Abstract
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.
1. 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 , and hazard rate The generalized entropy of is given by (see Varma [5]) Measure (1) reduces to Renyi entropy (see Renyi [3]) when and reduces to Shannon entropy (see Shannon [1]) when and 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 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 Also (2) reduces to Renyi’s residual entropy (see Asadi et al. [7]) when and reduces to residual entropy (see Ebrahimi [6]) when and 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 [8–11], 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 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 is said to be an upper record value if its value is greater than that of all the previous observations. For convenience, we denote and, for , . Then, is called a sequence of upper record values. The probability density function and the survival function of the th upper record value are given by respectively, where and , , . Note that 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 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.
2. 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 denote the th upper record value from a sequence of independent observations from . 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 .
Theorem 2. The generalized residual entropy of can be expressed as where .
Proof. From (2), (3), and (4) and using the transformation , we obtain since . This completes the proof.
As a consequence of Theorem 2, we get the following remark.
Remark 3. Let denote the th upper record value from a sequence of independent observations from a standard exponential distribution. Then, where .
In Table 1, we obtain expressions of for Weibull and Pareto distributions. It is easy to show that and are increasing functions in (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].
Theorem 4. The th upper record value is increasing (decreasing) generalized residual entropy (IGRE (DGRE)) if is IGRE (DGRE).
Note that for th and th upper record values, is increasing in (see Kochar [19]) and hence Therefore, the following corollary immediately follows from Theorem 4.
Corollary 5. The th upper record value is IGRE (DGRE) if th upper record value is IGRE (DGRE).
The following theorem provides bounds for the generalized residual entropy of the th upper record . We omit the proof as it follows in an analogous approach similar to that of Theorem 11 given in Kayal [11].
Theorem 6. (a) Let be finite. If is the mode of , then where
(b) Let , where is the mode of the distribution with density function . Also denote the th upper record value from a sequence of observation from . Then, for ,
As an application of Theorem 6, we obtain bounds of Pareto and exponential distributions presented in Table 2. The following theorem gives the monotone behaviour of the generalized residual entropy of upper record values in terms of .
Definition 7. Let and be two nonnegative random variables with survival functions and , respectively. Then, is said to be smaller than in the usual stochastic ordering, denoted by , if , for all .
Theorem 8. Let be a sequence of upper record values from a distribution with cumulative distribution function and probability density function . Also let be an increasing function. Then, is increasing (decreasing) in for .
Proof. From (8), we have Moreover, for , is an increasing (decreasing) function in Therefore, for and , we have for . Hence, from Theorem 12 of Kayal [11], it can be proved that is increasing (decreasing) for . Now, along the lines of the proof of Theorem 3.7 of Zarezadeh and Asadi [20], the proof follows. This completes the proof of the theorem.
3. 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.
Definition 9. A random variable with hazard rate is said to have a decreasing (increasing) failure rate (DFR (IFR)), if is decreasing (increasing) in .
Theorem 10. Let and be two random variables with cumulative distribution functions and , probability density functions and , survival functions and , and hazard rates and , respectively. If , for all , and either or is DFR, then , for .
Proof. The proof follows along the lines of that of Theorem of Asadi et al. [7].
Theorem 11. Under the assumptions of Theorem 10, if , for all , and either or is DFR, then , for .
Proof. Proof follows from that of Theorem of Asadi et al. [7].
Theorem 12. (a) Suppose , or , is decreasing. If or is DFR, then, for all , , for .
(b) Suppose , or , is increasing. If or is DFR, then, for all , , for .
Proof. It is not difficult to see that , for all , 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 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 , respectively. The following corollary is a consequence of Theorem 12.
Corollary 13. Let be DFR. Then, for all ,(a) for ;(b) for .
4. Estimation
In this section, we discuss the problem of estimation of the generalized residual entropy of a statistical distribution based on upper record values. Here, we consider exponential distribution. It has various applications in practice. Let follow exponential distribution with mean . Then, from (2), we obtain Based on the upper record values, the maximum likelihood estimator (mle) of can be obtained as , where is the th upper record value. Now, applying invariance property, we obtain the mle of as Also the uniformly minimum variance unbiased estimator (umvue) of can be obtained as where is a digamma function. To illustrate the estimation techniques developed in this section, we consider simulated data from exponential distribution with mean In this purpose, we use Monte-Carlo simulation.
Example 14. In this example, we consider a simulated sample of size from the exponential distribution with mean . The simulated upper records are as follows: Based on these upper record values, we have and when and .
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.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.