# On the Generalized Decreasing Mean Time to Failure or Replaced Ordering

Authors: Haiyan Wang; Diantong Kang; Lei Yan
Journal: Computational Intelligence and Neuroscience
 (2021)

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## Abstract

In this paper, we establish two new stochastic orders, DMTFR (decreasing mean time to failure or replaced) and GDMTFR (generalized decreasing mean time to failure or replaced), and mainly investigate properties of the GDMTFR order. Some characterizations of the GDMTFR order are given. The implication relationships between the DMTFR and the GDMTFR orders are considered. Also, closure and reversed closure properties of the new order GDMTFR are investigated. Meanwhile, several illustrative examples that meet the GDMTFR order are shown as well.


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## Body

## 1. Introduction and Preliminaries

In risk investment fields, we often need to compare or select two risk assets [1]. Similarly, two lifetimes of two systems or units also need to be compared in engineering technologies [2]. To do these things, some refined stochastic orders were defined in statistics. For more details on stochastic orders, one may refer to the studies by Maria Fernandez-Ponce et al. [3], Belzunce [4], Müller and Stoyan [5], Li and Yam [6], Shaked and Shanthikumar [7, 8], Zhao and Balakrishnan [9], Sunoj et al. [10], Kang [11, 12], Yan [13], Kang and Yan [14], Vineshkumar [15], and the references therein. However, sometimes, we need to compare two risk assets by the aid of the third referred system, such as when we compare two risk assets [16], their values are changeable with the settlement time or with the kind of valuation currency [17]. To solve this problem, we establish a new stochastic order by introducing a common measure factor, namely, referred function.LetX be a nonnegative and absolutely continuous random variable. X has distribution function FX with 0 being the left endpoint of its support, survival function F¯X=1−FX, and density function fX. Let again FX−1 be the right-continuous inverse function of FX defined by(1)FX−1p=supx∣FXx≤p,0<p<1.In many important real areas, such as reliability, economics, management sciences, information sciences, and other related fields, stochastic comparisons are paid much attention and have been used as sharp tools in dealing with some random problems in recent years. Based on comparisons of residual life, mean residual life, mean inactivity time, failure rate, and many aging concepts are presented earlier or later. In these aspects, interested readers can refer to the studies by Barlow and Proschan [18], Shaked et al. [7], Shaked and Shanthikumar [8], Müller and Stoyan [5], Kochar et al. [19], Ahmad et al. [20], Knopik [21], Kochar et al. [19], and Li and Shaked [22] for some existing results.In reality, a nonnegative random variableX is often called a life representing the lifetimes of a device. The life distribution classes IFR (increasing failure rate), IFRA (increasing failure rate average), DMRL (decreasing mean residual life), NBUE (new better than used in expectation), and MTFR (mean time to failure or replaced) classes of life distributions are commonly used to describe the ageing features of units or systems, for example, see Barlow and Proschan [18].In particular, it is well known that (see Barlow and Campo [23], Klefsjö [24], Marschall [25])(2)IFR⊂DMRL⊂NBUE.Knopik [21] introduced a new ageing class MTFR such that(3)IFR⊂MTFR⊂NBUE.It is worthwhile to mention that, in Barlow’s study [26], it is proved for the absolutely continuous distribution that(4)IFRA⊂MTFR.Kochar et al. [19] showed that, for any random variable, the inclusion (4) is also valid.Consider an age replacement police as the one in which a unit is replaced byt time units after installation or at failure, whichever occurs first, and then the expected value for the first time to an in-service is (see Barlow and Proschan [18])(5)MXt=∫0tF¯XxdxFXt, fort∈x:FXx>0.WhenMXt is monotonic, the case was considered by Barlow and Campo [23], Marschall and Proschan [27], and Klefsjö [24]. In Knopik [21], and the ageing class MTFR (mean time to failure or replaced) of lifetime distribution is introduced, and it is proved that the MTFR class is closed under the operation of maximum for independence. In Kochar et al. [19], it is showed that the class MTFR is closed under weak convergence of distributions and convolution, and the dual family MTFRD is closed under noncrossing mixtures.Definition 1.
The random variableX belongs to the mean time to failure with replacement (MTFR) class if the function X is decreasing for t∈x|FXx>0.
Assume thatX has finite mean EX=μX. The residual life of X at time t>0 is defined as Xt=X−t∣X>t. Then, the mean residual life of X at time t>0 is(6)μXt=EXt=∫t+∞F¯XxdxF¯Xt,fort≥0.
IfμXt is decreasing, then we say that X (or FX) is in the decreasing mean residual (DMRL) life distribution class and denoted by XorFX∈DMRL.
Emad-Edlin [28] proposed and studied the decreasing mean residual life (DMRL) ordering. Maria Fernandez-Ponce et al. [3] and Belzunce [4] proposed and studied the right spread order. Subsequently, Kochar et al. [19] proposed and studied the total time on test transform order and excess wealth order which is equivalent to the right spread order.
The total time on test (TTT) transform functions ofX and Y are defined, respectively, as(7)TXp≡∫0FX−1pF¯Xxdx,TYp≡∫0GY−1pG¯Yxdx,for allp∈0,1.
The right spread (RS, for short) functions ofX and Y are defined as, for all p∈0,1, respectively,(8)SXp=∫FX−1p∞F¯Xxdx=EX−FX−1p+=EmaxX−FX−1p,0,SYp=∫GY−1p∞G¯Yxdx=EY−GY−1p+=EmaxY−GY−1p,0,where F¯X=1−FX and G¯Y=1−GY are the survival functions of FX and GY, respectively, and x+=maxx,0.
Now, we recall several stochastic orders from Shaked and Shanthikumar [8].Definition 2.
LetX and Y be two nonnegative random variables with distribution functions FX,GY such that FX0=GY0=0, survival functions F¯X≡1−FX,G¯Y≡1−GY, right-continuous inverse functions FX−1,GY−1, and finite means μX,μY, respectively [8].(1)
X is said to be smaller than Y in the TTT transform order (denoted by X≤tttY) if(9)∫0FX−1pF¯Xxdx≤∫0GY−1pG¯Yxdx,for allp∈0,1.(2)
X is said to be smaller than Y in the right spread order (denoted by X≤rsY) if(10)∫FX−1p∞F¯Xxdx≤∫GY−1p∞G¯Yxdx,for allp∈0,1.(3)
X is said to be smaller than Y in the NBUE (new better than used in expectation) order (denoted by X≤nbueY) if(11)∫FX−1p∞F¯Xxdx∫GY−1p∞G¯Yxdx≤μXμY,for allp∈0,1.(4)
X is said to be smaller than Y in the location independent riskier order (denoted by X≤lirY) if(12)∫0FX−1pFXxdx≤∫0GY−1pGYxdx,for allp∈0,1.(5)
X is said to be smaller than Y in the DMRL order (denoted by X≤dmrlY) if(13)μYGY−1pμXFX−1p=∫GY−1p∞G¯Yxdx∫FX−1p∞F¯Xxdxis increasing inp∈0,1.
From Definition1, we easily have the following result without proof.Lemma 1.
LetX and Y be two continuous and nonnegative random variables with respective distribution functions FX and GY, density functions fX and gY, and the right-continuous inverse functions FX−1p and GY−1p of FX and GY, respectively. Then, the following statements are equivalent:(1)
X≤tttY.(2)
The inequality(14)∫0FX−1pF¯Xxdx≤∫0GY−1pG¯Yxdx,for allp∈0,1,is valid.(3)
The inequality(15)∫0tF¯XxfXxgYGY−1FXx−1dx≥0,for allt>0,is valid.Definition 3.
LetX and Y be two nonnegative random variables with distribution functions FX,GY such that FX0=GY0=0, survival functions F¯X≡1−FX,G¯Y≡1−GY, density functions fX,gY, and right-continuous inverse functions FX−1,GY−1, respectively.(1)
X is said to be smaller than Y in the usual stochastic order (denoted by X≤stY), if F¯Xx≤G¯Yx for all x>0.(2)
X is said to be smaller than Y in the star-shaped order (denoted by X≤∗Y), if the function GY−1FXx/x is increasing in x>0.(3)
