Journal of Statistical Theory and Applications

Volume 17, Issue 1, March 2018, Pages 172 - 192

On Some Aspects of Strong Risk Class and Associated Ordering

Authors
Mervat Mahdy*
*Corresponding author: Dept. of Statistics, Mathematics and Insurance, College of Commerce, Benha University, Egypt. Email. drmervat.mahdy@fcom.bu.edu.eg. Tel: +20-122-068-2460, fax: +20-13-323-0860.
Corresponding Author
Mervat Mahdy
Received 24 March 2016, Accepted 19 November 2017, Available Online 31 March 2018.
DOI
https://doi.org/10.2991/jsta.2018.17.1.13How to use a DOI?
Keywords
Stochastic orders, Mean inactivity time class, Strong mean inactivity time class, Variance inactivity time, Preservation, Risk measure
Abstract

The paper opens up a new aging class and a new stochastic orders which is depend on risk class, that plays vital role in the reliability theory, finance topics, stochastic orders, and the economic theory. The article presents some new interesting implications and characterizations concerning this class. In addition, we list a series of inequalities that provide bounds for strong risk and some aging classes. Furthermore, a sufficient condition for a probability distribution to have a new class is provided. In addition, The paper demonstrates the preservation properties of a new stochastic order under some reliability operations such as mixture, and convolution. Moreover, some new reliability concepts based on discrete lifetime random variable are studied.

Copyright
Copyright © 2018, the Authors. Published by Atlantis Press.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1 Introduction and Motivations

Consider a probability density function f(t) for a lifetime random variable X with distribution function F (t) and survival function F¯(t)=1F(t) , t ∈ ℝ+. In addition, we assume that the mean life μX=0F¯(u)du and the variance σX2 are finite. Likewise, let Y be the second lifetime random variable with the density function g (t), distribution function G (t) and survival function G¯(t)=1G(t) ; t ∈ ℝ+. Furthermore, the mean life μY=0G¯(u)du and variance σY2 are both assumed to be finite. Let lX = inf {t ∈ ℝ+ : F (t) > 0}, uX = sup {t ∈ ℝ+ : F (t) < 1}, ΩX = (lX, uX), lY = inf {t ∈ ℝ+ : G (t) > 0}, uY = sup {t ∈ ℝ+: G (t) < 1} and ΩY = (lY, uY). Let X has reversed hazard rate function r˜F(t)=f(t)/F(t) , t > lX and Y has reversed hazard rate function r˜G(t)=g(t)/G(t) , t > lY. Then, the mean inactivity lifetime functions, and variance inactivity lifetime functions respectively are defined by

mF(t)={1F(t)0tF(u)du,ift>lX0,iftlX
mG(t)={1G(t)0tG(v)dv,ift>lY0,iftlY
σF(t)2(t)={20t0yF(u)dudyF(t)mF2(t),ift>lX0,iftlX
and
σG(t)2(t)={20t0yG(u)dudyG(t)mG2(t),ift>lY0,iftlY.
The following definitions are essential for this study.

Definition 1.1.

The distribution function F (.) of the random variable X is said to have the following characteristics:

  1. (i)

    A decreasing reversed hazard rate (DRHR), if r˜F(t) is a decreasing function in t, or if F (t), is logarithmically concave in t ∈ ℝ+.

  2. (ii)

    An increasing strong mean past lifetime class if 0txF(x)dx/F(t) is non-decreasing for all t > lX.

The following stochastic orders are defined in Nanda et al. (2003); Shaked and Shanthikumar (2007); Mahdy (2012); Kayid and Izadkhah (2014):

Definition 1.2.

Let X1 and X2 be two non-negative and absolutely continuous random variables, with the distribution functions F1 (.) and F2 (.), density functions f1 (.) and f2 (.), reversed hazard rate functions r˜F1(.) and r˜F2(.) , mean past lifetime functions mF1 (.) and F2 (.), and the variance past lifetime functions σF1(x)2(.) and σF2(x)2(.) , respectively. Hence, X1 is smaller than or equal X2 in the following cases:

  1. (i)

    A reversed residual lifetime ordering (X1RHR X2), if r˜F1(x2)r˜F2(x2) , for all x2 > 0 or {F1 (x1) / F1 (x2)} ≥ F2 (x1) / F2 (x2)}, for all x1x2.

  2. (ii)

    A mean past lifetime order (X1MP X2), if mF1 (x2) ≥ mF2 (x2), for all x2 > 0.

  3. (iii)

    A variance past lifetime order (X1VP X2) if σF1(x)2(x2)σF2(x)2(x2) , for all x2 > 0.

  4. (iv)

    A strong mean past lifetime order (X1 SMP X2), if 0x2xF1(x)dx/F1(x2)0x2xF2(x)dx/F2(x2) for all x2 ∈ ℝ+.

  5. (v)

    A likelihood ratio order (X1LR X2), if f1 (u) f2 (v) ≥ f1 (v) f2 (u), for all uv.

Recently, Mahdy (2012) shown that, if X1VP X2, we can be get that X1 (φ) ≤VP X2 (φ) for any concave transformation φ. In addition, X1 (φ) ≤VP X2(φ) ⇒ X1VP X2 for any strictly increasing function (φ).

