Some Characterization Results on Dynamic Cumulative Residual Tsallis Entropy

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


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  as  () = − ∫ ∞ 0  () log ( ()) . ( For a residual lifetime   = [ −  |  > ], where  > 0, Ebrahimi [2] defined an entropy as a dynamic measure of uncertainty which is given by Alternative 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 as and called it the cumulative residual entropy (CRE).
For a residual lifetime   = [ −  |  > ], Asadi and Zohrevand [13] defined the dynamic measure of CRE as and 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  as 2

Journal of Probability and Statistics
Tsallis [18] defined the generalized entropy of order  as Both entropies ( 5) and ( 6) approach the Shannon entropy as  → 1.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.

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

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 𝐹 is increasing (decreasing) DCRTE if and only if for all
Proof.The proof of the theorem directly follows from (13).

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 by The 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.
In the following example we study the empirical cumulative Tsallis entropy for exponentially distributed random samples.
Hence from (60) we obtain the mean and variance of the empirical cumulative Tsallis entropy as follows: (62) Based on the empirical cumulative Tsallis entropy for random samples from exponential distribution with mean 1, we tabulated the values for mean and variance in Tables 1  and 2, respectively.It may be observed from tabulated data that the mean of empirical cumulative Tsallis entropy, that is, (  ( F )), is decreasing for different values of , whereas the variance of empirical cumulative Tsallis entropy, that is, Var(  ( F )), is increasing for different values of .

Conclusion
The dynamic generalized information measure based on cumulative distribution function is more stable rather than the density function.In this paper, we proposed the dynamic cumulative residual Tsallis entropy which is found to be monotonic in nature.Based on the proposed DCRTE, we characterized some well known lifetime distributions such as exponential, Weibull, Pareto, and finite range distributions which play a vital role in reliability modeling.Here we proposed weighted dynamic cumulative residual Tsallis entropy and examine its application in relation to weighted and equilibrium models.Finally, we introduce empirical cumulative Tsallis entropy for empirical samples.

Table 1 :
Mean of empirical cumulative Tsallis entropy for different values of  and .

Table 2 :
Variance of empirical cumulative Tsallis entropy for different values of  and .