Abstract

We propose a generalized cumulative residual information measure based on Tsallis entropy and its dynamic version. We study the characterizations of the proposed information measure and define new classes of life distributions based on this measure. Some applications are provided in relation to weighted and equilibrium probability models. Finally the empirical cumulative Tsallis entropy is proposed to estimate the new information measure.

1. Introduction

Shannon [1] introduced the concept of entropy which is widely used in the fields of communication theory, information theory, physics, economics, probability and statistics, and so forth.

Let be a random variable having probability density function , survival function , and hazard rate ; Shannon defined entropy for a random variable asFor a residual lifetime , where , Ebrahimi [2] defined an entropy as a dynamic measure of uncertainty which is given byAlternative entropy was introduced by Rao et al. [3] which is based on survival function instead of probability density function. Rao et al. [3] defined entropy asand called it the cumulative residual entropy (CRE).

For the study of the properties and applications of CRE, we refer to Asadi and Ebrahimi [4], Asadi et al. [5], Rao [6], Drissi et al. [7], Gupta [8], Di Crescenzo and Longobardi [9], Navarro et al. [10], Gupta and Taneja [11], and Taneja and Kumar [12].

For a residual lifetime , Asadi and Zohrevand [13] defined the dynamic measure of CRE asand called it as dynamic cumulative residual entropy (DCRE).

Abbasnejad et al. [14] proposed dynamic survival entropy of order and gave the relation of dynamic survival entropy with the mean residual life function. Further Sunoj and Linu [15] defined cumulative residual Renyi entropy of order and its dynamic version. Recently Kumar and Taneja [16] defined generalized cumulative residual information measure and its dynamic version based on Varma’s entropy function.

Renyi [17] defined the generalized entropy of order asTsallis [18] defined the generalized entropy of order asBoth entropies (5) and (6) approach the Shannon entropy as . There is a close relationship between the Renyi entropy and the Tsallis entropy given as It may be noted that Tsallis entropy is a nonextensive entropy and it is nonlogarithmic. However, Renyi entropy is an extensive entropy which is the major difference between them (cf. [19, 20]).

Tsallis entropy plays a central role in different areas such as physics, chemistry, biology, medicine, and economics. Cartwright [21] proposed applications of Tsallis entropy in various fields such as describing the fluctuation of magnetic field in solar wind, signs of breast cancer in mammograms, atoms in optical lattices, analysis in magnetic resonance imaging (MRI).

The aim of the paper is to study the cumulative residual information based on nonextensive entropy measures and characterize some well known lifetime distributions and probability models. The empirical form of this information measure is useful for real data problems. In Section 2, we propose a cumulative residual entropy based on Tsallis entropy of order and its dynamic version. Also, we study some characterization results using the relationship of dynamic cumulative residual Tsallis entropy (DCRTE) with hazard rate function and mean residual life function. In Section 3, we define new classes of life distribution based on these measures. In Section 4, we propose the weighted form of DCRTE and study its various properties. In Section 5 we introduce the empirical cumulative Tsallis entropy and express it in terms of the sample spacings. In order to study the empirical cumulative Tsallis entropy, an example is also being provided.

2. Dynamic Cumulative Residual Tsallis Entropy (DCRTE)

In this section, we define the cumulative residual Tsallis entropy and dynamic cumulative residual Tsallis entropy. We also give some characterization results of well known distributions in terms of DCRTE.

Definition 1. For a random variable with survival function (sf) , the cumulative residual entropy of order denoted by is defined as

Letting , (8) givesLet us consider a unit whose random life is represented by random variable . Let the unit survive up to time ; then the information based on entropy of the random variable may not be useful. In that case we may consider dynamic (time dependent) information based on the entropy of the random variable . The random variable has the survival function

Definition 2. For a random variable with survival function , the dynamic cumulative residual entropy of order denoted by is defined as

The following theorem shows that the dynamic cumulative residual entropy determines the survival function uniquely.

Theorem 3. Let the nonnegative random variable have the density function , the survival function , and the hazard rate . Assume that . Then for each (whereas ) uniquely determines the survival function .