In insurance, if we represent the distribution by the appropriate random variable X, and let θ present the risk measure functional, then

θ:X.

Let an insurance contract in a specified period (0, a) and let Ω be the state space. If none of the risks specified in the policy contract happen during the policy term, then the policy holder has no monetary compensation for the paid premiums. Then the loss that the difference between amount of compensation (that denoted by ℂ) and total losses resulting from achieve state α occurs (that denoted by φ (X(α)) = X2 is convex function) is [ℂ – φ (X(α)) | X(α) ≤ ℂ], where X is premiums state α. Suppose we associate a “risk” as Var[ℂ – φ (X(α)) | X(α) ≤ ℂ], that is convex. We called it a strong risk function, SR, it is natural to demand that a risk function has this non-decreasing. Then the variance of random variable [ℂ – φ (X(α)) | X(α) ≤ ℂ] base on convex function, φ (X(α)), can use both in studying the effects of investor and analyst beliefs on securities trading. Also, we can use SR in measure of dispersion of returns of investment portfolio. In addition, SR measures the variability of a security’s returns relative to market index or a particular benchmark. If Var[ℂ – φ (X(α)) | X(α) ≤ ℂ] is greater than that of the benchmark, then the financial instrument is thought to be more perilous than the benchmark price. Low Var[ℂ – φ (X(α)) | X(α) ≤ ℂ] may mean lower risk.

As a result, the objective study is to achieve two aims. The first aim is consider a new nonparametric class of distributions, depended on SR with introduce some characterizations, and some preservation. The second aim is suggest a new technique that improves the comparison between two distributions with some characterizations, preservation results, and applications for strong risk order. Section 2 provides some properties and application of new class. Furthermore, the behavior of SR is presented. In addition, we listed a series of inequalities that provide bound for SMP and SR functions. Also, a sufficient conditions for a probability distribution to have a new class are studied. In Section 4, the relationships between the strong risk order and others stochastic orders will be discussed. Furthermore, useful properties and characterization of the strong risk order is studied. In addition, we establish closure properties of strong risk order under relevant reliability operations such as convolution, mixture and transformation. Finally, Section 5 discusses the useful applications in statistical reliability theory involving the strong risk order and the increasing strong risk class.

2 Characterizations of SR Class

Let a random variable X have the density function f (.), distribution function F (.), and reversed hazard rate r˜F1(.) . Setting u(y)=0yF(x)dx , and E[U2|κ]=0u2dF[u|κ]dκ , and by using integration by parts one has SR function we obtain:

θF(κ)=(κ(1κ))2+40κx(κx2)F(x)dxF(κ)SF2(κ).
where SF(κ)=0κxF(x)dx/F(κ) , it is called strong mean past lifetime(SMP). Then, after some algebraic calculations, we obtain
E[(κX2)2|Xκ]=θF(κ)+SF2(κ)=(κ(1κ))2+4[κ0κ0xF(u)dudx0κx3F(x)dx]F(κ).
If we let πFi(x)=40xyiF(u)dy/F(x) , then (2.1) is equivalent to
θF(κ)=(κ(1κ))2+SF(κ)(2κSF(κ))πF3(κ).
In other words, (2.2) is equivalent to
θF(κ)=κ2(12κ)+SF(κ)(2κSF(κ))+E(X4|Xκ)/F(κ),=(κ(1κ))2+2κ(σF2(κ)+mF2(κ))4πF3(κ)SF2(κ).
In such cases we express F.

Definition 2.1.

The non-negative random variable X is said to be smaller than or equal to Y in strong risk order (Xθ(κ) Y) if

0κx(κx2)F(x)dx0κx(κx2)G(x)dx,isincreasinginκ,forallκ+.

Suppose we refer to strong risk with the symbol S, an increasing function with abbreviate I and for decreasing function with symbol D, we can define the following two classes of lifetime:

  1. (i)

    SD = (F : θF (κ), for all κ, is a D),

  2. (ii)

    SI = (F : θF (κ), for all κ, is a I).

Clearly SD and SI form a pair of dual classes base on the distinguishing θF (κ). In current section, we introduce and explain the properties and applications of the SD and SI classes. The sufficient conditions for F (.) to have the SD and SI are provided by next theorem.

Theorem 2.1:

Let T denote the lifetime of an equipment with SF(κ), πF3(κ) , r˜F(κ) and SF\(κ)=SF(κ)/κ . If

SF(κ)(1SF\(κ))+κSF\(κ)>()12(2κ(13κ)+r˜F(κ)πF3(κ)),κ>0,
or
r˜F(κ)θF(κ)<()2κ(12κ)+2(1κ)SF(κ)+r˜F(κ)[SF2(κ)+κ2(1κ)2].
Then TSI (SD).

Proof.

The first differentiation of SF (κ) can be rewritten as

SF\(κ)=κu(κ)/F(κ)=κf(κ)u(κ)F2(κ),=1r˜F(κ)SF(κ).
By differentiating (2.2) with respect to κ, we can obtain the following equation:
θF(κ)/κ=2κ(13κ)+2[SF(κ)(1SF\(κ))+κSF\(κ)]+r˜F(κ)πF3(κ),
by using (2.2), and (2.4) in (2.5), we can decide that:
θF(κ)/κ=2κ(12κ)+2(1κ)SF(κ)+r˜F(κ)[SF2(κ)+κ2(1κ2)θF(κ)].
In this way we obtain the complete proof.