X is said to be smaller than Y in the convex order (denoted by X≤cY), if the function GY−1FXx is convex for x>0.(4)
X is said to be smaller than Y in the dispersive order (denoted by X≤dispY), if fXFX−1p≥gYGY−1p for all p∈0,1.
LetX and X be two absolutely continuous nonnegative random variables with respective distribution functions FX and GY such that FX0=GY0=0, and density functions fX and gY, and right-continuous inverse functions of FX−1 and GY−1, respectively.
We say thatX is smaller than Y in the increasing concave order (denoted by X≤icvY) if(16)∫0tF¯Xxdx≤∫0tG¯Yxdx,for allt>0.
Here,(17)∫0tF¯Xxdx=EminX,t,∫0tG¯Yxdx=EminY,t.
Hence, the order≤icv is a direct comparison for the mean service times of X and Y. In insurance theory, ∫0dF¯Xxdx=EminX,d is the average losses undertaken by the insured when the deductible excess is d. Hence,(18)TXp≡∫0FX−1pF¯Xxdx=EminX,FX−1p,TYp≡∫0GY−1pG¯Yxdx=EminY,GY−1p,that is, TXp and TYp are the mean service times of X and Y before the respective equal probability time points. We wish to compare TXp and TYp by means of a ratio TYp/TXp Based on such an idea, we give Definition 4.
LetX and Z be two nonnegative continuous random variables with respective distribution functions FX and HZ such that FX0=HZ0=0, right-continuous inverse functions FX−1 and HZ−1, survival functions F¯X and H¯Z, and density functions fX and hZ, respectively.Definition 4.
X is said to be smaller than Z in the decreasing mean time to failure (DMTFR) order (denoted by X≤dmtfrZ), if(19)∫0HZ−1pH¯Zxdx∫0FX−1pF¯Xxdx,is increasing in p∈0,1.
From Definition3, the following lemma is obvious, the proof is omitted here.Lemma 2.
The following statements are equivalent:(1)
X≤dmtfrZ.(2)
The functionTZp/TXp is increasing in p∈0,1.(3)
The inequality(20)∫0tF¯XxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0,is valid.(4)
The inequality(21)tF¯XtfXthZHZ−1FXt−HZ−1FXtt−∫0txfXthZHZ−1FXt−HZ−1FXxxdF¯Xx≥0,for allt>0,is valid.Shaked and Shanthikumar [8] studied the generalized TTT transform and proposed the generalized TTT transform order. Bartoszewicz and Benduch [29] further studied some properties of the generalized TTT transform by iteration. Motivated by their excellent works, we establish the following two new stochastic orders, the RDMTFR and the GDMTFR orders, see Definitions 5 and 6, respectively.Definition 5.
LetX,Z, and Y be three random variables. X is said to be smaller than Z relative to Y in the relative DMTFR order (denoted by X≤rdmtfrZ r.t. Y), if(22)∫0HZ−1pgYGY−1HZxdx∫0FX−1pgYGY−1FXxdx,is increasing in p∈0,1.
From Definition5, the following lemma is obvious, and the proof is omitted here.Lemma 3.
The following statements are equivalent:(1)
X≤rdmtfrZ r.t. Y.(2)
The functionTZp;Y/TXp;Y is increasing in p∈0,1.(3)
The inequality(23)∫0tgYGY−1FXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0,is valid.(4)
The inequality(24)tgYGY−1FXtfXthZHZ−1FXt−HZ−1FXtt−∫0txfXthZHZ−1FXt−HZ−1FXxxdgYGY−1FXx≥0,for allt>0,is valid.It can be easily seen that whenY is an exponential random variable with a rate λ>0, that is, Y∼Eλ, there exists a simple relationship between the relative DMTFR order and the DMTFR order:(25)X≤rdmtfrZr.t.Y⟺X≤dmtfrZ.Assume thatψ is a real function defined on 0,1. Denote by(26)TXp;ψ=∫0FX−1pψFXxdx,TZp;ψ=∫0HZ−1pψHZxdx,for allp∈0,1.TXp;ψ and TZp;ψ are called the generalized TTT transforms of X and Z relative to ψ, respectively. Then, we give the following definition.Definition 6.
LetX and Z be two absolutely continuous nonnegative random variables; assume that ψ is a real function defined on 0,1. X is said to be smaller than Z with respect to ψ in the generalized DMTFR order (denoted by X≤gdmtfrZ w.r.t. ψ), if the function(27)∫0HZ−1pψHZxdx∫0FX−1pψFXxdx,is increasing in p∈0,1.Remark 1.
One can verify that the GDMTFR order is a partial order relation. The reflexive and transitive are evident; the antisymmetric is as follows (see Theorem1 (30) as follows).
X≤gdmtfrZ w.r.t. ψ and Z≤gdmtfrX w.r.t. ψ hold simultaneously, if and only if Y=kX+b almost surely, where k and b are any real numbers such that Y is nonnegative and k≠0.Remark 2.
It can be seen that the GDMTFR and DMRL orders do not have necessary implication relations each other. Especially, the DMTFR and DMRL orders do not have direct implication relationships each other.
From Definition5, the following lemma is obvious, and the proof is omitted here.
It can be easily seen that whenψu=gYGY−1u,u∈0,1, there exists a simple relationship between the GDMTFR order and the RDMTFR order:(28)X≤gdmtfrZw.r.t.ψ⟺X≤rdmtfrZr.t.Y.
Now, we consider a system composed of same components. We say that the system preserves some properties if the components of this system possess some properties, and we conclude from the structure of the system that the system also possesses the same property. And conversely, we say that this system has the reversed preservation for some property if the system has some property. According to the structure of the system, we conclude that the components of this system also have the same property.
For two systems, if their two components satisfy some stochastic order relation, by the structure of these two systems, we conclude that the two systems also satisfy the same stochastic order relation; then, we say that these systems of the structure preserve this stochastic order relation, or equivalently, we say that this stochastic order relation possesses closure property under the structure. Conversely, if two systems satisfy some stochastic order relation, by the structure of these two systems, we conclude that the two components of the two systems also satisfy the same stochastic order relation, and then we say that these systems of the structure reversely preserve this stochastic order relation, or equivalently, we say that this stochastic order relation possesses reversed closure property under the structure.
In reliability theory, such two problems are often of interest: one is to investigate the closure properties of a stochastic order, and the other is to examine the reversed closure properties of a stochastic order under several reliability operations, such as increasing convex transforms and taking of maxima and minima.
To prove our main results, we first introduce the following lemma from Barlow and Proschan [18], which plays a key role in the proofs of this paper and are repeatedly used in the sequel.Lemma 4.
LetW be a measure on the interval a,b, not necessarily nonnegative, where −∞≤a<b≤+∞. Let h be a nonnegative function defined on a,b. If ∫atdWx≥0. for all t∈a,b and if h is decreasing, then(29)∫abhxdWx≥0.In the next, we assume that all the random variables, under consideration, are nonnegative and continuous.Throughout this paper, the termincreasing stands for monotone nondecreasing and decreasing stands for monotone nonincreasing. Assume that all the random variables under considerations are nonnegative and absolutely continuous with 0 as the left endpoint of their supports, and that all the integrals and expectations involved are always finite. All the encountered ratios are always supposed to be well defined.In this paper, we devote our interest to the closure properties of the GDMTFR order. The paper is organized as follows. First, in Section2, we consider characterizations of GDMTFR order. We investigate the implication relationships between DMTFR and GDMTFR orders in Section 3. The closure and reversed closure properties of the GDMTFR order are studied in Section 4. Finally, in Section 5, we give two examples which meet the GDMTFR order.In the following, we always assume thatX and Z are two absolutely continuous and nonnegative random variables with distribution functions FX and HZ such that FX0=HZ0=0, survival functions F¯X≡1−F and H¯Z≡1−HZ, density functions fX and hZ, and right-continuous inverse functions FX−1 and HZ−1 of FX and HZ, respectively.


