# Some Characterization Results on Dynamic Cumulative Residual Tsallis Entropy

Authors: Madan Mohan Sati; Nitin Gupta
Journal: Journal of Probability and Statistics
 (2015)

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## 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.


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## Body

## 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.LetX be a random variable having probability density function f(x), survival function F¯(x), and hazard rate r(x)=f(x)/F¯(x); Shannon defined entropy for a random variable X as(1)HX=-∫0∞fxlog⁡fxdx.For a residual lifetime Xt=[X-t∣X>t], where t>0, Ebrahimi [2] defined an entropy as a dynamic measure of uncertainty which is given by(2)HX;t=-∫t∞fxF¯tlog⁡fxF¯tdx.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(3)ξX=-∫0∞F¯xlog⁡F¯xdxand 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 lifetimeXt=[X-t∣X>t], Asadi and Zohrevand [13] defined the dynamic measure of CRE as(4)ξX;t=-∫t∞F¯xF¯tlog⁡F¯xF¯tdxand called it as dynamic cumulative residual entropy (DCRE).Abbasnejad et al. [14] proposed dynamic survival entropy of order α 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 β 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 α as(5)HαX=11-αlog⁡∫0∞fxαdx,α>0,α≠1.Tsallis [18] defined the generalized entropy of order α as(6)SαX=1α-11-∫0∞fxαdx,α>0,α≠1.Both entropies (5) and (6) approach the Shannon entropy as α→1. There is a close relationship between the Renyi entropy and the Tsallis entropy given as (7)HαX=1α-1log⁡1-α-1SαX.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 Section2, we propose a cumulative residual entropy based on Tsallis entropy of order α 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 variableX with survival function (sf) F¯(x), the cumulative residual entropy of order α denoted by ηα(X) is defined as(8)ηαX=1α-11-∫0∞F¯xαdx,α>0,α≠1.Lettingα→1, (8) gives(9)limα→1⁡ηαX=ξX.Let us consider a unit whose random life is represented by random variable X. Let the unit survive up to time t; then the information based on entropy of the random variable X may not be useful. In that case we may consider dynamic (time dependent) information based on the entropy of the random variable Xt=[X-t∣X>t]. The random variable Xt=[X-t∣X>t] has the survival function(10)F¯tx=F¯xF¯t,wherex>t1,otherwise.Definition 2.
For a random variableXt with survival function F¯(x), the dynamic cumulative residual entropy of order α denoted by ηα(X;t) is defined as(11)ηαX;t=1α-11-∫t∞F¯xF¯tαdx,α>0,α≠1.The following theorem shows that the dynamic cumulative residual entropy determines the survival functionF¯(x) uniquely.Theorem 3.
Let the nonnegative random variableX have the density function f(x), the survival function F¯(x), and the hazard rate r(x). Assume that ηα(X;t)<∞;t≥0;∀α>0(≠1). Then for each α,ηα(X;t) (whereas ηα′(X;t)≠0) uniquely determines the survival function F¯(t).Proof.
Consider the dynamic cumulative residual Tsallis entropy of orderα as (11). Therefore, we have(12)α-1ηαX;t=1-∫t∞F¯xαdxF¯tα.Differentiating (12) with respect to t, we have (13)α-1ηα′X;t=1-αft∫t∞F¯xαdxF¯tα+1=1+αrtα-1ηαX;t-1,where the last step follows from (12) and r(x)=f(x)/F¯(x) is the hazard rate function of random variable X. Consider two survival functions F¯1(t) and F¯2(t) having dynamic entropies as ηα(X1;t) and ηα(X2;t) and hazard rate functions r1(t) and r2(t), respectively. Letting ηα(X1;t)=ηα(X2;t) implies that ηα′(X1;t)=ηα′(X2;t) which further implies(14)α-1ηα′X1;t=α-1ηα′X2;t,i.e.,1+αr1tα-1ηαX1;t-1=1+αr2tα-1ηαX2;t-1;last step follows using (13). Now as ηα(X1;t)=ηα(X2;t), therefore, (14) reduces to r1(t)=r2(t) or equivalently F¯1(t)=F¯2(t).Now we provide some characterization results in terms of relationship between DCRTE and hazard rate functionr(t).Theorem 4.
LetX be a nonnegative continuous random variable with survival function F¯(t) hazard rate function r(t) and dynamic cumulative residual Tsallis entropy ηα(X;t); then the relationship,(15)α-1ηα′X;t=crt,α>0,α≠1,gives survival function F¯(t)=exp⁡-∫0t(1/K-2αcx)dx, where K is the constant of integration and characterizes (i)
an exponential distribution forK>0 and c=0 with survival function(16)F¯t=e-λt,λ>0,t>0and(ii)
Weibull distribution forK=0 and c<0 with survival function(17)F¯t=e-γtβ,γ>0,β=12,t>0.Proof.
