Generalized Residual Entropy and Upper Record Values

In this communication, we deal with a generalized residual entropy of record values and weighted distributions. Some results on monotone behaviour of generalized residual entropy in record values are obtained. Upper and lower bounds are presented. Further, based on this measure, we study some comparison results between a random variable and its weighted version. Finally, we describe some estimation techniques to estimate the generalized residual entropy of a lifetime distribution.


Introduction
There have been several attempts made by various researchers to generalize the Shannon entropy (see Shannon [1]) since its appearance in Bell System Technical Journal.For various properties and applications of the generalized entropy measures, we refer to Kapur [2], Renyi [3], Tsallis [4], and Varma [5].In this paper, we consider generalized residual entropy due to Varma [5].Let  be a nonnegative random variable representing the lifetime of a system with an absolutely continuous cumulative distribution function (), probability density function (), survival function ()(= 1 − ()), and hazard rate   ().The generalized entropy of  is given by (see Varma [5]) Measure (1) reduces to Renyi entropy (see Renyi [3]) when  = 1 and reduces to Shannon entropy (see Shannon [1]) when  = 1 and  → 1.We often find some situations in practice where the measure defined by (1) is not an appropriate tool to deal with uncertainty.For example, in reliability and life testing studies, sometimes it is required to modify the current age of a system.Here, one may be interested to study the uncertainty of the random variable   = [ −  |  ≥ ].The random variable   is dubbed as the residual lifetime of a system which has survived up to time  ≥ 0 and is still working.Analogous to Ebrahimi [6], the generalized entropy of the residual lifetime   is given by which is also known as the generalized residual entropy.It reduces to (1) when  = 0. Also (2) reduces to Renyi's residual entropy (see Asadi et al. [7]) when  = 1 and reduces to residual entropy (see Ebrahimi [6]) when  = 1 and  → 1.Based on the generalized entropy measures given in ( 1) and ( 2), several authors obtained various results in the literature.In this direction, we refer to Kayal [8][9][10][11], Kayal and Vellaisamy [12], Kumar and Taneja [13], and Sati and Gupta [14].In this paper, we study some properties and characterizations of the generalized residual entropy given by (2) based on the upper record values.Let {  :  = 1, 2, . ..} be a sequence of identically and independently distributed nonnegative random variables having an absolutely continuous cumulative distribution function (), probability density function (), and survival function ().An observation   in an infinite sequence  1 ,  2 , . . . is said to be an upper record value if its value is greater than that of all the previous observations.For convenience, we denote  1 = 1 and, for  ≥ 2,   = min{;  −1 <  and   >   −1 }.
respectively, where () = − ln () and Γ(; ) = ∫ ∞   −  −1 ,  > 0,  ≥ 0. Note that Γ(; ) is known as incomplete gamma function.Also the hazard rate of    is Record values have wide spread applications in real life.For the applications of record values in destructive testing of wooden beams and industrial stress testing, one may refer to Glick [15] and Ahmadi and Arghami [16].Record values are also useful in meteorological analysis and hydrology.For an extensive study of record values and applications, we refer to Arnold et al. [17].The paper is arranged as follows.
In Section 2, we obtain various properties on the generalized residual entropy.It is shown that the measure given by (2) of the th upper record value of any distribution can be expressed in terms of that of the th upper record value from (0, 1) distribution.Upper and lower bounds are obtained.Monotone behaviour of (2) based on the upper record values is investigated.In Section 3, based on (2), we study comparisons between a random variable and its weighted version.We describe some estimation techniques to estimate the generalized residual entropy of a life distribution in Section 4. Some concluding remarks have been added in Section 5. Throughout the paper, we assume that the random variables are nonnegative.The terms increasing and decreasing stand for nondecreasing and nonincreasing, respectively.

Main Results
In this section, we study several properties of the generalized residual entropy given by (2) based on the upper record values.First, we state the following lemma.The proof is straightforward hence omitted.

Lemma 1. Let 𝑋 𝑈 *
denote the th upper record value from a sequence of independent observations from (0, 1).Then, In the following theorem, we show that the generalized residual entropy of the upper record value    can be expressed in terms of that of   *  .Let  be a random variable having truncated gamma distribution with density function For convenience, we denote  ∼ Γ  (, ).
As a consequence of Theorem 2, we get the following remark.
In Table 1, we obtain expressions of  , (   ; ) for Weibull and Pareto distributions.It is easy to show that     ()/  () and    ()/() are increasing functions in  ≥ 0 (see Li and Zhang [18]).Therefore, we have the following theorem whose proof follows along the lines similar to those in Theorem 8 of Kayal [11].

PDF Bounds
and hence     ≤   +1 .Therefore, the following corollary immediately follows from Theorem 4.

Weighted Distributions
To overcome the difficulty to model nonrandomized data set in environmental and ecological studies, Rao [21] introduced the concept of weighted distributions.Let the probability density function of  be (), and let () be the nonnegative function with   = (()) < ∞.Also let   () and   (), respectively, be the probability density function and survival function of the weighted random variable   , which are given by The hazard rate of   is For some results and applications on weighted distributions, one may refer to Di Crescenzo and Longobardi [22], Gupta and Kirmani [23], Kayal [9], Maya and Sunoj [24], Navarro et al. [25], and Patil [26].In the present section, we obtain some comparison results based on the generalized residual entropy between a random variable and its weighted version.We need the following definition in this direction.Proof.It is not difficult to see that   () ≤    (), for all  ≥ 0, when either (() |  ≥ ) or () is decreasing.Now, the proof of part (a) follows from Theorem 10.Part (b) can be proved similarly.This completes the proof of the theorem.
Let  be a random variable with density function () and cumulative distribution function ().Also let   = () > 0 be finite.Denote the length biased version of  by   .Then, the probability density function of   is given by The random variable   arises in the study of lifetime analysis and various probability proportional-to-size sampling properties.Associated with a random variable , one can define another random variable   with density function This distribution is known as equilibrium distribution of .
The random variables   and   are weighted versions of  with weight function   () =  and   () = 1/  (), respectively.The following corollary is a consequence of Theorem 12.

Estimation
In where () is a digamma function.To illustrate the estimation techniques developed in this section, we consider simulated data from exponential distribution with mean 1.
In this purpose, we use Monte-Carlo simulation.
Example 14.In this example, we consider a simulated sample of size  = 5 from the exponential distribution with mean 0.5.

Concluding Remarks
In this paper, we consider generalized residual entropy due to Varma [5] of record values and weighted distributions.We obtain some results on monotone behaviour of this measure in upper record values.Some bounds are obtained.Further, some comparison results between a random variable and its weighted version based on the generalized residual entropy are studied.Finally, two estimators of the generalized residual entropy of exponential distribution have been described.
Based on the  upper record values, the maximum likelihood estimator (mle) of  can be obtained as δ =    /, where    is the th upper record value.Now, applying invariance property, we obtain the mle of  , (; ) as ) ln ( +  − 1) .(21)Also the uniformly minimum variance unbiased estimator (umvue) of  , (; ) can be obtained as