## 2. Characterizations of the GDMTFR Order

Now, we explore some characterizations of the GDMTFR order. First, we give a result by Definition 1.6, which will be useful in the proofs of upcoming theorems.Theorem 1.
The following statements are equivalent:(1)
X≤gdmtfrZ w.r.t. ψ.(2)
The inequality(30)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0,is valid.(3)
The inequality(31)tψFXtfXthZHZ−1FXt−HZ−1FXtt−∫0txfXthZHZ−1FXt−HZ−1FXxxdψFXx≥0,for allt>0,is valid.Proof.
We only give proof for the case of1⟺2. Suppose that the function(32)∫0HZ−1pψHZxdx∫0FX−1pψFXxdx,is increasing in p∈0,1. By differentiating, we have that the numerator of the derivative of this ratio is(33)ψphZHZ−1p∫0FX−1pψFXxdx−ψpfXFX−1p∫0HZ−1pψHZxdx,for allp∈0,1.
Since the function(34)∫0HZ−1pψHZxdx∫0FX−1pψFXxdx,is increasing in p∈0,1, we have(35)∫0FX−1pfXFX−1phZHZ−1pψFXxdx−∫0HZ−1pψHZxdx≥0,for allp∈0,1.
LettingHZx=FXy in the second integral of the left-hand side of inequality (35) and then letting FX−1p=t, we obtain that(36)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0.
And, the above deduction is reversible. Therefore, the proof of the theorem is complete.Definition 7.
LetX and Z be nonnegative absolutely continuous random variables with distribution functions FX,HZ such that FX0=HZ0=0, density functions fX,hZ, and right-continuous inverse functions FX−1,HZ−1, respectively.(1)
X is said to be smaller than Z in the starshaped order (denoted by X≤∗Z), if the function HZ−1FXx/x is increasing in x>0.(2)
X is said to be smaller than Z in the convex order (denoted by X≤cZ), if the function HZ−1FXx is convex for x>0.(3)
X is said to be smaller than Z in the dispersive order (denoted by X≤dispZ), if fXFX−1p≥hZHZ−1p for all p∈0,1.Theorem 2.
Letψ be a nonnegative function defined on 0,1. Then, for any θ>0, X≤gdmtfrθX w.r.t. ψ.Theorem 3.
Letψ be a nonnegative function defined on 0,1. If X≤gdmtfrZ w.r.t. ψ, then for any θ>0, X≤gdmtfrθZ w.r.t. ψ.Proof.
Suppose thatX≤gdmtfrZ w.r.t. ψ, then from inequality (30), we have(37)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0.
In turn, from (30), X≤gdmtfrθZ w.r.t. ψ if and only if, for all t>0,(38)∫0tψFXxfXthθZHθZ−1FXt−fXxhθZHθZ−1FXxdx=θ∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0.
Clearly, if inequality (37) holds, then inequality (38) also holds, that is, X≤gdmtfrθZ w.r.t. ψ. Therefore, the proof of the theorem is complete.Remark 3.
Theorem3 states that the order ≤gdmtfr is scale invariant with respect to the compared variables.Theorem 4.
IfX≤gdmtfrZ w.r.t. ψ, then for any a>0, X≤gdmtfrZ w.r.t. aψ.Remark 4.
Theorem4 states that the ≤gdmtfr order is scale invariant with respect to the referred function.Theorem 5.
IfX≤gdmtfrZ w.r.t. ψ, then for any θ>0, θX≤gdmtfrθZ w.r.t. ψ.Remark 5.
Theorem5 states that the order ≤gdmtfr is scale equivalent with respect to the compared random variables.Theorem 6.
IfX≤gdmtfrZ w.r.t. ψ, then for any θ>0 and a>0, X≤gdmtfrθZ w.r.t. aψ.Remark 6.
Theorem6 states that the order ≤gdmtfr is scale invariant with respect to the compared random variables and the referred random variable.Theorem 7.
LetHZ0=FX0=0. X≤gdmtfrZ w.r.t. ψ and X≥gdmtfrZ w.r.t. ψ if and only if there exists some θ>0 such that FXx=HZθx, that is, Z=dθX, where “=d” means equality in distributions.Proof.
Suppose thatX≤gdmtfrZ w.r.t. ψ and X≥gdmtfrZ w.r.t. ψ. From Definition 6, we have that there exists some θ>0 such that(39)∫0HZ−1pψHZxdx∫0FX−1pψFXxdx=θ.
That is,(40)∫0HZ−1pψHZxdx=θ∫0FX−1pψFXxdx.
Differentiating both sides of above equality, we obtainHZ−1p′=FθX−1p′. Thus, HZ−1p=FθX−1p+c, where c is any real number. By the assumption of HZ0=FX0=0, c=0 and then HZ−1p=FθX−1p. Hence, Z=dθX, and this is the stated result.
The following theorem gives a sufficient condition for the GDMTFR order.Theorem 8.
Letψ be a nonnegative function defined on 0,1. If X≤cZ, then X≤gdmtfrZ w.r.t. ψ.Proof.
From (30) X≤gdmtfrZ w.r.t. ψ if and only if(41)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0.
IfX≤cZ, then the function fXx/gYGY−1FXx is increasing; hence,(42)fXxhZHZ−1FXx≤fXthZHZ−1FXt,for all0≤x≤t.
Moreover,ψFXx≥0 for all x≥0. From inequality (42), we see that inequality (41) holds. That is, X≤gdmtfrZ w.r.t. ψ. Therefore, the proof of the theorem is complete.