As mentioned earlier, the SR classes is important in reliability theory. The next example illustrated the application of theorem 2.1 in reliability theory whereas Theorem 1 in recognizing SI class.

Example 2.1:

Suppose Y be an Weibull random variable with the density function g (y) = βθβyβ−1 exp(−(θ(y)β), for y > 0, and β, θ > 0. One can easily prove that:

r˜G(y)=(βθβyβ1exp((θy)β))/[1exp((θy)β)],
SG(y)=1[1exp((θy)β)][(y2/2)+[exp((θy)β)βθ2(k=1(2/β)1ϖ(2)(θy)2β(k+1)+(θy)2β)]],
and
πG3(y)=1[1exp((θy)β)][y4+4(exp((θy)β)βθ4×(k=1(4/β)1ϖ(4)(θy)4β(k+1)(θy)4β))],
where ϖ(j) = (((j/β) – 1) ((j/β) – 2) … ((j/β) – k)). By using (2.4) it can easily check that
SG(y)(SG\(y)1)ySG\(y)<y(13y)+(r˜G(y)πG3(y))/2.
It follows from Theorem (2.1) that YSI. Also, according to YDRHR we obtain YSI.

Introduce the notation

G¯(r)(κ)=κG¯(r1)dI(<),forr=1,2,
where G¯(0)G¯ , and ΦrE(Yr). When normalized, these are the survival functions corresponding to the distribution function G(r) of Smith (1959, pp. 6), furthermore, by using Hall and Wellner (1981, pp. 173) we can prove that Φr=r!G¯(r)(0) , for r = 1, 2, … with G¯(r) finite if and only if Фr is finite. Hence dG¯(r)/dκ=G¯(r1) .

The following a list series of inequalities, those provide bounds for SMP and SR functions.

Since

κSG(κ)=E[Y2|Yκ],
we have
{κSG(κ)}G(κ)=E[Y2.I(Y<κ)]=Φ2E[Y2.I(Yκ)].
It is clear that
E[Y2.I(Y<κ)]κ2G(κ)
and
E[Y2.I(Y<κ)]Φ2,
and by using Jensen’s inequality, we get
E[Y2|Yκ](κmG(κ))2.
Further, Hölder’s inequality implies that
E[Y2.I(Y<κ)][E(Y2w)]1/w[G(κ)](11w),forw>1,
Similarly, E[Y2.I(Yκ)]κ2G¯(κ) is also true for all w > 1. In this way we obtain
E[Y2.I(Yκ)][E(Y2ω)]1/w[G¯(κ)](11w),forw>1.
It means
κ2=(E(Y2w))1/w(G¯(κ))1w.
Furthermore, we can prove that
θG(κ)=E[Y4|Yκ](E[Y2|Yκ])2.
We use below inequalities to obtain bound for SG(κ) and θG(κ).

Proposition 2.1.

If G is non-degenerate with ФrE(Yr) < ∞, then,

  1. (a)

    SG(κ) ≥ κ –2 / G (κ));

  2. (b)

    θG(κ) ≤ Ф4;

  3. (c)

    SG(κ)κ[Φ2wG(κ)]1/w , for all κ and any w > 1;

  4. (d)

    SG(κ)κ{(Φ2Φ2w1/w[G¯(κ)](11w))/G(κ)} , for Yκ and any w > 1.

Proof.

Since E[Y2.I(Y < κ)] = Ф2E [Y2.I(Yκ)] , it implies that G (κ) (κ – SG(κ)) Ф2. Then, we finished the proof (a). If

θG(κ)=0κ(κu2)2g(u)duG(κ)[κE[Y2|Yκ]]2,=E[Y4|Yκ](E[Y2|Yκ])2,
it follows that θ(κ) ≤ E [Y4|Yκ], which leads to complete proof of (b).

Furthermore, we have G(κ) {κ – SG(κ)} = E [Y2.I(Y < κ)], but

E[Y2.I(Y<κ)][E(Y2w)]1/w[G(κ)](11w).
Therefore,
G(κ)(κSG(κ))[Φ2w]1/wG(κ)[G(κ)]1w,
which leads to complete proof of (c). Now, we can provide that
E[Y2.I(Yκ)]E(Y2w)1/w[G¯(κ)](11w),forw>1,
In addition, (2.7) implies that
{κSG(κ)}G(κ)=E[Y2.I(Y<κ)]=Φ2E[Y2.I(Yκ)].
In this way we obtain
G(κ)(κSG(κ))Φ2E(Y2w)1/w[G¯(κ)](11w)=Φ2Φ2w1/w[G¯(κ)](11w),
which leads to complete proof of (d).

3 Preservation Properties of SR Class

We started by investigating the preservation properties of SR class closed under some reliability operations such as mixture.

For a family of distribution {M (θ | X), x ≥ 0} of random samples X and mixing distribution (prior distribution) D (θ), it is necessary to evaluate the mixture distribution (Ψ (.)). It obtained as follows

Ψ(κ)=0M(θ|κ)dD(θ),κ>0.
In order to verify the validity of SR class will be closed under mixture, we carried out the following theory.

Theorem 3.1.

