On extropy of past lifetime distribution
OOn extropy of past lifetime distribution
Osman Kamari
University of Human DevelopmentSulaymaniyah, Iraq
Francesco Buono
Universit`a di Napoli Federico IIItalySeptember 7, 2020
Abstract
Recently Qiu et al. (2017) have introduced residual extropy as measure of uncertaintyin residual lifetime distributions analogues to residual entropy (1996). Also, they obtainedsome properties and applications of that. In this paper, we study the extropy to measurethe uncertainty in a past lifetime distribution. This measure of uncertainty is called pastextropy. Also it is showed a characterization result about the past extropy of largest orderstatistics.
Keywords:
Reversed residual lifetime, Past extropy, Characterization, Order statistics.
AMS Subject Classification : 94A17, 62B10, 62G30
The concept of Shannon entropy as a seminal measure of uncertainty for a random variablewas proposed by Shannon (1948). Shannon entropy H ( f ) for a non-negative and absolutelycontinuous random variable X is defined as follows: H ( f ) = − E [log f ( x )] = − (cid:90) + ∞ f ( x ) log f ( x ) d x, (1)where F and f are cumulative distribution function (CDF) and probability density function(pdf), respectively. There are huge literatures devoted to the applications, generalizations andproperties of Shannon entropy (see, e.g. Cover and Thomas, 2006).Recently, a new measure of uncertainty was proposed by Lad et al. (2015) called extyopyas a complement dual of Shannon entropy (1948). For a non-negative random variable X theextropy is defined as below: 1 a r X i v : . [ m a t h . S T ] S e p ( X ) = − (cid:90) + ∞ f ( x )d x. (2)It’s obvious that J ( X ) ≤ . One of the statistical applications of extropy is to score the forecasting distributions usingthe total log scoring rule.Study of duration is an interest subject in many fields of science such as reliability, survivalanalysis, and forensic science. In these areas, the additional life time given that the componentor system or a living organism has survived up to time t is termed the residual life function ofthe component. If X is the life of a component, then X t = ( X − t | X > t ) is called the residuallife function. If a component is known to have survived to age t then extropy is no longeruseful to measure the uncertainty of remaining lifetime of the component.Therefore, Ebrahimi (1996) defined the entropy for resudial lifetime X t = ( X − t | X > t )as a dynamic form of uncertainty called the residual entropy at time t and defined as H ( X ; t ) = − (cid:90) + ∞ t f ( x ) F ( t ) log f ( x ) F ( t ) d x, where F ( t ) = P ( X > t ) = 1 − F ( t ) is the survival (reliability) funltion of X .Analogous to residual entropy, Qiu et al. (2017) defined the extropy for residual lifetime X t called the residual extropy at time t and defined as J ( X t ) = − (cid:90) + ∞ f X t ( x )d x = − F ( t ) (cid:90) + ∞ t f ( x )d x. (3)In many situations, uncertainty can relate to the past. Suppose the random variable X is the lifetime of a component, system or a living organism, having an absolutely continuousdistribution function F X ( t ) and the density function f X ( t ). For t >
0, let the random variable t X = ( t − X | X < t ) be the time elapsed after failure till time t , given that the component hasalready failed at time t . We denote the random variable t X , the reversed residual life (pastlifetime). For instance, at time t , one has under gone a medical test to check for a certaindisease. Suppose that the test result is positive. If X is the age when the patient was infected,then it is known that X < t . Now the question is, how much time has elapsed since thepatient has been infected by this disease? Based on this idea, Di Crescenzo and Longobardi(2002) introduced the entropy of the reversed residual lifetime t X as a dynamic measure ofuncertainty called past entropy as follows: H ( X ; [ t ]) = − (cid:90) t f ( x ) F ( t ) log f ( x ) F ( t ) d x. This measure is dual of residual entropy introduced by Ebrahimi (1996).2n this paper, we study the extropy for t X as dual of residual extropy that is called pastextropy and it is defined as below (see also Krishnan et al. (2020)): J ( t X ) = − (cid:90) + ∞ f t X ( x )d x = − F ( t ) (cid:90) t f ( x )d x, (4)where f t X ( x ) = f ( t − x ) F ( t ) , for x ∈ (0 , t ). It can be seen that for t ≥ J ( t X ) possesses all theproperties of J ( X ). Remark 1.