## 3. Implication Relationships between DMTFR and GDMTFR Orders

Recall from [5] that a function h:0,1⟶0,1 is called star-shaped (anti-star-shaped) if hx/x is increasing (decreasing). h is star-shaped (anti-star-shaped) with respect to the point 1,1 if 1−hx/1−x is increasing (decreasing) in x.Now, we propose the following notions about real functions.Definition 8.
For a positive integer numbern, we say that(1)
h is star-shaped (anti-star-shaped) with respect to the point 1,0 if hx/1−x is decreasing (increasing).(2)
h is dual (anti-star-shaped) star-shaped of order n if hx/1−xn is increasing (decreasing).(3)
h is star-shaped (anti-star-shaped) of order n with respect to the point 1,0 if hx/1−xn is decreasing (increasing) in x∈0,1.
LetX and Z be two nonnegative continuous random variables with respective distribution functions FX and HZ having 0 as the common left endpoint of their supports. Let again ψ be a nonnegative function defined on 0,1. Assume that X1,…,Xn and Z1,…,Zn are the i.i.d. copies of X and Z, respectively, and that Xk:n and Zk:n, k=1,2,…,n, are the order statistics of X and Z, respectively. Let Xk:n have distribution function FXk:nx, survival function F¯Xk:nx, and density function fXk:nx, then for all x≥0,(43)FXk:nx=Bk,n−k+1FXx,F¯Xk:nx=Bn−k+1,kF¯Xx,fXk:nx=βk,n−k+1FXxfXx=n!k−1!n−k!FXxk−1F¯Xxn−kfXx,where Bk,n−k+1⋅ and βk,n−k+1⋅ are the distribution function and density function of a beta distribution with parameters k and n−k+1, respectively.
The following Theorem9 gives some conditions under which the DMTFR order implies the GDMTFR order.Theorem 9.
(1)
Assume that the functionψ°Bk,n−k+1 is star-shaped with respect to the point 1,0. If X≤dmtfrZ, then(44)Xk:n≤gdmtfrZk:nw.r.t.ψ,for allk=1,2,…,n,whereψ°Bk,n−k+1 is the compound of ψ and Bk,n−k+1.(2)
Assume that the functionψ is star-shaped with respect to the point 1,0. If X≤dmtfrZ, then(45)X≤gdmtfrZw.r.t.ψ.(3)
Assume that the functionψ is star-shaped of order n with respect to the point 1,0. If X1:n≤dmtfrZ1:n, then(46)X≤gdmtfrZw.r.t.ψ.(4)
Assume that the functionψ is dual anti-star-shaped of order n. If Xn:n≤dmtfrZn:n, then(47)X≤gdmtfrZw. r. t.ψ.(5)
Assume that the functionψ star-shaped with respect to the point 1,0. If for some 1≤k≤n, Xk:n≤dmtfrZk:n, then(48)Xk:n≤gdmtfrZk:nw.r.t.ψ,for allk=1,2,…,n.Proof.
Barlow and Proschan [18] proved that, for all x≥0,(49)HZk:n−1FXk:nx=H¯Zk:n−1F¯Xk:nx=HZ−1FXx.
It can be proven that(50)fXk:nxhZk:nHZk:n−1FXk:nxfXxhZHZ−1FXx.(1)
Suppose thatX≤dmtfrZ. Then, we have, for all t>0,(51)∫0tF¯XxfXthZHZ−1FXt−fXxhZHZ−1FXxdx=∫0tF¯XxfXk:nthZk:nHZk:n−1FXk:nt−fXk:nxhZk:nHZk:n−1FXk:nxdx≥0.Sinceψ°Bk,n−k+1 is star-shaped with respect to the point 1,0, then ψ°Bk,n−k+1u/1−u is decreasing in u∈0,1. Hence, the function(52)hx=ψ°Bk,n−k+1FXx1−FXx=ψFXk:nxF¯Xx,is nonnegative decreasing inx≥0. From Lemma 4 and inequality (51), we have(53)∫0tψFXk:nxfXk:nthZk:nHZk:n−1FXk:nt−fXk:nxhZk:nHZk:n−1FXk:nxdx≥0,for allt>0,which is equivalent to(54)Xk:n≤gdmtfrZk:nw.r.t.ψ,for allk=1,2,…,n.(2)
Suppose thatX≤dmtfrZ. Then, for all t>0,(55)∫0tF¯XxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0.Since the functionψ is star-shaped with respect to the point 1,0, the function ψu/1−u is decreasing in u∈0,1. Thus, hx=ψFXx/F¯Xx is nonnegatively decreasing. From Lemma 4 and (55), we have(56)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0,which states thatX≤gdmtfrZ w.r.t. ψ.(3)
Suppose thatX1:n≤dmtfrZ1:n. Then, for all t>0,(57)∫0tF¯X1:nxfX1:nthZ1:nHZ1:n−1FX1:nt−fX1:nxhZ1:nHZ1:n−1FX1:nxdx≥0.Sinceψ is star-shaped of order n with respect to the point 1,0, the function(58)ψx1−xn,is decreasing inx, which leads to the function(59)hx=ψFXx1−FXxn=ψFXxF¯Xxn=ψFXxF¯X1:nx.being nonnegative decreasing. Moreover,(60)fX1:nxhZ1:nHZ1:n−1FX1:nx=fXxhZHZ−1FXx.From Lemma4, (57), and (60), we have(61)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0,which is equivalent to(62)X≤gdmtfrZw.r.t.ψ.(4)
Suppose thatXn:n≤dmtfrZn:n. Then, for all t>0,(63)∫0tFXn:nxfXn:nthZn:nHZn:n−1FXn:nt−fXn:nxhZn:nHZn:n−1FXn:nxdx≥0.Sinceψ is dual anti-star-shaped of order n, the function ψx/1−xn is nonnegative decreasing in x∈0,1, which leads to the function(64)hx=ψFXx1−FXxn=ψFXxF¯Xn:nx.being nonnegative decreasing. Moreover,(65)fXn:nxhZn:nHZn:n−1FX1:nx=fXxhZHZ−1FXx.From Lemma4, (63)–(65), we have(66)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0,and this asserts that(67)X≤gdmtfrZw.r.t.ψ.(5)
Let the functionψ be anti-star-shaped with respect to the point 1,0. Suppose that, for some 1≤k≤n, Xk:n≤dmtfrZk:n, then for all t>0,(68)∫0tFXk:nxfXk:nthZk:nHZk:n−1FXk:nt−fXk:nxhZk:nHZk:n−1FXk:nxdx≥0.