Suppose that M (θ | X) ∈ SI for any θ ∈ ℝ+. Then Ψ (κ) ∈ SI.

Proof.

Without loss of generality, suppose that M (θ | X) = Exp (θ) and D (θ) = Exp (1), that is, M (θ | κ) = Mθ (k) = 1 – eθκ, κ > 0, and D(θ) = 1 – e–θ, θ > 0, after some algebraic calculations, we have

SMθ(κ)=(κ2/2)[(κ/θ)exp(θκ)+(1/θ2)(exp(θκ)1)]1exp(θκ),κ+
and
0κx3Mθ(x)dxMθ(κ)=(κ4θ4/4)+exp(θκ)(κ2θ2(θ+3)+6θκ+6)6(1exp(θκ))θ4,
where SMθ (κ) is SMP of Mθ. Suppose
ς(x)=exp(θx)((4x2θ2(θ+3)+24θx+24θ4)+4(x2/θ)+4xθ2),
then we have
40κx(κx2)Mθ(x)dxMθ(κ)=κ3θ4(2κ)4κθ2θ4ς(κ)+24θ4[1exp(θκ)].
Hence, θMθ (κ) is increasing in κ, which leads to MθSI for all θ ∈ ℝ+.

Now, we checked for the presence of Ψ ∈ SI. Firstly, Ψ (κ) can be computed by the following equation:

Ψ(κ)=0Mθ(κ)dG(θ)=1(1+κ)1,forκ+.
Therefore, the density function of Ψ (denoted as ψ) is:
ψ(κ)=(1+κ)2,κ+,
and the reversed hazard rate (denoted as r˜Ψ(κ) ) is
r˜Ψ(κ)=(1+κ)2/[1+(1+κ)1],κ[0,).
It is evident that r˜Ψ(κ) is strictly decreasing in κ > 0, this mean that r˜Ψ(κ)DRHR . It follows from (2.6) that Ψ ∈ SI.

In order to verify the validity of the SI is closed under mixing of non-crossing distributions, we provide the following result.

Theorem 3.2.

Suppose {M (θ | X), x ≥ 0} be a family of life distribution achieving the following requirements:

  1. (i)

    Mθ (.) ∈ SI for each θ ∈ ℝ+.

  2. (ii)

    θ1θ2.

  3. (iii)

    The family {Mθ (k)} is non-crossing function for θ1 and θ2.

    Then ψ (κ) of {M (θ | X), x ≥ 0} is belong to the class SI.

Proof.

Equation (2.5) demonstrate that

SΨ(κ)(1SΨ\(κ))+κSΨ\(κ)κ(13κ)12r˜Ψ(κ)πΨ3(κ),
This true when Mθ(k) ∈ SI, for all κ ∈ ℝ+. This inequality can be written as
(0κuΨ(u)du)3Ψ3(κ)((13κ)rΨ˜(κ)+12κπΨ3(κ)+1r˜Ψ(κ))+1κ,
where SΨ(κ)=0κxΨ(x)dx/Ψ(κ) , r˜Ψ(κ)=Ψ(κ)/κΨ(κ) and πΨ3(κ)=40xy3Ψ(y)dy/Ψ(κ) .

According to (3.1), the property of the non-crossing function and Fubini’s theorem we have

(0κuΨ(u)du)3=(0[0κuFα(u)dG(α)]du)30[0κuFα(u)]3dG(α)du.
From now on we assume that M (θ | X) = Exp (θ) and D (θ) = Exp (1), and κ = 1. We started by investigating that Ψ ∈ SI. Firstly, we can derive that
(01uΨ(u)du)3=(01u(1(1+u)1)du)3=(ln212)3Ψ3(1)(2r˜Ψ(1)+12πΨ3(1)+1r˜Ψ(1))+1
Hence, the mixture Ψ satisfies (3.1), and then Ψ ∈ SI.

Theorem 3.3.

The class of lifetime distribution SI is not closed under convolution.

Proof.

We start with two functions U (κ) and V (κ), where

U(κ)={κ,ifκ(0,1),0,ifκ(0,1),andV(x)={1eκ,ifκ>0,0,otherwise..
Clearly, Uunif (0, 1), while VExp (1). Now, consider the following mixture of two distributions U and V :
Eθ(x)=θU(x)+θ¯V(x),whereθ(0,1)andθ¯=1θ.
A random variable T is the lifetime of some devices with the distribution function Eθ. We started by investigating EθSI or EθSD. Hence, by using (2.5) we have
θEθ(κ)/κ=2κ(13κ)+2[SEθ(κ)(1SEθ\(κ))+κSEθ\(κ)]+r˜Eθ(κ)πEθ3(κ),
where SEθ(κ)=0κxEθ(x)dx/Eθ(κ) , r˜Eθ(κ)=Eθ(κ)/κEθ(κ) and πEθ3(κ)=40xy3Eθ(y)dy/Eθ(κ) . Then, we derive
SEθ(κ)=θκ33+θ¯(κ22+κeκ+eκ1)θκ+θ¯(1eκ)andr˜Eθ(κ)=θ+θ¯(eκ)θκ+θ¯(1eκ).
Furthermore
πEθ3(κ)=θ4κ55+θ¯(κ4+4κ2eκ+12κ2eκ+24(κeκ+eκ1))θκ+θ¯(1eκ).
Additionally, Eq. (2.4) and Eq. (3.3) satisfy the following relation
SEθ\(κ)=1(θ+θ¯(eκ))(θκ33+θ¯(κ22+κeκ+eκ1))(θκ+θ¯(1eκ))(θκ+θ¯(1eκ)).
Thus by setting (κ = 1, and θ = 1/2), we have θEθ (κ) is decreasing. Hence, we can decide that Eθ (κ) =SI i.e. Eθ (κ) =SD.

