Some Characterization Results Based on the Mean Vitality Function of the First-Order Statistics

A. Toomaj; A. Shirvani

doi:10.2991/jsta.d.190514.004

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Volume 18, Issue 2, June 2019, Pages 123 - 128

Some Characterization Results Based on the Mean Vitality Function of the First-Order Statistics

Authors

A. Toomaj¹^{, *}, A. Shirvani²

¹Department of Mathematics and Statistics, Gonbad Kavous University, Gonbad Kavous, Iran

²Faculty of Basic Sciences and Engineering, Department of Statistics, Velayat University, Iranshahr, Iran

^*Corresponding author. Email: ab.toomaj@gmail.com

Corresponding Author

A. Toomaj

Received 20 February 2016, Accepted 2 January 2018, Available Online 18 June 2019.

DOI: 10.2991/jsta.d.190514.004 How to use a DOI?
Keywords: MRL; MVF; Proportional hazard rate model; Weibull distribution; Mixed system; Stochastic orders
Abstract: In this paper, we investigate some new properties of the mean vitality function (MVF) of a random variable, proposed by A. Toomaj, M. Doostparast, J. Stat. Theory Appl. 13 (2013), 189–195. Specifically, we explore properties of MVF and study under what conditions the MVF of the first-order statistics can uniquely determines the parent distribution. We show that in all distributions the Weibull family, which is commonly used in several fields of applied probability, is characterized through the ratio of the MVF of the first-order statistics to its expectation.
Copyright: © 2019 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The notion of entropy as a measure of uncertainty, introduced by Shannon [1], has a fundamental importance in different areas such as probability and statistics, financial analysis, engineering, and information theory; see, for example, Cover and Thomas [2]. For a continuous random variable X with probability density function (pdf) f and cumulative distribution function (cdf) F, the differential entropy of X is defined as HX=−E(log fX). Throughout this paper “log” stands for the natural logarithm. Recently Rao et al. [3] introduced an alternate measure of uncertainty, called it cumulative residual entropy (CRE) and is used for measuring the residual uncertainty of a random variable. For a nonnegative random variable X, the CRE is defined by

(1)E(X)=−∫0∞F¯(x)log⁡F¯(x)dx,

where F¯x=1−Fx stands for the reliability function. This measure, in general, is more stable since the distribution function is more regular than the density function and has a lot of mathematical properties. The applications and other properties and some new version of the mentioned measure can be found in Asadi and Zohrevand [4], Baratpour [5], Baratpour and Habibi rad [6], Navarro et al. [7], Psarrakos and Navarro [8], Rao [9], and the references therein.

Let X be a random lifetime with cdf F. Moreover, let us suppose that Ft=0 for t<0, with a finite moment. The mean residual life (MRL) is defined as

(2)mt=EX−t|X>t=∫t∞F¯xF¯tdx,

where F¯t>0 for t>0. If F is absolutely continuous with the pdf f, then MRL can be rewritten as

(3)mt=∫t∞xfxF¯tdx−t=vt−t, t>0.

The function vt=EX|X〉t=∫t∞xfxF¯tdx is known as the vitality function (VF) or life expectancy which introduced by Kupka and Loo [10]. The VF and MRL play important roles in engineering reliability, biomedical science, and survival analyzes; see Kotz and Shanbhag [11], Murari and Sujit [12], Ruiz and Navarro [13], and Bairamov et al. [14] and the references therein. It is worth mentioning that the rapid ageing on average of a component needs to low vitality relatively, whereas high vitality implies relatively slow (or even possibly “negative”) ageing during the time period. By using (3), Toomaj and Doostparat [15] defined a new measure which called it mean vitality function (MVF) and defined as

(4)Ev(X)=EX+μ,

where EX is given in (1) and μ=EX denotes the expectation of a random variable. It is worth pointing out that the MVF expresses as the sum of two terms, the CRE and the mean value of X. Hence to compute the MVF, one can compute the mentioned measures. Another useful representation for the MVF is given as

EvX=∫0∞xfxψF¯xdx,

where ψv=−log v, 0<v<1. Also, the probability integral transformation U=F¯X provides a useful expression as follows:

(5)EvX=∫01F¯−1uψudu=μEψZF,

where F¯−1u=supx:F¯x≥u and ZF has a pdf

hFu=F¯−1uμ, 0<u<1.