Since the functionψ is star-shaped with respect to the point 1,0, the function ψx/1−x is nonnegatively decreasing in x∈0,1. Thus, φx=ψFXk:nx/F¯Xk:nx is nonnegatively decreasing. From Lemma 4 and (68), we have(69)∫0tψFXk:nxfXk:nthZk:nHZk:n−1FXk:nt−fXk:nxhZk:nHZk:n−1FXk:nxdx≥0,for allt>0,which asserts that(70)Xk:n≤gdmtfrZk:nw.r.t.ψ,for allk=1,2,…,n.
Therefore, the proof of the theorem is complete.
Theorem10 gives some conditions under which the GDMTFR order implies the DMTFR order.Theorem 10.
(1)
Assume that the functionψ is anti-star-shaped with respect to the point 1,0. If for some 1≤k≤n, Xk:n≤gdmtfrZk:n w.r.t. ψ, then(71)Xk:n≤dmtfrZk:n,for allk=1,2,…,n.(2)
Assume that the functionψ is anti-star-shaped with respect to the point 1,0. If X≤gdmtfrZ w.r.t. ψ, then(72)X≤dmtfrZ.(3)
Assume that the functionψ is dual star-shaped of order n. If X≤gdmtfrZ w.r.t. ψ, then(73)Xn:n≤dmtfrZn:n.(4)
Assume that the functionψ is anti-star-shaped of order n with respect to the point 1,0. If X≤gdmtfrZ w.r.t. ψ, then(74)X1:n≤dmtfrZ1:n.(5)
Assume that the functionψ is increasing in u∈0,1. If X≤dmtfrZ w.r.t. ψ, then(75)Xk:n≤gdmtfrZk:n,for allk=1,2,…,n.Proof.
(1)
Suppose that, for some1≤k≤n, Xk:n≤gdmtfrZk:n w.r.t. ψ. Then, for all t>0,(76)∫0tψFXk:nxfXk:nthZk:nHZk:n−1FXk:nt−fXk:nxhZk:nHZk:n−1FXk:nxdx≥0.Since the functionψ is anti-star-shaped with respect to the point 1,0, the function 1−u/ψu is nonnegatively decreasing in u∈0,1. Thus, hx=F¯Xk:nx/ψFXk:nx is nonnegatively decreasing. From Lemma 4 and (76), we have(77)∫0tF¯Xk:nxfXk:nthZk:nHZk:n−1FXk:nt−fXk:nxhZk:nHZk:n−1FXk:nxdx≥0,for allt>0,which asserts that(78)Xk:n≤dmtfrZk:n,for allk=1,2,…,n.(2)
Suppose thatX≤gdmtfrZ w.r.t. ψ. Then, for all t>0,(79)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0.Since the functionψ is anti-star-shaped with respect to the point 1,0, the function 1−u/ψu is decreasing in u∈0,1. Thus, hx=F¯Xx/ψFXx is nonnegatively decreasing. From Lemma 4 and (79), we have(80)∫0tF¯XxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0,which states that(81)X≤dmtfrZ.(3)
Suppose thatX≤gdmtfrZ w.r.t. ψ. Then, for all t>0,(82)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0.Sinceψ is dual star-shaped of order n, the function 1−un/ψu is nonnegative decreasing in u∈0,1, which leads to the function(83)φx=1−FXxnψFXx=F¯Xn:nxψFXx,being nonnegative decreasing. Moreover,(84)fXxhZHZ−1FXx=fXn:nxhZn:nHZn:n−1FX1:nx.From Lemma4, (82)–(84), we have(85)∫0tF¯Xn:nxfXn:nthZn:nHZn:n−1FXn:nt−fXn:nxhZn:nHZn:n−1FXn:nxdx≥0,for allt>0,and this asserts that(86)Xn:n≤dmtfrZn:n.(4)
Suppose thatX≤gdmtfrZ w.r.t. ψ. Then, for all t>0,(87)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0.Sinceψ is anti-star-shaped of order n with respect to the point 1,0, the function 1−un/ψu is decreasing in u∈0,1, and this leads to the function(88)hx=1−FXxnψFXx=F¯X1:nxψFXx,being nonnegatively decreasing. Moreover,(89)fX1:nxhZ1:nHZ1:n−1FX1:nx=fXxhZHZ−1FXx.From Lemma4, (87)–(89), we have(90)∫0tF¯X1:nxfX1:nthZ1:nHZ1:n−1FX1:nt−fX1:nxhZ1:nHZ1:n−1FX1:nxdx≥0,for allt>0,which is equivalent to(91)X1:n≤dmtfrZ1:n.(5)
Suppose thatX≤dmtfrZ w.r.t. ψ. Then, we have, for all t>0,(92)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx=∫0tψFXxfXk:nthZk:nHZk:n−1FXk:nt−fXk:nxhZk:nHZk:n−1FXk:nxdx≥0.
Sinceψ is increasing in u∈0,1, then F¯Xk:nx/ψFXx is nonnegatively decreasing in x>0. From Lemma 4 and inequality (92), we have(93)∫0tψFXk:nxfXk:nthZk:nHZk:n−1FXk:nt−fXk:nxhZk:nHZk:n−1FXk:nxdx≥0,for allt>0,which is equivalent to(94)Xk:n≤gdmtfrZk:n,for allk=1,2,…,n.
Therefore, the proof of the theorem is complete.
Denote byX1:N=minX1,X2,…,XN and XN:N=maxX1,X2,…,XN. Z1:N and ZN:N are similar. Let φN⋅ be the probability generating function of N.Theorem 11.
Assume thatX1,X2,…, and Z1,Z2,…, are the i.i.d. copies of X and Z, respectively.(1)
Let the functionψ∘φN be star-shaped with respect to the point 1,0. If X≤dmtfrZ, then(95)XN:N≤gdmtfrZN:Nw.r.t.ψ.(2)
Let the functionψ∘φN be anti-star-shaped with respect to the point 1,0. If XN:N≤gdmtfrZN:N w.r.t. ψ, then(96)X≤dmtfrZ.(3)
Letψ be a function defined on the interval 0,1 such that the function ψ1−φNu/u is increasing in u∈0,1. If X≤dmtfrZ, then(97)X1:N≤gdmtfrZ1:Nw.r.t.ψ.(4)
Letψ be a function defined on the interval 0,1 such that the function u/ψ1−φNu is increasing in u∈0,1. If X1:N≤gdmtfrZ1:N w.r.t. ψ, then(98)X≤dmtfrZ.Proof.
(1)
Suppose thatX≤dmtfrY. This asserts that(99)∫0tF¯XxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0.Shaked and Shanthikumar [8] showed that(100)HZN:N−1FXN:Nx=HZ−1FXx.It can be proven that(101)fXN:NxhZN:NHZN:N−1FXN:Nx=fXxhZHZ−1FXx.Since the functionψ∘φN is star-shaped with respect to the point 1,0, the function ψφNu/1−u is decreasing in u∈0,1, and this leads to the function(102)hx=ψφNFXx1−FXx=ψFXN:NxF¯Xx,being nonnegative decreasing. From Lemma4, (99), and (101), we have for all t>0,(103)∫0tψFXN:NxfXN:NthZN:NHZN:N−1FXN:Nt−fXN:NxhZN:NHZN:N−1FXN:Nxdx≥0.That is,XN:N≤gdmtfrZN:N w.r.t. ψ.(2)