Let F1 and F2 be two independent distributions function of T1 and T2. Suppose F1 and F2 are SI and the distribution of T1 + T2 can be represented as

FT1+T2(κ)=(F1*F2)(κ)=0κF1(κu)dF2(u),κ+.
Let us take T1, T2Eθ (x) and the G is convolution of Eθ, i.e. C (κ) = Eθ * Eθ (κ). Equations (3.2) and (3.4) demonstrate that:
eθ(κ)=θ+θ¯exp(κ),
where eθ (.) is the density function of Eθ (.), and
C(κ)=0.5θ2κ2+θ¯[αβ],
where
α=2θ(exp(κ)+κ1),andβ=θ¯exp(κ)(1κ)+θ¯
Thus, SMP of C can present as follow:
SC(κ)=23κ3θ232κ2θ2+0.125κ4θ2+(2θθ¯θθ¯)ɛκ+θθ¯(κ2exp(κ)+2ɛκ)0.5θ2κ2+θ¯(αβ).
Moreover,
r˜C(κ)=θ+θ¯exp(κ)0.5θ2κ2+θ¯[αβ],
and
πC3(κ)=4(ρ(κ)+η(κ)+0.5θ2κ66+2θθ¯κ552θθ¯κ44θθ¯κ44)0.5θ2κ2+θ¯[αβ].
where SC(κ)=0κxC(x)dx/C(κ) , r˜C(κ)=C(κ)/κC(κ) , and πC3(κ)=40xy3C(y)dy/C(κ) , εκ = (−κ exp (−κ) − exp (−κ) + 1), ρ(κ) = (1 − θ)2(24−24κeκ − 16κ2eκκ4eκ − 24eκ) and η (κ) = (2θ (1 – θ) (1 – θ)2) (6 – 6κe–κ 4κ2e–κ 6e–κ) Then we can decide that ∂θC (κ) / ∂κ is I. Thus CSI. Therefore, T1 and T2SI. But their convolution C is belong to SI. It mean SI is not closed under convolution, in general. Hence, the proof is complete.

4 Properties of SR Ordering Under Operations

In this section, we focus our attention on a new stochastic comparison defined in terms of SR function. The next theorem provides that ≤θ(t) order lies between ≤RHR and ≤ θ(t).

Theorem 4.1.

Let X1 ≥ 0 and X2 ≥ 0 be two continuous random variables, with the distribution functions F1 (.) and F2 (.), and let αi(t)=0tviF1(v)dv/F1(t) and βi(t)=0tviF2(v)dv/F2(t) . Then we have

  1. 1.

    X1RHR X2X1θ(t) X2,

  2. 2.

    X1SMP X2X1θ(t) X2,

  3. 3.

    X1θ(t) X2X1VP X2.

Proof:

  1. (1)

    By definition of ≤RHR we can obtain

    0tα0(x)F1(x)F1(t)β0(x)F2(x)F2(t)dx0,forallt+.
    which means that X1θ(t) X2.

  2. (2).

    Note that X1SMP X2 implies

    α3(t)F1(t)β3(t)F2(t)isincreasingint.
    which, by using the same type of argument as used in proving theorem 3.1 of Nanda et al. 2003, gives
    0tα0(u)F1(u)du0tβ0(u)F2(u)duisincreasingint+.
    Hence, by (1.1), we have our result.

As follows from the theorem shown above, the next example indicates that Xθ(t) YX ≤RHR Y and Xθ(t) YXSMP Y. In addition, XVP YXθ(t) Y.

Example 4.1 :

Let X1 and X2 be to non-negative random variables with the distribution functions F1, and F2, respectively, which are given by

F1(u)={u20u21u>2
and
F2(u)={u220u1u2+261<u21u>2
Then, after some algebraic calculations, we deduce that
0tu(tu2)F1(u)du0tu(tu2)F2(u)du={81053t3t2t20t1t4(106t)(5t525t63+10t35t4t252+203)1<t21t>2,
and
0tα0(u)F1(u)du0tβ0(u)F2(u)du={2t0t16tt2+121<t21x>t.
By (4.1) we can conclude that X1θ(t) X2 does not hold in this case. While in parallel, Kayid and Izadkhah (2014, pp.595) showed in their counterexample 1 that X1SMP X2. Also, (4.2) is increasing in t, and hence X1VP X2.

The next result provides useful characterization of the SR order.

Theorem 4.2.

Suppose X1 ∈ ℝ+ and X2 ∈ ℝ+ be two continuous random variables with distribution functions F1 and F2, respectively, and let αi(x)=0xuiF1(u)du/F1(x) and βi(x)=0xuiF2(u)du/F2(x) , hence X1SMP X2 holds, if X1θ(t) X2 holds, therefore

mF1(t)mF2(t){SF1(t)SF2(t)}/t,
and
X1MPX2.