It is worth pointing out that hFu=0 if either u≥1 or u≤0 and it has a decreasing probability density function. The aim of the present paper is to investigate some new properties of the MVF measure. Specifically, we provide some stochastic ordering properties and characterization results.

2. MAIN RESULTS

First, we observe that MVF is shift and scale dependent under linear transformation. In other terms, EvX has the same properties of the expectation of a random variable. First, we have the following lemma for CRE.

Lemma 2.1.

Let a,b>0. It holds that

(6)EaX+b=aEX.

Proof.

The result follows by noting that F¯aX+bx=F¯x−ba, x≥0, and using (1).

From (4), Lemma 2.1 and the properties of the expected value we have the following proposition.

Proposition 2.1.

Let a,b>0. It holds that

(7)EvaX+b=aEvX+b.

Let Xθ⋆ be an absolutely continuous nonnegative random variable with survival function F¯θ⋆x. The survival function of the proportional hazard rate model with proportionality constant θ>0, is defined as

(8)F¯θ⋆x=F¯xθ, x≥0.

For more details on the applications and properties of proportional hazards rate model, see, for example, Gupta et al. [16], Gupta et al. [17], and Mudholkar et al. [18], and the references therein. In the next proposition, we provide an upper bound for the MVF of Xθ⋆ depending on Evx.

Proposition 2.2.

For n=1,2,…, we have

EvXθ⋆≤θ EvX if θ≥1,

and the inequality being reversed if 0<θ≤1.

Proof.

Recalling (1) and (8), we have

EXθ⋆=−θ∫0∞F¯xθlog F¯xdx.

Since F¯xθ≤F¯x, when θ≥1, we obtain

EXθ⋆≤θEX.

On the other hand, we obtain

EXθ⋆≤EX≤θEX.

Hence representation (4) completes the proof. For 0<θ≤1, we have F¯x≤F¯xθ, and hence the desired result follows.

Hereafter, we obtain some stochastic ordering properties and characterization results of the MVF. For this purpose, we consider two absolutely continuous nonnegative random variables X and Y with cdfs F and G and pdfs f and g, respectively. First, we recall that a random variable X is said to be smaller than Y in the usual stochastic order, denoted by X≤stY, if EϕX≤EϕY for all increasing functions ϕ such that the expectations exist. A random variable X is said to be smaller than Y in the increasing convex order, denoted by X≤icxY, if EϕX≤EϕY for all increasing convex functions ϕ such that the expectations exist. A random variable X is said to be smaller than Y in the convex order, denoted by X≤cxY, if EϕX≤EϕY for all convex functions ϕ such that the expectations exist. Further properties and applications can be found in the book of Shaked and Shanthikumar [19]. Also, we recall the following definition given by Toomaj and Doostparast [15].

Definition 2.1.

A random variable X is said to be smaller than Y in the MVF, denoted by X≤mvfY, if EvX≤EvY such that the expectations exist.

It is worth pointing out that the MVF order is not a stochastic order in a strict sense, since it does not satisfy the antisymmetric property, that is, X=mvfY does not imply X=stY. Note that from Corollary 2.2 of Toomaj and Doostparast [15] we have that X≤stY implies X≤mvfY. Therefore, X≤mvfY can be seen as a necessary condition for X≤stY. Also, we show that X≤mvfY can be seen as a necessary condition for X≤icxY as well. First, consider the following proposition.

Proposition 2.3.

