Estimating the Entropy and Residual Entropy of a Lomax Distribution under Generalized Type-II Hybrid Censoring

The Lomax distribution (or Pareto II) was ﬁrst introduced by K. S. Lomax in 1954. It can be readily applied to a wide range of situations including applications in the analysis of the business failure life time data, economics, and actuarial science, income and wealth inequality, size of cities, engineering, lifetime, and reliability modeling. In his pioneering paper, Shannon 1948 deﬁned the notion of entropy as a mathematical measure of information, which is sometimes called Shannon entropy in his honor. He laid the groundwork for a new branch of mathematics in which the notion of entropy plays a fundamental role over different areas of applications such as statistics, information theory, ﬁnancial analysis, and data compression. [Ebrahimi and Pellerey 14] introduced the residual entropy function because the entropy shouldn’t be applied to a system that has survived for some units of time, and therefore, the residual entropy is used to measure the ageing and characterize, classify and order lifetime distributions. In this paper, the estimation of the entropy and residual entropy of a two parameter Lomax distribution under a generalized Type-II hybrid censoring scheme are introduced. The maximum likelihood estimation for the entropy is provided and the Bayes estimation for the residual entropy is obtained. Simulation studies to assess the performance of the estimates with different sample sizes are described, ﬁnally conclusions are discussed.


Introduction
Censoring schemes are commonly used in reliability qualification and reliability acceptance tests. Type-I censoring and Type-II censoring are the most popular and widely used in practice. While each Type has its merits, it has its drawbacks and limitations. Hybrid censoring schemes (HCS) are mixtures of Type-I and Type-II censoring and can be described as follows. Consider a life testing experiment in which a random sample of n units, from a distribution with a cumulative distribution function cdf F (x) and probability density function pdf f (x), are put on the test such that successive failure times are recorded as X (1:n) , . . . , X (n:n) . The life-test is terminated either at a prefixed number of failures r (r < n) or a prefixed time, T , whichever is reached first. Let T * = min(T, X r ) be the actual time of the termination of the experiment. Referred to this scheme as Type-I hybrid censoring scheme (Type-I HCS). The disadvantage of Type-I HCS is that there is a possibility that very few failures may occur before time T . In that case, the efficiency of the estimator(s) might be low. For this reason, [Childs et al,2] proposed a new HCS that terminate the experiment at the random time T * = max(T, X r ), and is referred to as Type-II hybrid censoring scheme (Type-II HCS). In this scheme, more than r failures may be observed at the termination time, and this will result in a more efficient estimation procedure. On other hand, the termination time is a random variable, which is consider a disadvantage. So, [Chandrasekar et al,3] proposed the generalized Type-II HCS (G Type-II HCS) as a modification of Type-II HCS. The reason behind the proposed modification is to fix the underlying disadvantages inherent in both Type-I and Type-II HCS. G Type-II HCS can be described as follows. Assume n items are put on a test. Fix r ∈ {1, 2, . . . , n}, and T 1 , T 2 ∈ (0, ∞), where T 1 < T 2 . Then we are faced with one of three situations: -If the r th failure occurs before the time point T 1 , terminate the experiment at T 1 .
-If the r th failure occurs between T 1 and T 2 , terminate the experiment at x r . -Otherwise, terminate the experiment at T 2 . The following is a schematic diagram of the three possibilities Case I

Case II
Case III The likelihood functions for the three different cases are as follows, see [Chandrasekar et al,3].
D j denote the number of failures up to time T j , j ∈ (1, 2).
[ Shannon 5]defined the notion of entropy as a mathematical measure of information which measures the average reduction of uncertainty of random variable X following a discrete distribution. [Shannon 5] defined the entropy as Sometimes we write H X instead of H(f ) to indicate our interest in the entropy as a measure of information contained in the random variable X.
Consider a Lomax distribution with the cdf : pdf : and the reliability function: where λ is the scale parameter and α is a shape parameter.
The wider the spread of a distribution, the lower the value of its entropy. Conversely, the narrower the spread, the higher the value of the entropy. [Ebrahimi and Pellerey 14], considered a dynamic version of the classic Shannon entropy and defined the entropy of the residual lifetime by WhereF (x) = P r(X > t) is the reliability function of X. In particular, H(f, 0) = H(X), the residual entropy is used to measure the ageing and characterize, classify and order lifetime distributions see [ From equation (7) and after some calculations, the residual entropy function associated with Lomax distribution is: In the analysis of lifetime data, the quality of its statistical procedure used depends, to a high extent, on the assumed probability model or distribution. The proper choice of a parametric model provides useful information about reliability measures and failure characteristics of devices under study resulting in sound conclusions and decisions. Among models successfully use in analysing a wide range of real-world situations is the Lomax model. It has been extensively used in reliability modelling and life testing, see [Balkema and de Haan 1]. Lomax distribution arises as a Limiting distribution of residual lifetime at old age, see [Balkema and de Haan 1]. It is a continuous long-tailed distribution proposed as a heavy-tailed alternative to the exponential, Weibull, and longtailed gamma distributions, see [Bryson 12]. Furthermore, Lomax distribution is related to Burr family of distributions, see [Tadikamalla 19], and can be obtained as a special case from compound gamma distributions, see [Durbey 21]. This distribution plays a fundamental role in statistics, probability and some related areas, such as socioeconomics, engineering, medical and biological sciences, and actuarial science, see [Johnson et al. 17] and [Kleiber and Kotz 6].
The primary purpose of the present paper is to estimating the entropy and residual entropy of a Lomax distribution under G Type-II HCS. To the best of our knowledge, no attempt has been made to estimate the entropy and residual entropy of a Lomax distribution under G Type-II HCS. In Section 2, we obtain the maximum likelihood estimation. In Section 3, we obtain the Bayes estimation for the residual entropy. Simulation studies to assess the performance of the estimates with different sample sizes are described in Section 4, while conclusions are discussed in Section 5.