Proof.

If X1θ(t) X2 holds, we obtain

tF2(t)0tαi(u)F1(u)dutF1(t)0tβi(u)F2(u)duF1(t)F2(t)(α3(t)β3(t)).
It evident that
0t(tu){F1(u)F2(t)F2(u)F1(t)}du0tu3{F1(u)F2(t)F2(u)F1(t)}du/t,
and
t2(mF1(t)mF2(t))+t(SF2(t)SF1(t))0tx3F1(x)dx/F1(t)0tx3F2(x)dx/F2(t),
since X1SMP X2, Thus the following result is obtained:
0tu3F1(u)F2(t)u3F2(u)F1(t)du0,
consequently,
tmF1(t)tmF2(t)SF1(t)SF2(t),
which is equivalent to the required statement.

In next result develop some preservation of SMP and SR orders and SI class.

Proposition 4.1.

Let X1 ∈ ℝ+ be continuous random variable with distribution function F1 and SMP function SF1 (t), hence SF1 (t) is increasing in t if X1SMP X1 + X2 for any continuous random variable X2 ∈ ℝ+ independent of X1.

Proof:

By apply Fubini’s theorem and using change of order double integration, it is obvious that

SX1+X2(t)=0t0yyF1(yv)dF2(v)dy0tF1(tv)dF2(v),=0t0tv(y+v)F1(y)dydF2(v)0tF1(tv)dF2(v),=0tF1(tv)SX(tv)dF2(v)+0tvF1(tv)mX1(tv)dF2(v)0tF1(tv)dF2(v)SX1(t)+0tvF1(tv)mX1(tv)dF2(v)0tF1(tv)dF2(v).
This is true for SF1 (t) is increasing in t ∈ ℝ+. Therefore, we shall write the above expression as
SX1+X2(t)SX1(t),
when a=0tuF1(tu)mX1(tu)dF2(u)/0tF1(tu)dF2u and aSX1 + X2 (t). Hence, we have the required results.

Theorem 4.3.

Let X1 ∈ ℝ+ be continuous random variable with distribution function F1 and SR function θF1 (t), then X1SI if X1θ(t) X1 + X2 for any continuos random variable X2, that is independent of X1.

Proof:

Based on Fubini’s theorem and using change of order double integration, it is clear that

θX1+X2(t)+SX1+X22(t)=(t(1t))2+4t0t0y0xF1(xv)dF2(v)dxdy0tF1(tv)dF2(v)4t0t0vz3F1(zv)dF2(v)dz0tF1(tv)dF2(v)=(t(1t))2+4[t0t0tv0yvF1(x)dxdydF2(v)0tvtz3F1(zv)dzdF2(v)]0tF1(tv)dF2(v).
This is true for θF1 (t) increasing in t ∈ ℝ+. According to (4.3) we obtain
(t(1t))2+0tF1(tv)[θX1(tv)+SX12(tv)((tv)(1(tv)))2]dF2(v)40t0tv[(y+v)314y3]F1(y)dydF2(v)0tF1(tv)dF2(v)θX1(t)+SX12(t),
where
0t0tv[4v3+12vy2+12yv2]F1(y)dydF2(v)0t0tvy3F1(y)dydF2(v).
By apply proposition (4.1) we can prove that SX1+X22(t)SX12(t) . This completes the proof.

Theorem 4.4.

Suppose X1 ∈ ℝ+ and X2 ∈ ℝ+ are two random variables with distribution functions U1 and U2 respectively, and let ψ is strictly increasing and convex upwards and ψ (0) = 0. If X1θ(t) X2, then we have ψ (X1) ≤ θ(t) ψ (X2).

Proof:

Let ψ is differentiable with derivative ψ\. If X1θ(t) X2 then for any t > 0, we have