Let X be an absolutely continuous nonnegative random variable with EX<+∞. Then, we have

(9)EX=EgX,

where

(10)gx=∫0xΛtdt, x≥0.

Proof.

From (1) and Fubini's theorem, we obtain

EX=∫0∞∫t∞fxdxΛtdt=∫0∞fx∫0xΛtdtdx,

which immediately follows (9) by using (10).

The next proposition shows that the increasing convex order implies the CRE order.

Proposition 2.4.

If X and Y are nonnegative random variables such that X≤icxY, then it holds that gX≤icxgY, where the function g⋅ is defined in Eq. (10). In particular, X≤icxY implies EX≤EY.

Proof.

Since the function g⋅ is an increasing convex function, it follows by Theorem 4.A.8. of Shaked and Shantikumar [19] that gX≤icxgY. In particular, recalling the definition of increasing convexorder, we have EX≤EY.

Proposition 2.5.

Let X and Y be two nonnegative random variables with cdfs F and G, respectively. If X≤icxY, then X≤mvfY.

Proof.

From Eq. (4), it is sufficient to prove that EX+EX≤EX+EX. First, the condition X≤icxY, implies EX≤EY due to relation 4.A.2 of Shaked and Shanthikumar [19]. Also, Proposition 2.4 yields EX≤EY and hence the desired result follows.

Proposition 2.6.

Let X and Y be two nonnegative random variables with cdfs F and G, respectively. Then

If X≤cxY, then X≥mvfY.
If ZF≤cxZG, then X≤mvfY.

Proof.

It is known that X≤cxY implies μ=EX=EY. Now, since ψv is a decreasing function of v, hence the proof of (i) can be obtained from Lemma 2.3 of Navarro and Rychlik [20], expression Eq. (5) and the definition of usual stochastic order. The proof of (ii) is an immediate consequence of Eq. (5) and the definition of the convex order.

Proposition 2.7.

Let X1:n and Y1:n be two lifetimes of series systems with i.i.d. components having the common cdfs F and G, respectively. If X≤mvfY, then X1:n≤mvfY1:n.

Proof.

Since X≤mvfY, from, we have

(11)EvX−EvY=∫01ψuF¯−1(u)−G¯−1udu≤0.

On the other hand, we have

EvX1:n−EvY1:n=n∫01un−1ψuF¯−1u−G¯−1udu,

(12)≤n∫01ψuF¯−1u−G¯−1udu≤0.

The first inequality is obtained by noting that un−1≤1, 0<u<1, while the second inequality is obtained from Eq. (11).

Example 2.1.

Let us consider a Weibull distribution with survival function

F¯x=e−λxα, x>0,

where α>0, and λ>0 are scale and shape parameters, respectively. It is easy to verify that (see Baratpour [5]), EX=1λΓ1+1α and EX=1αλΓ1+1α and hence we have

EvX=α+1αλΓ1+1α.

It is not hard to verify that

EvX1:n=α+1n1ααλΓ1+1α.

On the other hand, we have EX1:n=1n1αλΓ1+1α and thus

EvX1:nEX1:n=α+1α.

Therefore, it shows that in the Weibull family this ratio is constant for all n.

In the next theorem, we show that, in all distributions, only in Weibull distribution the ratio EvX1:nEX1:n is constant analogue Theorem 2.1 of Baratpour [5]. First, we recall the following lemma due to Müntz-Szász Theorem (see Kamps [21]) which is used in the proofs of this paper. Recently, many authors applied the Müntz-Szász Theorem giving characterization results by recurrence relations for moments of order statistics; see for example Khan and Zia [22].

Lemma 2.2.

For any increasing sequence of positive integers nj,j≥1, the sequence of polynomials xnj is complete on L0,1, if and only if,

∑j=1+∞nj−1=+∞, 0<n1<n2<⋯.

Theorem 2.1.