It’s clear that J ( + ∞ X ) = J ( X ).Past extropy has applications in the context of information theory, reliability and sur-vival analysis, insurance, forensic science and other related fields beceuse in that a lifetimedistribution truncated above is of utmost importance.The paper is organized as follows: in section 2, an approach to measure of uncertainty inthe past lifetime distribution is proposed. Then it is studied a characterization result with thereversed failure rate. Following a characterization result is given based on past extropy of thelargest order statistics in section 3. Analogous to residual extropy (Qiu et al. (2017)), the extropy for t X is called past extropyand for a non-negative random variable X is as below: J ( t X ) = − (cid:90) + ∞ f t X ( x )d x = − F ( t ) (cid:90) t f ( x )d x, (5)where f t X ( x ) = f ( t − x ) F ( t ) , for x ∈ (0 , t ) is the density function of t X . Its clear that J ( t X ) ≤ −∞ , + ∞ ]. Also, J ( + ∞ X ) = J ( X ). Example 1. a) If X ∼ Exp ( λ ), then J ( t X ) = − λ − λt − e − λt for t >
0. This shows that thepast extropy of exponential distribution is an increasing function of t.b) If X ∼ U (0 , b ), then J ( t X ) = − t .c) If X has power distribution with parameter α >
0, i.e. f ( x ) = αx ( α − , 0 < x <
1, then J ( t X ) = − α α − t .d) If X has Pareto distribution with parameters θ > , x >
0, i.e. f ( x ) = θx x θ +10 x θ +1 , x > x ,then J ( t X ) = θ θ +1)( t θ − x θ ) (cid:104) x θ t − t θ x (cid:105) .3here is a functional relation between past extropy and residual extropy as follows: J ( X ) = F ( t ) J ( t X ) + F ( t ) J ( X t ) , ∀ t > . In fact F ( t ) J ( t X ) + F ( t ) J ( X t ) = − (cid:90) + ∞ t f ( x )d x − (cid:90) t f ( x )d x = − (cid:90) + ∞ f ( x )d x = J ( X ) . From (4) we can rewrite the following expression for the past extropy: J ( t X ) = − τ ( t )2 f ( t ) (cid:90) t f ( x )d x, where τ ( t ) = f ( t ) F ( t ) is the reversed failure rate. Definition 1.
A random variable is said to be increasing (decreasing) in past extropy if J ( t X )is an increasing (decreasing) function of t . Theorem 2.1. J ( t X ) is increasing (decreasing) if and only if J ( t X ) ≤ ( ≥ ) − τ ( t ) .Proof. From (5) we get dd t J ( t X ) = − τ ( t ) J ( t X ) − τ ( t ) . Then J ( t X ) is increasing if and only if2 τ ( t ) J ( t X ) + 12 τ ( t ) ≤ , but τ ( t ) ≥ J ( t X ) ≤ − τ ( t ) . Theorem 2.2.
The past extropy J ( t X ) of X is uniquely determined by τ ( t ) .Proof. From (5) we get dd t J ( t X ) = − τ ( t ) J ( t X ) − τ ( t ) . So we have a linear differential equation of order one and it can be solved in the following way J ( t X ) = e − (cid:82) tt τ ( s )d s (cid:20) J ( t X ) − (cid:90) tt τ ( s )e (cid:82) st τ ( y )d y d s (cid:21) , where we can use the boundary condition J ( + ∞ X ) = J ( X ), so we get J ( t X ) = e (cid:82) + ∞ t τ ( s )d s (cid:20) J ( X ) + (cid:90) + ∞ t τ ( s )e − (cid:82) + ∞ s τ ( y )d y d s (cid:21) . (6)4 xample 2. Let X ∼ Exp ( λ ), with reversed failure rate τ ( t ) = λ e − λt − e − λt . It follows from (6)that J ( t X ) = e (cid:82) + ∞ t λ e − λs − e − λs d s (cid:34) J ( X ) + (cid:90) + ∞ t λ e − λs (1 − e − λs ) e − (cid:82) + ∞ s λ e − λy − e − λy d y d s (cid:35) = (cid:16) − e − λt (cid:17) − (cid:20) J ( X ) + 12 (cid:90) + ∞ t λ e − λs d s (cid:21) = λ − λt − − e − λt ) = − λ − λt − e − λt , so we find again the same result of example 1.Using the following definition (see Shaked and Shanthikumar, 2007), we show that J ( t X )is increasing in t . Definition 2.