Under the assumption that (15) holds; using (13), we have (18)1+αrtα-1ηαX;t-1=crtor equivalently, using (12),(19)F¯tft-c=α∫t∞F¯αxdxF¯αt.Differentiating (19) with respect to t, we have(20)-1+F¯tft2f′t=α-1+αftF¯t∫t∞F¯αxdxF¯αt,-1+1rt·f′tft=α-1+rtF¯tft-c,1+1rt·f′tft=αcrt,αcrt2-rt=f′tft,αcrt2=ddtlog⁡rt.Now letting log⁡r(t)=y, that is, r(t)=ey, (20) reduces to (21)αce2y=dydt,i.e.,e-2ydy=αcdt.Integrating on both sides, we have(22)e-2y=K-2αct,where K is a constant. Therefore, r(t)=ey provides(23)rt=1K-2αct.Since the hazard rate uniquely determines the survival function using the relationship F¯(t)=exp⁡-∫0tr(x)dx, consider the following case.(i)
ForK>0 and c=0, F¯t=exp⁡-∫0t(1/K)dx or equivalently F¯(t)=e-λt, where λ=1/K>0.(ii)
Forc≠0, survival function F¯(t)=e-K/αceK-2αct/αc. Further, for K=0 and c<0(c=-a,a>0), F¯(t)=e-(γt)β, where γ=2/aα>0, β=1/2>0.
Converse. We assume that random variable X is distributed exponentially with pdf λe-λx,λ>0,x≥0. Using (12), we have α-1ηα(X;t)=1-1/αλ from which (15) follows with c=0.
WhenX is distributed as Weibull with pdf βγβxβ-1e-(γx)β,γ>0, β=1/2, x≥0. Using (12), we have α-1ηα(X;t)=1-(2/α2γ)1+α(γt)1/2 which on differentiation yields α-1ηα′(X;t)=-1/α(γt)1/2=cr(t) with r(t)=(1/2)(γ/t)1/2 and c=-a<0.The following theorem characterize the distributions using relationship between DCRTE and mean residual life (MRL)mF(t).Theorem 5.
LetX be a nonnegative continuous random variable with survival function F¯(t), MRL mF(t), and dynamic cumulative residual Tsallis entropy of order α, ηα(X;t); if(24)α-1ηαX;t=1-KmFt,α>0,α≠1,then X has (i)
an exponential distribution iffK=1/α,(ii)
a Pareto distribution iffK<1/α,(iii)
a finite range distribution iffK>1/α.Proof.
(i) If random variableX denotes an exponential distribution, then it has pdf, sf, and MRL, respectively, as (25)ft=λe-λt,λ>0,t>0,F¯t=e-λt,mFt=1λ.Therefore, after simplification, using (12), (26)α-1ηαX;t=1-KmFt,where K=1/α and mF(t)=1/λ.
(ii) If random variableX denotes the Pareto distribution, then it has pdf, sf, and MRL, respectively, as (27)ft=1+ta-a-1,a>1,t>0,F¯t=1+ta-a,mFt=a+ta-1.Therefore, after simplification, using (12), (28)α-1ηαX;t=1-KmFt,where K=a-1/αa-1<1/α if α>1 and mF(t)=a+t/a-1.
(iii) If random variableX denotes the finite range distribution, then it has pdf, sf, and MRL, respectively, as (29)ft=b1-tb-1,b>0,0<t<1,F¯t=1-tb,mFt=1-tb+1.Therefore, after simplification, using (12), (30)α-1ηαX;t=1-KmFt,where K=b+1/αb+1>1/α if α>1 and mF(t)=1-t/b+1.
Converse. Let (24) holds. Using (12), we get(31)KmFt=∫t∞F¯xαdxF¯tα;differentiating (31) with respect to t, we obtain(32)KmF′t=-1+αrt∫t∞F¯xαdxF¯tα=-1+αrtKmFt.Using the relation between mean residual life and hazard rate, that is, r(t)mF(t)=1+mF′(t), we have(33)mF′t=Kα-1K1-α.Integrating (32) on both sides with respect to t over (0,x), we get(34)mFx=Kα-1K1-αx+mF0.Equation (33) is linear in MRL function mF(x) of continuous random variable X, if and only if the underlying distribution is exponential (K=1/α), Pareto (K<1/α), or finite range (K>1/α); 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 DCRTEηα(X;t).Definition 6.
The distribution functionF is said to be increasing (decreasing) DCRTE, denoted by IDCRTE (DDCRTE), if ηα(X;t) is an increasing (decreasing) function of t.The following theorem gives the necessary and sufficient condition forηα(X;t) to be increasing (decreasing) DCRTE.Theorem 7.
The distribution functionF is increasing (decreasing) DCRTE if and only if for all t≥0(35)ηαX;t≥≤1α-11-1αrt,∀α>0≠1⁡.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.
LetX and Y be two nonnegative absolutely continuous random variables with survival functions F¯(t) and G¯(t) and hazard rate functions rF(t) and rG(t), respectively. If X≥hrY, that is, rF(t)≤rG(t) for all t≥0, then (36)ηαX;t≤≥ηαY;t,∀α>10<α<1.Proof.
The assumption thatrF(t)≤rG(t) implies F¯Xt(x)≥G¯Xt(x):(37)F¯xF¯tα≥G¯xG¯tα,∀α>0,1-∫t∞F¯xF¯tαdx≤1-∫t∞G¯xG¯tαdx.For α>1(38)1α-11-∫t∞F¯xF¯tαdx≤1α-11-∫t∞G¯xG¯tαdx,ηαX;t≤ηαY;t.For 0<α<1(39)11-α∫t∞F¯xF¯tαdx-1≥11-α∫t1∞G¯xG¯tαdx-1,ηαX;t≥ηαY;t.Hence (40)ηαX;t≤≥ηαY;t,∀α>10<α<1.In the following lemma, we discuss the effect of linear transformation on DCRTE.Lemma 9.
For any nonnegative random variableX, let Z=aX+b, where a>0 and b≥0; then (41)ηαZ;t=1-aα-1+aηαX;t-ba,t≥b.Proof.
The result follows using the fact thatF¯aX+b(x)=F¯X(x-b)/a, x∈R and (11).