Suppose thatXN:N≤gdmtfrZN:N w.r.t. ψ. This asserts that, for all t>0,(104)∫0tψFXN:NxfXN:NthZN:NHZN:N−1FXN:Nt−fXN:NxhZN:NHZN:N−1FXN:Nxdx≥0.Since the functionψ∘φN⋅ is anti-star-shaped with respect to the point 1,0, the function 1−u/ψφNu is decreasing in u∈0,1, and this leads to the function(105)hx=1−FXxψφNFXx=F¯XxψFXN:Nx,being nonnegatively decreasing. From Lemma4, (104), and (101), we have(106)∫0tF¯XxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0.That is,X≤dmtfrY.(3)
Suppose thatX≤dmtfrZ. Then,(107)∫0tF¯XxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0.Shaked and Shanthikumar [8] showed that(108)HZ1:N−1FX1:Nx=HZ−1FXx.It can be proven that(109)fX1:NxhZ1:NHZ1:N−1FX1:Nx=fXxhZHZ−1FXx.Since the functionψ1−φNu/u is increasing in u∈0,1, the function(110)hx=ψφNF¯XxF¯Xx=ψFX1:NxF¯Xx,is nonnegative decreasing. From Lemma4, (107), and (109), we have, for all t>0,(111)∫0tψFX1:NtfX1:NthZ1:NHZ1:N−1FX1:Nt−fX1:NxhZ1:NHZ1:N−1FX1:Nxdx≥0.That is,X1:N≤gdmtfrZ1:N w.r.t. ψ.(4)
Suppose thatX1:N≤gdmtfrZ1:N w.r.t. ψ. Then, for all t>0,(112)∫0tψFX1:NxfX1:NthZ1:NHZ1:N−1FX1:Nt−fX1:NxhZ1:NHZ1:N−1FX1:Nxdx≥0.
Since the functionu/ψ1−φNu is increasing in u∈0,1, the function(113)hx=F¯XxψφNF¯Xx=F¯XxψFX1:Nx,is nonnegatively decreasing. From Lemma 4, (112), and (109), we have for all t>0,(114)∫0tF¯XxfX1:NthZ1:NHZ1:N−1FX1:Nt−fX1:NxhZ1:NHZ1:N−1FX1:Nxdx≥0.
That is,X1:N≤gdmtfrZ1:N. Therefore, the proof of the theorem is complete.Theorem 12.
Assume thatX≤gdmtfrZ w.r.t. ψ, ψu=Bn−k+1,ku,u∈0,1, for some 1≤k≤n, where Bn−k+1,k is the distribution function of a beta distribution with parameters n−k+1 and k. Then, we have(115)Xk:n≤dmtfrZk:n,for allk=1,2,…,n.Proof.
Suppose thatX≤gdmtfrZ w.r.t. ψ. This asserts that(116)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0.
By takingψu=Bn−k+1,ku,u∈0,1, we get that(117)∫0tB¯n−k+1,kFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0.
In view of the fact that(118)Bn−k+1,kF¯Xx=F¯Xk:nx,fXxhZHZ−1FXx=fXk:nxhZk:nHZk:n−1FXk:nx,thus(119)∫0tF¯Xk:nxfXk:nthZk:nHZk:n−1FXk:nt−fXk:nxhZk:nHZk:n−1FXk:nxdx≥0,for allt>0,which is equivalent to (120)Xk:n≤dmtfrZk:n,for allk=1,2,…,n.
Hence, the proof is complete.
Letk=1 and k=n in Theorem 12, respectively, we have the following corollary.Corollary 1.
Assume thatX≤gdmtfrZ w.r.t. ψ.(i)
Ifψu=Bn,1u=∫0unxn−1dx=un, u∈0,1. Then, X1:n≤dmtfrZ1:n.(ii)
Ifψu=B1,nu=∫0un1−xn−1dx=1−1−un, u∈0,1. Then Xn:n≤dmtfrZn:n.Theorem 13.
Assume thatX1,X2,…, and Z1,Z2,…, are the i.i.d. copies of X and Z, respectively. Let N be a positive integer-valued random variable that is independent of Xi’s and Yi’s.(1)
Let the functionψ be star-shaped with respect to the point 1,0.(i)
IfX1:N≤dmtfrZ1:N, then X1:N≤gdmtfrZ1:N w.r.t. ψ.(ii)
IfXN:N≤dmtfrZN:N, then XN:N≤gdmtfrZN:N w.r.t. ψ.(2)
Let the functionψ be anti-star-shaped with respect to the point 1,0.(i)
IfX1:N≤gdmtfrZ1:N w.r.t. ψ, then X1:N≤dmtfrZ1:N.(ii)
IfXN:N≤gdmtfrZN:N w.r.t. ψ, then XN:N≤dmtfrZN:N.Proof.
We only give the proof for the case (1) (b), the proofs for other cases are similar and hence omitted. IfXN:N≤dmtfrZN:N, from (30), we obtain that, for all t≥0,(121)∫0tF¯XN:Nx⋅fXN:NthZN:NHZN:N−1FXN:Nt−fXN:NxhZN:NHZN:N−1FXN:Nxdx≥0.
Since the functionψ is star-shaped with respect to the point 1,0, then the function(122)ψu1−u,is decreasing in u∈0,1. Hence, ψFXN:Nx/F¯XN:Nx is decreasing in x≥0; from Lemma 4 and inequality (121), we have for all t≥0,(123)∫t∞ψFXN:Nx⋅fXN:NthZN:NHZN:N−1FXN:Nt−fXN:NxhZN:NHZN:N−1FXN:Nxdx≥0.
That is,XN:N≤gdmtfrZN:N w.r.t. ψ as claimed.Theorem 14.
(1)
Let the functionψ be star-shaped of order n with respect to the point 1,0. If X1:n≤dmtfrZ1:n, then X≤gdmtfrZ w.r.t. ψ.(2)
Let the functionψ be dual anti-star-shaped of order n. If Xn:n≤dmtfrZn:n, then X≤gdmtfrZ w.r.t. ψ.Proof.
We only give the proof for case (1), the proof for case (2) is similar, and hence omitted. IfX1:n≤dmtfrZ1:n, from (30), we obtain that(124)∫0tF¯X1:nx⋅fX1:nthZ1:nHZ1:n−1FX1:nt−fX1:nxhZ1:nHZ1:n−1FX1:nxdx≥0,for allt≥0.
Shaked and Shanthikumar [8] showed that HZn:n−1FXn:nx=HZ−1FXx. It can be proven that(125)fX1:nxhZ1:nHZ1:n−1FX1:nx=fXxhZHZ−1FXx,for allx≥0.
Since the functionψx is star-shaped of order n with respect to the point 1,0, then the function ψx/1−xn is decreasing in x∈0,1. Hence, the function(126)ψFXxF¯Xxn,is nonnegatively decreasing in x≥0. From Lemma 4, inequality (124), and equation (125), we have(127)∫0tψFXx⋅fXxhZHZ−1FXx−fXthZHZ−1FXtdx≥0,for allt>0.
That is,X≤gdmtfrZ w.r.t. ψ as claimed.
At the end of this section, we show the relation between the orders DMTFR, RDMTFR, and GDMTFR:(128)X≤gdmtfrZw.r.t.ψ,ψu=1−u,u∈0,1⟺X≤dmtfrZ,X≤gdmtfrZw.r.t.ψ,ψu=gYGY−1u,u∈0,1⟺X≤rdmtfrZ,X≤rdmtfrZw.r.t.Y,Yis exponential⟺X≤dmtfrZ.