Let X1,⋯,Xn be n iid absolutely continuous nonnegative random variables with the common pdf f and cdf F. Then F belong to Weibull family, if and only if

(13)EvX1:nEX1:n=k, k>1,

for all n=nj,j≥1, such that ∑j=1+∞nj−1=+∞.

Proof.

The necessity is trivial. Hence it remains to prove the sufficiency part. Since EvX1:n=EX1:n+EX1:n, hence it is sufficient to prove that EX1:nEX1:n=c, where c=k−1, k>1. Therefore, Theorem 2.1 of Baratpour [5] completes the proof.

In the next theorems, we characterize the distributions based on the MVF of the first-order statistics.

Proposition 2.8.

Let X and Y be two nonnegative random variables with pdfs fx and gx and absolutely continuous cdfs Fx and Gx, respectively. Then F and G belong to the same family of distributions if and only if

EvX1:n=EvY1:n,

for n=nj, j≥1 such that ∑j=1∞nj−1 is infinite.

Proof.

The necessity is trivial and hence it remains to prove the sufficiency part. By using the probability integral transformation U=F¯X, we have

EvX1:n=n∫01F¯−1uun−1ψudu.

For EvY1:n can be obtain in a similar way. Since EvX1:n=EvY1:n, we obtain

(14)∫01un−1ψuF¯−1u−G¯−1udu=0.

If Eq. (14) holds for n=nj, j≥1, such that ∑j=1∞nj−1=∞, then from Lemma 2.2 we can conclude that F¯−1u=G¯−1u, 0<v<1, and this completes the proof.

Proposition 2.9.

Let X and Y be two nonnegative random variables with pdfs fx and gx and absolutely continuous cdfs Fx and Gx, respectively. Then F and G belong to the same family of distributions, but for a change in location, if and only if

(15)EvX1:nEX=EvY1:nEY,

for n=nj, j≥1 such that ∑j=1∞nj−1 is infinite.

Proof.

The necessity is trivial and hence it remains to prove the sufficiency part. By using Eq. (4) it is sufficient to prove that EX1:nEX=EY1:nEY. We have

EX1:nEX=∫01unψu/fF¯−1udu∫01F¯−1udu.

If Eq. (15) holds, then we get

(16)∫01unψu/fF¯−1udu∫01F¯−1udu=∫01unψu/gG¯−1udu∫01G¯−1udu.

Let us suppose that c=∫01G¯−1udu/∫01F¯−1udu, then Eq. (16) can be expressed as

(17)∫01unψu1fF¯−1u−1cgG¯−1udu=0.

If Eq. (17) holds for n=nj, j≥1, such that ∑j=1∞nj−1= ∞, then from Lemma 2.2 we can conclude that fF¯−1u=cgG¯−1u, 0<v<1, and hence it follows that F¯−1u=G¯−1u+d. Since X and Y have a common support 0,+ ∞, we can conclude that d=0, which means that F and G belong to the same family of distributions, but for a change in scale.

3. CONCLUSIONS

Some new properties of the MVF are investigated. It is explored that under what conditions the MVF of the first-order statistics can uniquely determines the parent distribution. It is shown that the Weibull family is characterized through ratio of the MVF of the first-order statistics to its expectation.

ACKNOWLEDGMENTS

The authors are grateful to the Editor-in-Chief and anonymous referees for careful reading of this manuscript.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 18 - 2
Pages: 123 - 128
Publication Date: 2019/06/18
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.190514.004 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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ris enw bib

TY  - JOUR
AU  - A. Toomaj
AU  - A. Shirvani
PY  - 2019
DA  - 2019/06/18
TI  - Some Characterization Results Based on the Mean Vitality Function of the First-Order Statistics
JO  - Journal of Statistical Theory and Applications
SP  - 123
EP  - 128
VL  - 18
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.190514.004
DO  - 10.2991/jsta.d.190514.004
ID  - Toomaj2019
ER  -

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