## 4. Weighted Dynamic Cumulative Residual Tsallis Entropy

LetX be a random variable with probability density function f(t) and survival function F¯(t). Let XW be weighted random variable associated with X and their probability density function and survival function denoted by fw(t) and F¯w(t), given by(42)fwt=wtfXtEwX,F¯wt=EwX∣X>tF¯XtEwX;0<EwX<∞.The weighted dynamic cumulative residual Tsallis entropy denoted by ηα(W;t) is proposed as(43)ηαW;t=1α-11-∫t∞F¯wxF¯wtαdt,α>0≠1.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.
IfE(w(X)∣X>x)≤E(w(X)∣X>t) for all x≥t, then ηα(W;t)≤(≥)ηα(X;t) for 0<α<1(α>1). If E(w(X)∣X>x)≥E(w(X)∣X>t) for all x≥t, then ηα(W;t)≥(≤)ηα(X;t) for 0<α<1(α>1).Proof.
Rewriting (43), we have(44)ηαW;t=1α-11-∫t∞F¯wxF¯wtαdx=1α-11-∫t∞EwX∣X>xF¯xαEwX∣X>tF¯tαdx=1α-11-∫t∞EwX∣X>xEwX∣X>tαF¯αxF¯αtdx.
IfE(w(X)∣X>x)≤E(w(X)∣X>t) for all x≥t, that is, E(w(X)∣X>x)/E(w(X)∣X>t)≤1 for all x≥t, then we have(45)∫t∞EwX∣X>xEwX∣X>tαF¯αxF¯αtdx≤∫t∞F¯αxF¯αtdx,which further implies the following: for all 0<α<1(α>1), (46)11-α∫t∞EwX∣X>xEwX∣X>tαF¯αxF¯αtdx≤≥11-α∫t∞F¯αxF¯αtdx,1α-1+11-α∫t∞EwX∣X>xEwX∣X>tαF¯αxF¯αtdx≤≥1α-1+11-α∫t∞F¯αxF¯αtdx,1α-1-1α-1∫t∞EwX∣X>xEwX∣X>tαF¯αxF¯αtdx≤≥1α-1-1α-1∫t∞F¯αxF¯αtdx,1α-11-∫t∞EwX∣X>xEwX∣X>tαF¯αxF¯αtdx≤≥1α-11-∫t∞F¯αxF¯αtdx.Now using (44) and (11), from (46) we get(47)ηαW;t≤≥ηαX;t,∀0<α<1α>1.Similarly if E(w(X)∣X>x)≥E(w(X)∣X>t) for all x≥t, then(48)ηαW;t≥≤ηαX;t,∀0<α<1α>1.Particularly, when the weight function is defined asw(t)=F¯(t)/f(t), the corresponding weighted distribution becomes the equilibrium distribution. Let XE be a random variable corresponding to equilibrium distribution with probability density function fE(t)=F¯(t)/μ, t>0, and survival function F¯E(t)=(rt/μ)F¯(t), where μ=E(X)<∞; then dynamic cumulative residual Tsallis entropy of XE is proposed as(49)ηαE;t=1α-11-∫t∞F¯ExF¯Etαdx,α>0≠1.Theorem 11.
IfF¯(t) has decreasing hazard rate, then ηα(E;t)≥(≤)ηα(X;t) for 0<α<1(α>1). If F¯(t) has increasing hazard rate, then ηα(E;t)≤(≥)ηα(X;t) for 0<α<1(α>1).Proof.
Note that(50)ηαE;t=1α-11-∫t∞F¯ExF¯Etαdx,α>0≠1=1α-11-∫t∞rxrtαF¯αxF¯αtdx.Now if F¯(t) has decreasing hazard rate, we have r(x)≥r(t)∀x≤t; that is, r(x)/r(t)≥1∀x≤t; then we have(51)∫t∞rxrtαF¯αxF¯αtdx≥∫t∞F¯αxF¯αtdx,which further implies the following: for all 0<α<1, (52)11-α∫t∞rxrtαF¯αxF¯αtdx≥11-α∫t∞F¯αxF¯αtdx,1α-1+11-α∫t∞rxrtαF¯αxF¯αtdx≥1α-1+11-α∫t∞F¯αxF¯αtdx,1α-11-∫t∞rxrtαF¯αxF¯αtdx≥1α-11-∫t∞F¯αxF¯αtdx.Now using (50) and (11), we have(53)ηαE;t≥ηαX;t,∀0<α<1.Also when α>1, using (51), we have(54)ηαE;t≤ηαX;t,∀α>1.Similarly it follows that if F¯(t) has increasing hazard rate, then(55)ηαE;t≤≥ηαX;t,∀0<α<1α>1.