## 4. Closure and Reversed Closure Properties of the GDMTFR Order

In this section, we investigate the closure and reversed closure properties of the GDMTFR order with respect to the referred functions and the compared random variables, respectively.The following Theorem15 indicates that the GDMTFR order has the closure property with respect to the referred function under multiplication of functions.Theorem 15.
IfX≤gdmtfrZ w.r.t. ψ, and φ is nonnegative decreasing in x∈0,1, then(129)X≤gdmtfrZw.r.t.φ⋅ψ.Proof.
Suppose thatX≤gdmtfrZ w.r.t. ψ. Then, from (30), we have(130)∫0tψFXxfXtgYGY−1FXt−fXxgYGY−1FXxdx≥0,for allt>0.
SinceφFXt is nonnegatively decreasing in x∈0,1, from (130) and Lemma 4, we obtain for all t>0,(131)∫0tφFXx⋅ψFXxfXtgYGY−1FXt−fXxgYGY−1FXxdx≥0,which is equivalent to that X≤gdmtfrZ w.r.t. ψ⋅φ. Therefore, the proof of the theorem is complete.
From Theorem15, we have the following corollary.Corollary 2.
IfX≤gdmtfrZ w.r.t. ψ, and ψ is decreasing in x∈0,1, then(132)X≤gdmtfrZw.r.t.ψn.Theorem 16.
Letψ1 and ψ2 be two continuous nonnegative functions defined on 0,1, X and Z be two continuous nonnegative random variables. Then, the following statements are true.(1)
IfX≤gdmtfrZ w.r.t. ψi, i=1,2, then(133)X≤gdmtfrZw.r.t.ψ1+ψ2.(2)
IfX≤gdmtfrZ w.r.t. ψ1 and ψ2 is increasing in x∈0,1, then(134)X≤gdmtfrZw.r.t.ψ1ψ2.(3)
IfX≤gdmtfrZ w.r.t. ψ2, ψ1 is star-shaped and ψ2 is decreasing, then(135)X≤gdmtfrZw.r.t.ψ1°ψ2,whereψ1°ψ2 is the compound of ψ1 and ψ2.(4)
IfX≤gdmtfrZ w.r.t. ψ2, ψ1 is anti-star-shaped and ψ2 is increasing, then(136)X≤gdmtfrZw.r.t.ψ1°ψ2,(5)
IfX≤gdmtfrZ w.r.t. ψ1/ψ2 and ψ2 is decreasing in x∈0,1, then(137)X≤gdmtfrZw.r.t.ψ1.(6)
IfX≤gdmtfrZ w.r.t. ψ2 and ψ1/ψ2 is decreasing in x∈0,1, then(138)X≤gdmtfrZw.r.t.ψ1.Proof.
We only give the proofs for the case of (3); the proofs for the other cases are similar and omitted. (3) Suppose thatX≤gdmtfrZ w.r.t. ψ2. Then, from (30), we have(139)∫0tψ2FXxfXxhZHZ−1FXx−fXthZHZ−1FXtdx≥0,for allt>0.
Since the functionψ2 is decreasing, ψ2FXx is then decreasing. Moreover, ψ1 is star-shaped on the interval 0,1, and then ψ1u/u is increasing in u∈0,1; thus, the function ψ1ψ2FXx/ψ2FXx is decreasing in x≥0. From Lemma 4 and inequality (139), we have for all t>0,(140)∫0tψ1ψ2FXxfXxhZHZ−1FXx−fXthZHZ−1FXtdx≥0,which asserts that X≤gdmtfrZ r.t. ψ1∘ψ2 as claimed.
LetX be a nonnegative absolutely continuous random variable and ϕ be a nonnegative increasing function defined on 0,∞ with ϕ0=0. We call that ϕ is a generalized scale function and ϕX is the generalized scale transform of X˙. In the accelerated life testing, the ϕ is called an accelerating factor.
If a functionϕ is increasing convex with ϕ0=0, then ϕ is called a risk preference function, and ϕX is called the risk preference transform of X. If ϕ is increasing concave with ϕ0=0, then ϕ is called a risk aversion function, and ϕX is called the risk aversion transform of X. If ϕ is a risk aversion function, we say that −ϕ″/ϕ′ is the absolute risk aversion coefficient of ϕ. If ϕ is a risk preference function, we say that −ϕ″/ϕ′ is the risk preference coefficient of ϕ.Theorem 17.
Letψ be a nonnegative function defined on 0,1 and ϕ be a nonnegative function defined on 0,+∞ with ϕ0=0. Suppose that the function ϕ is increasing concave with −ϕ″/ϕ′ being decreasing in x≥0, and X≥dispZ. Then,(141)X≤gdmtfrZw.r.t.ψ⇒ϕX≤gdmtfrϕZw.r.t.ψ.Proof.
Suppose thatX≤gdmtfrZ w.r.t. ψ. Then, from (30), we have for all t>0,(142)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0.
Since the functionϕx is increasing concave, ϕ′x is nonnegatively decreasing in x≥0. From Lemma 4 and (142), we have(143)∫0tψFXxϕ′xfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0.
On the other hand,ϕX≤gdmtfrϕZ w.r.t. ψ if and only if for all t>0,(144)∫0tψFϕXxfϕXthϕZHϕZ−1FϕXt−fϕXxhϕZHϕZ−1FϕXx=∫0tψFXxϕ′xfXthZHZ−1FXt⋅ϕ′αtϕ′t−fXxhZHZ−1FXx⋅ϕ′αxϕ′xdx≥0,where αx=HZ−1FXx. Under the given conditions of the theorem, we have that the function(145)ϕ′αxϕ′x,is increasing and nonnegative for all x>0. Hence, from (143), we obtain for all t>0,(146)∫0tψFXxϕ′xfXthZHZ−1FXt⋅ϕ′αtϕ′t−fXxhZHZ−1FXx⋅ϕ′αxϕ′xdx≥ϕ′αtϕ′t∫0tψFXxϕ′xfXxhZHZ−1FXx−fXxhZHZ−1FXtdx≥0.That is to say, inequality (144) is valid, so the proof of the theorem is complete.Remark 7.
Theorem17 says, under some appropriate conditions, that the order ≤gdmtfr has the closure property under the concave generalized scale transform.Remark 8.
Theorem17 also says, under some appropriate conditions, that the order ≤gdmtfr has the closure property under the action of an increasing concave accelerating factor.Remark 9.
Theorem17 also says, under the condition of X≥dispZ, that the GDMTFR order has the closure property under the risk averse transform ϕ with an increasing absolute risk aversion coefficient −ϕ″/ϕ′.
With a similar manner of the proof of Theorem17, Theorem 18 can be proven, and the detailed proof is omitted.Theorem 18.
Letψ be a nonnegative function defined on 0,1 and ϕ be a nonnegative function defined on 0,+∞ with ϕ0=0. Suppose that the function ϕ is increasing and convex with ϕ″/ϕ′ being increasing in x≥0. If ϕX≤gdmtfrϕZ w.r.t. ψ and X≥dispZ, then(147)X≤gdmtfrZw.r.t.ψ.Remark 10.
Theorem18 says, under some appropriate conditions, that the order ≤gdmtfr has the reversed closure property under the convex generalized scale transform.Remark 11.
Theorem18 also says, under some appropriate conditions, that the order ≤gdmtfr has the reversed closure property under the action of an increasing and convex accelerating factor.Remark 12.
Theorem18 also says, under the condition of X≥dispZ, that the GDMTFR order has the reversed closure property under the risk preference transform ϕ with an increasing risk preference coefficient ϕ″/ϕ′.Theorem 19.
Letψ be a nonnegative increasing function defined on 0,1 and let ϕ be a nonnegative increasing concave function defined on 0,+∞ with ϕ0=0 and with ϕ″/ϕ′ being decreasing in x>0. Suppose that X≥dispZ. Then,(148)X≤gdmtfrZw.r.t.ψ⇒ϕX≤gdmtfrϕZw.r.t.ϕ∘ψ.Proof.
Suppose thatX≤gdmtfrZ w.r.t. ψ. Then, from (30), we have(149)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0.
Since the functionϕx is increasing concave, ϕ′x is nonnegatively decreasing in x>0. From Lemma 4 and (149), we have(150)∫0tψFXxϕ′xfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for allt>0.
Sinceϕ is nonnegatively increasing concave, ϕx/x is nonnegatively decreasing and ψ being nonnegatively increasing leads to increasing ψFXx, so ϕψFXx/ψFXx is nonnegatively decreasing. From Lemma 4 and (150), we obtain that(151)∫0tϕψFXxϕ′xfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0,for all t>0.
On the other hand,ϕX≤gdmtfrϕZ w.r.t. ϕ∘ψ if and only if for all t≥0,(152)∫0tϕψFϕXxfϕXthϕZHϕZ−1FϕXt−fϕXxhϕZHϕZ−1FϕXx=∫0tϕψFXxϕ′xfXthZHZ−1FXt⋅ϕ′αtϕ′t−fXxhZHZ−1FXx⋅ϕ′αxϕ′xdx≥0,where αx=HZ−1FXx. Since ϕ″/ϕ′ is decreasing, and X≥dispZ, we have that the function(153)ϕ′αxϕ′x,is nonnegatively increasing in x>0. Therefore, for all t>0,(154)∫0tϕψFXxϕ′xfXthZHZ−1FXt⋅ϕ′αtϕ′t−fXxhZHZ−1FXx⋅ϕ′αxϕ′xdx≥ϕ′αtϕ′t∫0tϕψFXxϕ′xfXxhZHZ−1FXx−fXxhZHZ−1FXtdx≥0.
From (151) and (154), we see that inequality (152) is valid. That is to say, ϕX≤gdmtfrϕZ w.r.t. ϕ∘ψ. Therefore, the proof of the theorem is complete.Remark 13.
Theorem19 says that, under some appropriate conditions, the order ≤gdmtfr has the closure property with respect to both the compared random variables and the referred function under the concave generalized scale transforms.Remark 14.
Theorem19 also says that, under some appropriate conditions, the order ≤gdmtfr has the closure property with respect to both the compared random variables and the referred function under the action of a concave accelerating factor.Remark 15.
Theorem19 also says that, under the condition of X≥dispZ, the GDMTFR order has the closure property with respect to both the compared random variables and the referred function under the risk aversion transform ϕ with an increasing absolute risk aversion coefficient −ϕ″/ϕ′.Theorem 20.
Letψ be a nonnegative increasing function defined on 0,1, and let ϕ be a nonnegative increasing convex function defined on 0,+∞ with ϕ0=0 and with −ϕ″/ϕ′ being decreasing in x>0. Assume that X≥dispZ. Then,(155)ϕX≤gdmtfrϕZw.r.t.ϕ∘ψ⇒X≤gdmtfrZw.r.t.ψ.Proof.
Suppose thatϕX≤gdmtfrϕZ w.r.t. ϕ∘ψ. Then, from (30), we have, for all t>0,(156)∫0tϕψFϕXxfϕXthϕZHϕZ−1FϕXt−fϕXxhϕZHϕZ−1FϕXx=∫0tϕψFXxϕ′xfXthZHZ−1FXt⋅ϕ′αtϕ′t−fXxhZHZ−1FXx⋅ϕ′αxϕ′xdx≥0.
Sinceϕ is increasing convex, ϕy/y is increasing, and ψ is increasing. This leads to ψFXx is nonnegatively increasing. Combining these facts, we obtain that the function(157)ψFXxϕψFXx,is nonnegative decreasing in x>0. Moreover, since ϕ is increasing convex, 1/ϕ′ is nonnegatively decreasing. So,(158)ψFXxϕψFXxϕ′x,is nonnegatively decreasing in x>0. From Lemma 4 and (156), we obtain that, for all t>0,(159)∫0tψFXxfXthZHZ−1FXt⋅ϕ′αtϕ′t−fXxhZHZ−1FXx⋅ϕ′αxϕ′xdx≥0.
Under the given conditions thatϕ is increasing convex with ϕ″/ϕ′ being decreasing in x≥0, and X≥dispZ, we get that ϕ′αx/ϕ′x is nonnegatively decreasing in x>0. Hence,(160)0≤∫0tψFXxfXthZHZ−1FXt⋅ϕ′αtϕ′t−fXxhZHZ−1FXx⋅ϕ′αxϕ′xdx≤ϕ′αtϕ′t∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx,for allt>0.
Thus,(161)∫0tψFXxfXthZHZ−1FXt−fXxhZHZ−1FXxdx≥0.
That is to say,ϕX≤gdmtfrϕZ w.r.t. ϕ∘ψ. Therefore, the proof of the theorem is complete.Remark 16.
Theorem20 says that, under some appropriate conditions, the order ≤gdmtfr has the reversed closure property with respect to both the compared random variables and the referred function under the convex generalized scale transforms.Remark 17.
Theorem20 also says that, under some appropriate conditions, the order ≤gdmtfr has the reversed closure property with respect to both the compared random variables and the referred function under the action of a convex accelerating factor.Remark 18.
Theorem20 also says that, under the condition of X≥dispZ, the GDMTFR order has the reversed closure property with respect to both the compared random variables and the referred function under the risk preference transform ϕ with a decreasing risk preference coefficient ϕ″/ϕ′.