Let X and Y be two non-negative variables with reliability functions F , G andpdfs f, g respectively. X is smaller than Y a) in the likelihood ratio order, denoted by X ≤ lr Y , if f ( x ) g ( x ) is decreasing in x ≥ X ≤ st Y if F ( x ) ≤ G ( x ) for x ≥ Remark 2.
It is well known that if X ≤ lr Y then X ≤ st Y and X ≤ st Y if and only if E ( ϕ ( Y )) ≤ ( ≥ ) E ( ϕ ( X )) for any decreasing (increasing) function ϕ . Theorem 2.3.
Let X be a random variable with CDF F and pdf f . If f (cid:0) F − ( x ) (cid:1) is decreasingin x ≥ , then J ( t X ) is increasing in t ≥ .Proof. Let U t be a random variable with uniform distribution on (0 , F ( t )) with pdf g t ( x ) = F ( t ) for x ∈ (0 , F ( t )), then based on (4) we have J ( t X ) = − F ( t ) (cid:90) F ( t )0 f (cid:0) F − ( u ) (cid:1) d u = − F ( t ) (cid:90) F ( t )0 g t ( u ) f (cid:0) F − ( u ) (cid:1) d u = − F ( t ) E (cid:2) f (cid:0) F − ( U t ) (cid:1)(cid:3) . Let 0 ≤ t ≤ t . If 0 < x ≤ F ( t ), then g t ( x ) g t ( x ) = F ( t ) F ( t ) is a non-negative constant. If F ( t ) < x ≤ F ( t ), then g t ( x ) g t ( x ) = 0. Therefore g t ( x ) g t ( x ) is decreasing in x ∈ (0 , F ( t )), whichimplies U t ≤ lr U t . Hence U t ≤ st U t and so0 ≤ E (cid:2) f (cid:0) F − ( U t ) (cid:1)(cid:3) ≤ E (cid:2) f (cid:0) F − ( U t (cid:1)(cid:3) using the assumption that f (cid:0) F − ( U t ) (cid:1) is a decreasing function. Since 0 ≤ F ( t ) ≤ F ( t ) then J ( t X ) = − F ( t ) E (cid:2) f (cid:0) F − ( U t ) (cid:1)(cid:3) ≤ − F ( t ) E (cid:2) f (cid:0) F − ( U t ) (cid:1)(cid:3) = J ( t X ) . emark 3. Let X be a random variable with CDF F ( x ) = x , for x ∈ (0 , f (cid:0) F − ( x ) (cid:1) = 2 √ x is increasing in x ∈ (0 , J ( t X ) = − t is increasing in t ∈ (0 , f (cid:0) F − ( x ) (cid:1) is decreasing in x is sufficient but not necessary. Let X , X , . . . , X n be a random sample with distribution function F , the order statistics ofthe sample are defined by the arrangement X , X , . . . , X n from the minimum to the maximumby X (1) , X (2) , . . . , X ( n ) . Qiu and Jia (2017) defined the residual extropy of the i − th orderstatistics and showed that the residual extropy of order statistics can determine the underlyingdistribution uniquely. Let X , X , . . . , X n be continuous and i.i.d. random variables with CDF F indicate the lifetimes of n components of a parallel system. Also X n , X n , . . . , X n : n bethe ordered lifetimes of the components. Then X n : n represents the lifetime of parallel systemwith CDF F X n : n ( x ) = ( F ( x )) n , x >