## 5. Empirical Cumulative Tsallis Entropy

LetX1,X2,…,Xn be nonnegative, absolutely continuous, independent, and identically distributed random variables with distribution function F(x). According to (8), we define the empirical cumulative Tsallis entropy as(56)ηαF^n=1α-11-∫0∞F^nxαdx,α>0,α≠1,where (57)F^nx=1n∑i=1nIXi≤x,x∈R,is the empirical distribution of the sample and X(1)<X(2)<⋯<X(n) are the order statistic. Equation (56) can be expressed as(58)ηαF^n=1α-11-∑j=1n-1∫XjXj+1F^nxαdx.Recalling that (59)F^nx=0,x<X1,jn,Xj≤x<Xj+1,j=1,2,…,n-11,x≥Xnfrom (58), we get(60)ηαF^n=1α-11-∑j=1n-1Uj+1jnα,α>0,α≠1,where (61)Ui=Xi-Xi-1,i=1,2,…,nare the sample spacings (cf. [9, 28]).In the following example we study the empirical cumulative Tsallis entropy for exponentially distributed random samples.Example 12.
LetX1,X2,…,Xn be a random sample of exponentially distributed random variables with parameter λ. By Pyke [29], the sample spacings are independent, with U(j+1) exponentially distributed with parameter λ(n-j). Hence from (60) we obtain the mean and variance of the empirical cumulative Tsallis entropy as follows:(62)EηαF^n=1α-11-∑j=1n-11λn-jjnα,α>0,α≠1,Var⁡ηαF^n=1α-12∑j=1n-11λ2n-j2jn2α,α>0,α≠1.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 Tables1 and 2, respectively. It may be observed from tabulated data that the mean of empirical cumulative Tsallis entropy, that is, EηαF^n, is decreasing for different values of n, whereas the variance of empirical cumulative Tsallis entropy, that is, VarηαF^n, is increasing for different values of n.Table 1
Mean of empirical cumulative Tsallis entropy for different values ofn and α.
n
α
=
0.25
α
=
0.5
α
=
0.75
α
=
0.99
α
=
1.01
α
=
1.25
α
=
1.5
10
−8.63
−13.43
−27.64
−706.51
707.69
28.83
14.65
50
−59.81
−90.24
−181.31
−4549.44
4550.71
182.59
91.55
100
−125.56
−188.86
−378.55
−9480.62
9481.90
379.84
190.18
500
−656.74
−985.64
−1972.13
−49320.07
49321.36
1973.43
986.97
1000
−1322.48
−1984.26
−3969.36
−99250.80
99252.09
3970.66
1985.59Table 2
Variance of empirical cumulative Tsallis entropy for different values ofn and α.
n
α
=
0.25
α
=
0.5
α
=
0.75
α
=
0.99
α
=
1.01
α
=
1.25
α
=
1.5
10
2.46
5.03
18.45
10673.12
10606.53
15.77
3.66
50
2.80
6.14
23.97
14664.52
14638.81
22.95
5.62
100
2.86
6.33
24.98
15420.17
15404.57
24.35
6.02
500
2.91
6.52
25.97
16179.85
16175.45
25.79
6.43
1000
2.92
6.55
26.13
16300.87
16298.39
26.03
6.49


## 6. 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.



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