The maximum likelihood estimation
Assume n items with lifetime distribution that are i.i.d. Lomax random variables with cdf (3) and pdf (4), put on a test. Assume also that we subject the experiment to G Type-II HCS described in section 1. Let D denotes the number of failures that occur by time point T , then based on the G Type-II HCS, the likelihood functions of α and λ are given by: Therefore, Cases I, II and III can be combined and can be written as: where U = D 1 and Q = T 1 for Case I, U = r and Q = x r for Case II and U = D 2 and Q = T 2 for Case III.
The logarithm of (9) can be written as: Taking derivatives with respect to α and λ of (10) From (11) we obtain: Using (13) in (12) λ can be written as: The estimate of α and λ for case I, case II and III in a G Type-I HCS can be written as: where . where ) , ) , ) , Equation (15) can be solved numerically to obtain λ and substitute it in equation (14) to obtain α. We used Mathematica 11 to solve equation (15). After we obtainα andλ we use equation (6) to get the MLEs of the entropy Lomax distribution as:Ĥ 3 The Bayes estimation for the residual entropy for the Lomax distribution under generalized Type II hybrid censored sample [Ashour et al 20] considered the Bayesian inference for the two-parameter α and λ on a HCS. They assumed that α and λ have the joint prior density as π(α, λ) = π(α)π(λ) Let X = (x 1 , . . . , x U ) be the observed sample by the end the experiment. Based on the above joint prior distribution, the joint density of the α, λ and X can be written as follows.
The posterior distribution of α and λ, given X, is obtained as: Useing Equation (9) and Equation (16) the joint posterior density functions of α and λ under G Type-II HCS will be: The Bayes estimation of α and λ is obtained as: Using Equation (8) and Equation (17) the Bayes estimation for the residual entropy for the Lomax distribution under G Type-II HCS will be: H B (x) will be computed numerically after substitutingλ andα obtained from solving Equations (14) and (15).
The estimation of the entropy of the G-Type-II HCS is presented in Table 1.

Simulation Study
In this section, we present the results of two simulation studies that were carried out for the following. Again Mathematica 11 will be used to compute Equation (20).
1-To assess the performance of M LE estimation of the Lomax entropy. The assessment is carried out through measures of the entropy and the mean square error (M SE) of entropy under different choices of the G Type-II HCS and different combinations of n, T 1 , T 2 and r values. In each case the process was replicated N = 1000 times for a particular G Type-II HCS. The MLEs for the entropy were obtained as described before in Section 2. We were able to express α in terms of λ as in formula (9) therefore obtaining the M LE estimates is attained by solving the equation (10). The computational system Mathematica 11 was used to solve equation (10) in λ. We substituted these values in (9) to obtain the values of α. These values of α and λ constitute their maximum likelihood estimates. We substituted these values in (13) to obtain the MLE estimates of the entropy of the Lomax distribution under G Type-II HCS in Tables 2 and 3. 2-To assess the performance of the Bayes estimation for the residual entropy of the Lomax distribution. The assessment is carried out through measures of the residual entropy, Relative Bias (RB) and the mean square error (M SE) of residual entropy under different choices of the G Type-II HCS and different combination of n, T 1 , T 2 and r values. In each case process was replicated N = 1000 times for a particular G Type-II HCS. The Bayes estimation for the residual entropy were obtained as described before in formula (19) in Tables 4  and 5.
In Table 2, when λ is fixed, and α is increase the RB and MSE are decrease, and when fixed n, r and T 1 , and the time T 2 is increase we observe the M SE and RB are increases, also, when the sample size n increase, the values of H(x), M SE and RB are decreases.
In Table 3, when α is fixed, and λ is increase the RB and M SE are decrease, and when the sample size n increase, the values ofĤ(x), M SE and RB are decreases.
In Table 4, when λ is fixed, and α is increase the SD is increase, the RB and M SE are decrease, and when the time T 2 is increase the SD is increase, the RB and M SE are decrease.
In Table 5, for a fixed n, α and r is increase theH(B) is decrease. When n, r are increases the SD is decrease, when T 1 is increase the SD is decrease. Also, T 2 is fixed and n, T 1 are increase, the M SE and RBias are decrease.

Summary
In this article, entropy estimates for the Lomax distribution were computed using the M LE of α and λ based on G Type-II HCS. The estimates were assessed in terms of their mean square error (M SE), and relative bias (RBias) of entropy. Also, we performed simulation studies with different sample sizes focusing on the residual entropy estimate of the Lomax distribution under the G Type-II HCS. The estimates we were assessed in terms of their standard deviation (SD), the mean square error (M SE), and relative bias (RB), of residual entropy under different chaise of the G Type-II HCS.  Table 2. Entropy estimatesĤ and its MSEs and relative bias, for selected values of n,T 1 ,T 2 and r.