θU1(t)=4[t0t0xU1(u)dudx0tv3U1(v)dv]U1(t)(0tuU1(v)dvU1(t))24[t0t0xU2(u)dudx0tv3U2(v)dv]U2(t)(0tuU2(u)duU2(t))2=θU2(t).
It lead to
4ψ1(t)[0ψ1(t)0ψ1(v)U1(u)dudvU1(ψ1(t))0ψ1(t)0ψ1(v)U2(u)dudvU2(ψ1(t))]4[0ψ1(t)x3U1(x)dxU1(ψ1(t))0ψ1(t)x3U2(x)dxU2(ψ1(t))][(0ψ1(t)uU1(u)duU1(ψ1(t)))2(0ψ1(t)uU2(u)duU2(ψ1(t)))2]0.
According Kayid and Izadkhah (2014, Theorem 5, pp. 596) we can prove
X1SMPX2ψ(X1)SMPψ(X2)SU12(t)SU22(t)SU12(ψ(t))SU22(ψ(t))
since t ≥ 0 and SU1(t) and SU2 (t) are non-negative functions. According Mahdy (2012) we can decide that
0ψ1(t)0ψ1(v)U1(u)dudvU1(ψ1(t))0ψ1(t)0ψ1(v)U2(u)dudvU2(ψ1(t))=0t0vU1(ψ1(u))dudvU1(ψ1(t))0t0vU2(ψ1(u))dudvU2(ψ1(t)).
In addition, X1θ(t) X2 implies that,
0tx3U1(x)dxU1(t)0tx3U2(x)dxU2(t)
where t ∈ ℝ+. It is clear that
0ψ1(t)x3U1(x)dxU1(ψ1(t))0ψ1(t)x3U2(x)dxU2(ψ1(t)),
and
0ψ1(t)[v3U1(v)U1(ψ1(t))v3U2(v)U2(ψ1(t))]dv0.
On the other hand, ψ (X1) ≤θ(t) ψ (X2) iff, for all t > 0 and ψ (0) = 0,
0tv3Pr(ψ(X1)v)dvPr(ψ(X1)v)0tv3Pr(ψ(X2)v)dvPr(ψ(X2)v),
since
Pr(ψ(X1)x)=Pr(X1ψ1(x))=U1(ψ1(x)),
therefore,
0tx3U1(ψ1(x))dxU1(ψ1(t))0tx3U2(ψ1(x))dxU2(ψ1(t)).
If we let θ = ψ−1 (x) ⇒ x = ψ(θ) ⇒ dx/ = ψ/ (θ) then we obtain
0ψ1(t)ψ3(θ)ψ/(θ)U1(dθ)U1(ψ1(t))0ψ1(t)ψ3(θ)ψ/(θ)U2(dθ)U2(ψ1(t)),
and
0ψ1(t)β(x)[x3U1(θ)dθU1(ψ1(t))x3U2(θ)dθU2(ψ1(t))]0,
where β(s) = ψ3 (s) ψ/ (s) / s3. Its obvious that, if ψ (s) is non-negative and decreasing, we can conclude that ψ3 (s) / s3 is decreasing in s, it is implies that β (s) is decreasing in s. Therefore, the complete proof is given by apply lemma 7.1(b) of Barlow and Proschan (1981). .

5 Reliability Applications of SR Ordering

In the following results, we explore the possibility of apply a new techniques in statistical reliability theory. Let U1, U2, …, Un denote the component lifetimes of the system and assume that U1:n, U2:n, …, Un:n represent the ordered lifetimes of the components.

5.1 Order Statistics

Theorem 5.1.

Let U1, U2, …, Un and V1, V2, …, Vn be independent and identically distributed (i.i.d) copies of U and V, with distribution functions ℍ1 and ℍ2 respectively. If Un:nθ(t) Vn:n, then we have Uiθ(t) Vi.

Proof.

Let Un:nθ(t) Vn:n hold. Then we implies that

t0t0x{1n(u)2n(t)2n(u)1(n)(t)}dudx0tx3{1n(x)2n(t)2n(x)1n(t)}
Since,
k(u)=[i=1n[2ni(t)1ni(u)][1i1(t)2i1(u)]]1,
is non-negative and decreasing in u ≥ 0 for any t > 0. Now, we can derive
2(t)1(u)1(t)2(u)={1n(u)2n(t)2n(u)1n(t)}k(u),
and
t0t0x{1(u)2(t)2(u)1(t)}dudx0tv3{1(v)2(t)2(u)1(t)}dvforanyt>0,
based on Lemma 7.1(b) of Barlow and Proschan 1981. It lead to Uiθ(t) Vi, for i ∈ ℕ.

5.2 Discrete SR order

Definition 5.1.

Suppose X ∈ ℝ+ and Y ∈ ℝ+ be two random variables, with distribution functions FX and FY, and discrete strong mean past lifetime Sd,X (x) and Sd,Y (x) respectively. Then, it is said that X is smaller than or equal Y in discrete SR order (Xdθ(x) Y), if

Sd,X(x)+j=lxx1j3FX(j)2xFX(x)Sd,Y(x)+j=lxx1j3FY(j)2xFY(x),forxN+.
Given a sequence of absolutely continuous non-negative random variables Z1, Z2, …are i.i.d. random variables with common cumulative distribution function FZ1 and a common density function fZ1 Let 𝕄1N+ and 𝕄2N+ be two random variables which are independent of the Zi’s. Denote Z1:𝕄jmin{Z1, Z2, ..., Z𝕄j} and Z𝕄j:𝕄jmax{Z1, Z2, ..., Z𝕄j}, j = 1, 2. Below we consider the SR order between two such extreme order statistics.

Theorem 5.2.

Let 𝕄1d–θ(z) 𝕄2, then we have Z𝕄1:𝕄1θ(z) Z𝕄2:𝕄2.

Proof.