0. The CDF of [ t − X n : n | X n : n < t ] is 1 − (cid:16) F ( t − x ) F ( t ) (cid:17) n where[ t − X n : n | X n : n < t ] is called reversed residual lifetime of the system. Now past extropy forreversed residual lifetime of parallel system with distribution function F X n : n ( x ) is as follows: J ( t X n : n ) = − n F ( t )) n (cid:90) t f ( x )[ F ( x )] n − d x. Theorem 3.1. If X has an increasing pdf f on [0 , T ] , with T > t , then J ( t X n : n ) is decreasingin n ≥ .Proof. The pdf of ( X n : n | X n : n ≤ t ) can be expressed as g tn : n ( x ) = nf ( x ) F n − ( x ) F n ( t ) , x ≤ t. We note that g t n − n − ( x ) g t n +1:2 n +1 ( x ) = 2 n − n + 1 F ( t ) F ( x )is decreasing in x ∈ [0 , t ] and so ( X n − n − | X n − n − ≤ t ) ≤ lr ( X n +1:2 n +1 | X n +1:2 n +1 ≤ t )which implies ( X n − n − | X n − n − ≤ t ) ≤ st ( X n +1:2 n +1 | X n +1:2 n +1 ≤ t ). If f is increasingon [0 , T ] we have E [ f ( X n − n − ) | X n − n − ≤ t ] ≤ E [ f ( X n +1:2 n +1 ) | X n +1:2 n +1 ≤ t ] . From the definition of the past extropy it follows that J ( t X n : n ) = − n F n ( t ) (cid:90) t f ( x ) F n − ( x )d x = − n n − F ( t ) (cid:90) t (2 n − F n − ( x ) f ( x ) F n − ( t ) f ( x )d x = − n n − F ( t ) E [ f ( X n − n − ) | X n − n − ≤ t ] . J ( t X n : n ) J ( t X n +1: n +1 ) = n ( n + 1) n − n + 1 E [ f ( X n − n − ) | X n − n − ≤ t ] E [ f ( X n +1:2 n +1 ) | X n +1:2 n +1 ≤ t ] ≤ E [ f ( X n − n − ) | X n − n − ≤ t ] E [ f ( X n +1:2 n +1 ) | X n +1:2 n +1 ≤ t ] ≤ . Since the past extropy of a random variable is non-negative we have J ( t X n : n ) ≥ J ( t X n +1: n +1 )and the proof is completed. Example 3.
Let X be a random variable distribuited as a Weibull with two parameters, X ∼ W α, λ ), i.e. f ( x ) = λαx α − exp ( − λx α ). It can be showed that for α > T = (cid:0) α − λα (cid:1) α . Let us consider the case in which X has a Weibull distributionwith parameters α = 2 and λ = 1, X ∼ W ,
1) and so T = √ . The hypothesis of thetheorem 3.1 are satisfied for t = 0 . < T = √ . Figure 1 shows that J ( . X n : n ) is decreasingin n ∈ { , , . . . , } . Moreover the result of the theorem 3.1 does not hold for the smallestorder statistic as shown in figure 2.Figure 1: J ( . X n : n ) of a W ,
1) for n = 1 , , . . . , X has an increasing pdf on [0 , T ] with T > t we give a lower boundfor J ( t X ). Theorem 3.2. If X has an increasing pdf f on [0 , T ] , with T > t , then J ( t X ) ≥ − τ ( t )2 . J ( . X n ) of a W ,
1) for n = 1 , , . . . , Proof.
From the definition we get J ( t X ) = − F ( t ) (cid:90) t f ( x )d x = − f ( t )2 F ( t ) + 12 F ( t ) (cid:90) t F ( x ) f (cid:48) ( x )d x ≥ − τ ( t )2 . Example 4.
