samples of sizes n = 50, 100, 150, 200 and 300 are generated from the IL distribution based on multiple censored sample.
For the failures at censoring level (CL), we arbitrary chose CL = 0.5 and 0.7 (for instance, CL = 0.7 means that the observations are based on 30% failed units and 70% censored units).

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The true values for I δ (X; α, θ) and H q (X; α, θ) given by (5) and (6), and the average estimateŝ I δ (X) andĤ q (X) given by (7) and (8)  All the numerical results are presented in Tables 1-4 for the Rényi entropy, and Tables 5-8 for the q-entropy. All is calculated by the use of the mathematical software Mathcad.        Here, some remarks can be formulated about the behavior of the Rényi and q-entropies estimates according to Tables 1-8, and Figures 1-8: • The MSEs ofÎ δ (X) decrease as the sample size increases.

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The ALs ofÎ δ (X) decrease as the sample size increases.

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The MSEs ofÎ δ (X) increase when the value of δ increases.

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The ALs ofÎ δ (X) increase when the value of δ increases.

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The MSEs ofĤ q (X) decrease as the sample size increases.

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The ALs ofĤ q (X) decrease as the sample size increases.

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The MSEs ofĤ q (X) decrease when the value of q increases.

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The ALs ofĤ q (X) decrease when the value of q increases.

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In almost situations the MSE ofÎ δ (X) at CL = 0.5 is less than the MSE ofÎ δ (X) at CL = 0.7.

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In almost situations the MSE ofĤ q (X) at CL = 0.5 is less than the MSE ofĤ q (X) at CL = 0.7.
These facts prove the good accuracy of our entropy estimates, which are logically recommended for further practical purposes.

Application
In this Section, two real life data sets are used to illustrate the finding, both described below. The first data set is a physiological data set extracted from Reference [33]. It concerns twenty Duchenne patients (6-18 years age) with classical type of the muscular dystrophy. The Electrocardiography of these 20 patients based on the heart rate is given below in Table 9. The second data set is an economic data set extracted from the following electronic address: https://tradingeconomics.com/pakistan/consumer-price-index-cpi. It refers to Pakistan Consumer Price Index (CPI) in Pakistan from May 2019 to April 2020. The data are collected in Table 10. Then, based on the data, adopting the multiple censored data scheme, we applyÎ δ (X) andĤ q (X) to estimate I δ (X; α, θ) and H q (X; α, θ) where X denotes the considered random variables of interest, assuming to follow the IL distribution. Different values for CL, δ and q are considered. The obtained numerical results are displayed in Tables 11 and 12 for the first and second data sets, respectively. Table 11. Estimated of Rényi and q-entropies at CL = 0.5 and CL = 0.7 for the first data set.  Thus, Tables 11 and 12 show some numerical values of estimated entropies in a concrete scenario, following the multiple censored data scheme. We see that the results depends on the entropy parameters (δ or q), and also, the value for CL, beyond the standard complete standard (which can be obtained by taking CL = 0).

Concluding Remarks
This article studies the estimation of the Rényi and q-entropies for inverse Lomax distribution under multiple censored data. We propose an efficient estimation strategy by using the maximum likelihood and plugging methods. The behavior of the Rényi and q-entropies estimates are calculated in terms of their mean squared errors and average lengths (depending on two-sided approximate confidence intervals). Numerical results are provided, showing that, as the sample size increases, the mean squared errors of our estimates decrease. Also, it can be observed that, as the sample size increases, the average lengths of our estimates decreases. Thus, the proposed estimates reveal to be efficient, providing new useful tools with potential applicability in many applied situations dealing with entropy of the inverse Lomax distribution. The article ends by presenting an application to two real life data sets.