The density function of Z𝕄j:𝕄j is given by

fZ𝕄j:𝕄j(z)=n=1nFZ1n1(z)fZ1(z)Pr(𝕄j=n),
and the distribution function of Z𝕄j:𝕄j which is given by
H𝕄j:𝕄j(z)=n=1Fn(z)Pr(𝕄j=n),forallz>0,
for all z > 0, and nN. Thus, by Shaked & Shanthikumar (2007), we obtain
ψ(z,j)=0z0y{H𝕄j:𝕄j(u)y2H𝕄j:𝕄j(y)}dudy=n=1ϕ(z,n)τ(n,j),forallz>0andj=1,2,
where ϕ(z,n)=0z0y{Fn(u)y2Fn(y)}dudy and in addition, τ(n, j) = Pr (𝕄j = n). Let k(n,j)=i=nτ(i,j) for each nN and j = 1, 2. We observe that 𝕄1LR 𝕄2, implies 𝕄1θ(z) 𝕄2 by using Theorem 1.C.1 on page 43 of Shaked and Shanthikumar (2007), and Li and Zhang (2008). If 𝕄1LR 𝕄2, and thus k(n, j) is TP2 in (n, j) ∈ N × {1, 2}. But it is not hard to see that φ(z, n) isn’t TP2 in (n, j) ∈ N × {1, 2}, where det [φ(z, n)] is negative for any strictly increasing convex function and zFn(z)0zufZ𝕄j:𝕄j(u)duz3Fn(z) , for zR+. Also, therefore, by the basic composition formula (Theorem 5.1 on page 123 of Karlin (1968)) we can be seen that ψ(z, j) is not TP2 in z ≥ 0 and j ∈ {1, 2}. That is, Z𝕄1:𝕄1θ(z) Z𝕄2:𝕄2.

5.3 Inactivity time of random at a random

Let U and V be two the lifetimes of system C1 and system C2, respectively. Then the residual lifetime at random time V can be represented as

UV=UV|U>V.
It represents the residual life of system C1 at a time when system C2 fails. Dequan and Jinhua (2000) established a number of stochastic orders for UV under a lot of assumptions of U and V. In addition, Misra et al. (2008) studied adequate permissions for log-concavity and log-convexity of the residual life at random time. Stochastic orders are conducted under particular permissions on the concerned total life and random time and their preservation properties are established by Li and Zuo (2004). Dewan and Khaledi (2014) presented now stochastic orders between multivariate residual lifetime at random time and studies some characterizations. Furthermore suppose G (.) be distribution function of random variable V and Vis hold for V as well. Moreover, let U and V are independent. The random variable U(V) = VU |UV is called the inactivity time of U at a random time V. It has the distribution function
GU(V)(x)=0[G1(y)G1(yx)]dG2(y)/0G1(y)dG2(y),x0.
when U and V are U-independent. The following results shows that U(t)SMP U does imply U(V)SMP U, in addition, U(t)θ(t) U does imply U(V)θ(t) U.

Theorem 5.3:

U(V)SMP U for any V that is d-independent of U iff U(t)SMP U for all t ≥ 0.

Proof.

When U(t)SMP U for all t, u ≥ 0, we obtain the following:

0uv[G1(t)G1(tv)]dvdG2(y)G1(t)G1(tu)0uvG1(v)dvG1(u).
From this we deduce that
00uv[G1(t)G1(tv)]dvdG2(y)0[G1(t)G1(tu)]dG2(t)0G1(t)G1(tu)G1(u)0uxG1(x)dxdG2(y)0G1(t)dG2(t),=0[G1(t)G1(tu)]S(u)dG2(y)0[G1(t)G1(tu)]dG2(t)=S(u),foranyu0
by using (5.1), we can decide that U(V)SMP U. Further, let U(V)SMP U holds for all value of y and let y equal to constant then U(t)SMP U for all t ≥ 0.

Proposition 5.1.

U(V)θ(t) U for any V that is d-independent of U iff U(t)θ(t) U for all t ≥ 0.

Proof.

With same steps in theorem 8, and by using (2.4) and (5.1) and we can achieve its proof.

In the following example, we can illustrate Theorem 5.3 and Proposition 5.1.

Example 5.1:

Let U, and V denote the random variable have distribution functions G1, and G2 respectively, which are given by

G1(u)={ubforu[0,b)1forubandG2(u)=1exp(λu),u+.
and thus GU(V) (u) = . It is clear that
0uvGU(V)(v)dvGU(V)(u)=0uvG1(v)dvG1(u)=0uvG(u)(v)dv=u23.
Hence, U(V)SMP UU(t)SMP U. In addition, by using (2.2), we get that U(t)θ(t) U, also, U(V)θ(t) U.

References

[1]RE Barlow and F Proschan, In: Statistical Theory of Reliability and Life Testing, McArdle Press, Silver Spring, 1981.
[3]WJ Hall and JA Wellner, In: M Csorgo, JNK Rao, and AKMdE Saleh (editors), Mean Residual Life, in: Statistics and Related Topics, North Holland, Amesterdam, 1981, pp. 169OE184.
[4]S Karlin, In: Total Positivity, Stanford University Press, Stanford, California, Vol. I, 1968.
[12]WL Smith, In: Biometrika, Vol. 46, 1959, pp. 1-29. on the cumulants of renewal processes.
Journal
Journal of Statistical Theory and Applications
Volume-Issue
17 - 1
Pages
172 - 192
Publication Date
2018/03
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
https://doi.org/10.2991/jsta.2018.17.1.13How to use a DOI?
Copyright
Copyright © 2018, the Authors. Published by Atlantis Press.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Mervat Mahdy
PY  - 2018
DA  - 2018/03
TI  - On Some Aspects of Strong Risk Class and Associated Ordering
JO  - Journal of Statistical Theory and Applications
SP  - 172
EP  - 192
VL  - 17
IS  - 1
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.2018.17.1.13
DO  - https://doi.org/10.2991/jsta.2018.17.1.13
ID  - Mahdy2018
ER  -