Let X ∼ W , , T ]with T = √ . The hypothesis of the theorem 3.2 are satisfied for t < T = √ . Figure 3 showsthat the function − τ ( t )2 (in red) is a lower bound for the past extropy (in black). We remarkthat the theorem gives information only for t ∈ [0 , T ], in fact for larger values of t the function − τ ( t )2 could not be a lower bound anymore, as showed in figure 3.Qiu (2016), Qiu and Jia (2017) showed that extropy of the i − th order statistics and residualextropy of the i − th order statistics can characterize the underlying distribution uniquely. Inthe following theorem, whose proof requires next lemma, we show that the past extropy of thelargest order statistic can uniquely characterize the underlying distribution. Lemma 3.1.
Let X and Y be non-negative random variables such that J ( X n : n ) = J ( Y n : n ) , ∀ n ≥ . Then X d = Y . J ( t X ) (in black) and − τ ( t )2 (in red) of a W , Proof.
From the definition of the extropy, J ( X n : n ) = J ( Y n : n ) holds if and only if (cid:90) + ∞ F n − X ( x ) f X ( x )d x = (cid:90) + ∞ F n − Y ( x ) f Y ( x )d x i.e. if and only if (cid:90) + ∞ F n − X ( x ) τ X ( x )d F X ( x ) = (cid:90) + ∞ F n − Y ( x ) τ Y ( x )d F Y ( x ) . Putting u = F X ( x ) in the left side of the above equation and u = F Y ( x ) in the right side wehave (cid:90) u n − τ X (cid:0) F − X ( √ u ) (cid:1) d u = (cid:90) u n − τ Y (cid:0) F − Y ( √ u ) (cid:1) d u. that is equivalent to (cid:90) u n − (cid:2) τ X (cid:0) F − X ( √ u ) (cid:1) − τ Y (cid:0) F − Y ( √ u ) (cid:1)(cid:3) d u = 0 ∀ n ≥ . Then from Lemma 3.1 of Qui (2017) we get τ X (cid:0) F − X ( √ u ) (cid:1) = τ Y (cid:0) F − Y ( √ u ) (cid:1) for all u ∈ (0 , √ u = v we have τ X (cid:0) F − X ( v ) (cid:1) = τ Y (cid:0) F − Y ( v ) (cid:1) and so f X (cid:0) F − X ( v ) (cid:1) = f Y (cid:0) F − Y ( v ) (cid:1) forall v ∈ (0 , F − X ) (cid:48) ( v ) = ( F − Y ) (cid:48) ( v ) i.e. F − X ( v ) = F − Y ( v ) + C , for all v ∈ (0 ,
1) with C constant. But for v = 0 we have F − X (0) = F − Y (0) = 0 and so C = 0.9 heorem 3.3. Let X and Y be two non-negative random variables with cumulative distributionfunctions F ( x ) and G ( x ) , respectively. Then F and G belong to the same family of distributionsif and only if for t ≥ , n ≥ , J ( t X n : n ) = J ( t Y n : n ) . Proof.
It sufficies to prove the sufficiency. J ( t X n : n ) is the past extropy for X n : n but it is also theextropy for the variable t X n : n . So through lemma 3.1 we get t X d = t Y . Then F ( t − x ) F ( t ) = G ( t − x ) G ( t ) ,for x ∈ (0 , t ). If exists t (cid:48) such that F ( t (cid:48) ) (cid:54) = G ( t (cid:48) ) then in (0 , t (cid:48) ) F ( x ) = αG ( x ) with α (cid:54) = 1. Butfor all t > t (cid:48) , exists x ∈ (0 , t ) such that t − x = t (cid:48) and so F ( t ) (cid:54) = G ( t ) and as in the precedentstep we have F ( x ) = αG ( x ) for x ∈ (0 , t ). Letting t to + ∞ we have a contradiction because F and G are both distribution function and their limit is 1. In this paper we studied a measure of uncertainty, the past extropy. It is the extropy of theinactivity time. It is important in the moment in which with an observation we find our systemdown and we want to investigate about how much time has elapsed after its fail. Moreover westudied some connections with the largest order statistic.
Francesco Buono is partially supported by the GNAMPA research group of INdAM (IstitutoNazionale di Alta Matematica) and MIUR-PRIN 2017, Project ”Stochastic Models for ComplexSystems” (No. 2017 JFFHSH).On behalf of all authors, the corresponding author states that there is no conflict of interest.
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