Proof. Consider the dynamic cumulative residual Tsallis entropy of order as (11). Therefore, we haveDifferentiating (12) with respect to , we have where the last step follows from (12) and is the hazard rate function of random variable . Consider two survival functions and having dynamic entropies as and and hazard rate functions and , respectively. Letting implies that which further implieslast step follows using (13). Now as , therefore, (14) reduces to or equivalently .

Now we provide some characterization results in terms of relationship between DCRTE and hazard rate function .

Theorem 4. Let be a nonnegative continuous random variable with survival function hazard rate function and dynamic cumulative residual Tsallis entropy ; then the relationship,gives survival function , where is the constant of integration and characterizes (i)an exponential distribution for and with survival function and(ii)Weibull distribution for and with survival function

Proof. Under the assumption that (15) holds; using (13), we have or equivalently, using (12),Differentiating (19) with respect to , we haveNow letting , that is, , (20) reduces to Integrating on both sides, we havewhere is a constant. Therefore, providesSince the hazard rate uniquely determines the survival function using the relationship , consider the following case.(i)For and , or equivalently , where .(ii)For , survival function . Further, for and , , where , . Converse. We assume that random variable is distributed exponentially with pdf . Using (12), we have from which (15) follows with .
When is distributed as Weibull with pdf , , . Using (12), we have which on differentiation yields with and .

The following theorem characterize the distributions using relationship between DCRTE and mean residual life (MRL) .

Theorem 5. Let be a nonnegative continuous random variable with survival function , MRL , and dynamic cumulative residual Tsallis entropy of order , ; ifthen has (i)an exponential distribution iff ,(ii)a Pareto distribution iff ,(iii)a finite range distribution iff .

Proof. (i) If random variable denotes an exponential distribution, then it has pdf, sf, and MRL, respectively, as Therefore, after simplification, using (12), where and .
(ii) If random variable denotes the Pareto distribution, then it has pdf, sf, and MRL, respectively, as Therefore, after simplification, using (12), where if and .
(iii) If random variable denotes the finite range distribution, then it has pdf, sf, and MRL, respectively, as Therefore, after simplification, using (12), where if and .
Converse. Let (24) holds. Using (12), we getdifferentiating (31) with respect to , we obtainUsing the relation between mean residual life and hazard rate, that is, , we haveIntegrating (32) on both sides with respect to over , we getEquation (33) is linear in MRL function of continuous random variable , if and only if the underlying distribution is exponential (), Pareto (), or finite range (); refer to Hall and Wellner [22]. This completes the theorem.

3. New Class of Life Distributions

In this section, we define new class of life distribution based on the DCRTE .

Definition 6. The distribution function is said to be increasing (decreasing) DCRTE, denoted by IDCRTE (DDCRTE), if is an increasing (decreasing) function of .

The following theorem gives the necessary and sufficient condition for to be increasing (decreasing) DCRTE.

Theorem 7. The distribution function is increasing (decreasing) DCRTE if and only if for all

Proof. The proof of the theorem directly follows from (13).

In the following theorem, we give the hazard rate ordering using the DCRTE.

Theorem 8. Let and be two nonnegative absolutely continuous random variables with survival functions and and hazard rate functions and , respectively. If , that is, for all , then

Proof. The assumption that implies :For For Hence

In the following lemma, we discuss the effect of linear transformation on DCRTE.

Lemma 9. For any nonnegative random variable , let , where and ; then

Proof. The result follows using the fact that , and (11).

4. Weighted Dynamic Cumulative Residual Tsallis Entropy

Let be a random variable with probability density function and survival function . Let be weighted random variable associated with and their probability density function and survival function denoted by and , given byThe weighted dynamic cumulative residual Tsallis entropy denoted by is proposed as

The importance of weighted distribution can be seen in Patil and Rao [23], Gupta and Kirmani [24], Nair and Sunoj [25], Di Crescenzo and Longobardi [26], and Maya and Sunoj [27]. For the weighted distribution, we obtain the following result based on MRL ordering.

Theorem 10. If for all , then for . If for all , then for .

Proof. Rewriting (43), we haveIf for all , that is, for all , then we havewhich further implies the following: for all , Now using (44) and (11), from (46) we getSimilarly if for all , then