## 5. Examples

We give several illustrative examples that meet the GDMTFR order.Example 1.
Letψ be a nonnegative function defined on the interval 0,1. Now, we consider comparing two Weibull random variables in the GDMTFR order. Let Xi, i=1,2, be two Weibull random variables with respective survival function F¯Xi. Specifically, let Xi∼Wαi,λ, i=1,2, F¯Xix=e−λxαi, x≥0. Hence, the density functions of Xi are, respectively,(162)fXix=λαixαi−1e−λxαi,x≥0.
Then, the GDMTFR order is determined by the parametersαi. Indeed, it is easy to verify that(163)FX2−1FX1x=xα1/α2,fX2x=λα2xα2−1e−λxα2.
Hence,(164)fX1xfX2FX2−1FX1x=α1α2xα1/α2−1.(1)
Ifα1≥α2>0, then from (164), it is clear that the function fX1x/fX2FX2−1FX1x is increasing in x>0. From this fact and (30), we have that X1≤gdmtfrX2 w.r.t. ψ.(2)
If0<α1≤α2, then from (164), it is clear that the function fX1x/fX2FX2−1FX1x is decreasing in x>0. From this fact and (21), we have that X1≥gdmtfrX2 w.r.t. ψ.Example 2.
Consider a comparison of two beta random variables in the GDMTFR order. Letψ be a nonnegative function defined on the interval 0,1. Let Xi, i=1,2, be two Beta random variables with respective distribution functions FXi. Specifically, let Xi∼Betaαi,1, FXix=xαi, x∈0,1. Hence, the density functions of Xi are given by, respectively,(165)fXix=αixαi−1,x∈0,1.
It is readily verified that(166)FX2−1FX1x=xα1/α2,fX2x=λα2xα2−1e−λxα2,x∈0,1.
Hence,(167)fX1xfX2FX2−1FX1x=α1α2xα1/α2−1.(1)
Ifα1≥α2>0, it can be seen that the function fX1x/fX2FX2−1FX1x is increasing in x≥0. From this fact and (30), we see that X1≤gdmtfrX2 w.r.t. ψ.(2)
If0<α1≤α2, from (171), we see that the function fX1x/fX2FX2−1FX1x is decreasing x≥0. From this fact and (30), we have that X1≥gdmtfrX2 w.r.t. ψ.Example 3.
Consider a comparison of another two beta random variables in the GDMTFR order. Letψ be a nonnegative function defined on the interval 0,1. Let Xi, i=1,2, be two beta random variables with respective distribution function FXi. Specifically, let Xi∼Beta1,βi, then the density functions of Xi are given by, respectively,(168)fXix=βi1−xβi−1,x∈0,1,i=1,2.
The distribution functions ofXi are thus(169)FXix=1−1−xβi,x∈0,1,i=1,2,respectively.
It can be verified that(170)FX2−1FX1x=1−1−xβ1/β2,fX2x=β21−xβ2−1,x∈0,1.
Hence,(171)fX1xfX2FX2−1FX1x=β1β21−xβ1/β2−1.(1)
Ifβ1≥β2>0, it is clear that the function fX1x/fX2FX2−1FX1x is decreasing x>0. From this fact and (30), we see that X1≥gdmtfrX2 w.r.t. ψ.(2)
Assume that0<β1≤β2. From (171), we see that fX1x/fX2FX2−1FX1x is increasing x>0. From this fact and (30), we see that X1≤gdmtfrX2 w.r.t. ψ.Example 4.
Letψ be a nonnegative function defined on the interval 0,1; we consider to compare two Pareto random variables in the GDMTFR order. Specifically, assume that(172)X∼ℙα1,λ,Y∼ℙα2,λ,where α1,α2,λ are positive real numbers. That is, for all x≥0,(173)F¯Xx=λλ+xα1,G¯Yx=λλ+xα2, (174)fXx=α1λλλ+xα1+1,gYx=α2λλλ+xα2+1.
Then,(175)GY−1FXx=G¯Y−1F¯Xx=λ1+xλα1/α2−λ.
Hence,(176)fXxgYGY−1FXx=α1α21+xλα1/α2−1.(1)
Ifα1≥α2>0, then from (176), we see that the function fXx/gYGY−1FXx is increasing in x>0; hence, from (30), we have X≤gdmtfrY w.r.t. ψ.(2)
If0<α1≤α2, then fXx/gYGY−1FXx is decreasing in x>0; from (30), we get that X≥gdmtfrY w.r.t. ψ.



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