DISPERSION-TYPE VARIABILITY ORDERS

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Probability in the Engineering and Informational Sciences, 2003, 17(3), 305-334.

Dispersion-type Variability Orders

Felix Belzunce

Dpto. de Estadıstica e Investigacion Operativa

Universidad de Murcia

30100 Espinardo (Murcia), Spain

Taizhong Hu

Department of Statistics and Finance

University of Science and Technology of China

Hefei, Anhui 230026

People’s Republic of China

Baha-Eldin Khaledi

Department of Statistics

College of Sciences

Razi University

Kermanshah, Iran

�

Dispersion-type orders are introduced and studied. The new orders can be used to compare

the variability of the underlying random variables, among which are the usual dispersive order

and the right spread order. Connections among the new orders and other common stochastic

orders are examined and investigated. Some closure properties of the new orders under the oper-

ation of order statistics, transformations and mixtures are derived. Finally, several applications

of the new orders are given

Mathematics Subject Classification (2000): 60E15; 60K10

Keywords: Likelihood ratio order, hazard rate order, mean residual life order, usual stochastic

order, increasing convex order, Laplace transform order, dispersive order, right spread order,

order statistics, spacing, imperfect repair, non-homogeneous Poisson process, mixture, aging

classes

§1 Introduction

Stochastic orders have been proven to be very useful in applied probability, statistics, reliabil-

ity, operations research, among others. Various types of stochastic orders and associated properties

1


have been developed rapidly over the years, resulting in a large body of literature.

The purpose of this paper is to introduce and study a family of dispersion-type variability

orders, and to highlight their meaning, properties and applications. The new family of dispersion-

type orders can be used to compare the variability of the underlying random variables, among

which are the usual dispersive order and the right spread order. Such a study is meaningful

because it throws an important light on the understanding of properties of the dispersive and the

right spread orders and of relationships among these two orders and other orders. On the other

hand, the properties of the new orders have potential applications; for example, we prove that IFR

(increasing failure rate) and DFR (decreasing failure rate) are, respectively, equivalent to IFR(2)

(increasing failure rate of degree 2) and DFR(2) (decreasing failure rate of degree 2) by using the

properties of the new orders.

The paper is organized as follows. The new orders are defined in Section 2. Relationships of

the new orders to other common stochastic orders are given in Section 3. In Section 4, we derive

some closure properties of the new orders under the operations of order statistics, transformations

and mixtures. Several applications are presented in Section 5.

Throughout, the terms ‘increasing’ and ‘decreasing’ mean ‘non-decreasing’ and ‘non-increasing’,

respectively. a/0 is understood to be ∞ whenever a > 0 and 0/0 is not defined. All expectations

and integrals are implicitly assumed to exist whenever they are written, and all ratios are well

defined. For any random variable X with distribution function F , F = 1 − F denotes its survival

function, and the inverse F−1 of F is taken to be the right continuous version of it defined by

F−1(u) = sup{x : F (x) ≤ u} for u ∈ [0, 1].

§2 Definitions and notations

In this section, we first recall the definitions of some well known stochastic orders, and then

give the definitions of dispersion-type variability orders.

Definition 2.1. (cf. Shaked and Shanthikumar, 1994, Chapter 1) Let X and Y be two random

variables with distribution functions F and G, respectively. Then X is said to be smaller than Y

(1) in the usual stochastic order (denoted by X ≤st Y ) if F (x) ≤ G(x) for all x, or if E [h(X)] ≤

E [h(Y )] for all increasing functions h for which the expectations exist;

(2) in the hazard rate order (denoted by X ≤hr Y ) if G(x)/F (x) is increasing in x;

(3) in the reversed hazard rate order (denoted by X ≤rh Y ) if G(x)/F (x) is increasing in x;

(4) in the likelihood ratio order (denoted by X ≤lr Y ) if F and G have density functions f and

g, respectively, and if g(x)/f(x) is increasing in x;

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(5) in the mean residual life order (denoted by X ≤mrl Y ) if E [X − t|X > t] ≤ E [Y − t|Y > t]

for all t for which the expectations exist.

The relationships among these orderings are shown in the following diagram:

X ≤lr Y =⇒ X ≤hr Y =⇒ X ≤mrl Y

⇓ ⇓

X ≤rh Y =⇒ X ≤st Y

Definition 2.2. Let X and Y be two random variables with distribution functions F and G,

respectively. Then X is said to be smaller than Y

(1) in the increasing convex [concave] order (denoted by X ≤icx [≤icv] Y ) if E [h(X)] ≤ E [h(Y )]

for all increasing convex [concave] functions h for which the expectations exist (cf. Shaked

and Shanthikumar, 1994, Sect. 3.A);

(2) in the dispersive order (denoted by X ≤disp Y ) if F−1(β) − F−1(α) ≤ G−1(β) − G−1(α) for

all 0 < α < β < 1 (cf. Shaked and Shanthikumar, 1994, Sect. 2.B);

(3) in the right spread order or in the excess wealth order (denoted by X ≤rs Y or X ≤ew Y ) if

E [(X−F−1(p))+] ≤ E [(Y −G−1(p))+] for all p ∈ (0, 1), provided that the expectations exist,

where x+ = max(x, 0) (cf. Fernandez-Ponce et al., 1998; Shaked and Shanthikumar, 1998);

(4) in the Laplace transform order (denoted by X ≤Lt Y ) if X and Y are nonnegative and if

E[e−sX

]≥ E

[e−sY

]for every s ≥ 0 (cf. Shaked and Shanthikumar, 1994, Sect. 3.B).

For properties and applications of the dispersive order and the right spread order, one can refer

to Belzunce et al. (1997), Belzunce (1999), Fagiuoli et al. (1999), Fernandez-Ponce et al. (1998),

Kochar et al. (2001), Munoz-Perez (1990), Shaked and Shanthikumar (1998), among others.

Surveys on stochastic orders and their applications can be found in Shaked and Shanthikumar

(1994), Szekli (1995) and Muller and Stoyan (2002).

Now, we give the definitions of the dispersion-type variability orders, in terms of some of which

Hu et al. (2002) characterized some aging notions by using the residual lifetime of the underlying

random variable.

Definition 2.3. Let X and Y be two random variables with distribution functions F and G,

respectively. Then X is said to be smaller than Y

(1) in the ∗-order of the dispersion-type (denoted by X ≤disp−∗ Y ) if

(X − F−1(p)

)+≤∗

(Y − G−1(p)

)+

, ∀ p ∈ (0, 1), (2.1)

where ∗ = hr, rh, st, mrl, icx, icv and Lt.

(2) in the lr-order of the dispersion-type (denoted by X ≤disp−lr Y ) if

[X − F−1(p)

∣∣X > F−1(p)]≤lr

[Y − G−1(p)

∣∣Y > G−1(p)]

, ∀ p ∈ (0, 1). (2.2)

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Let X be a random variable with distribution function F , and let

Xt = [X − t|X > t], t ∈ {x : F (x) < 1},

denote a random variable whose distribution is the same as the conditional distribution of X − t

given that X > t. When X is the lifetime of a device, Xt can be regarded as the residual lifetime

of the device at time t, given that the device has survived up to time t. Note that if F and G in

Definition 2.3 are continuous then we have

X ≤disp−∗ Y ⇐⇒ XF−1(p) ≤∗ YG−1(p) for all p ∈ (0, 1), (2.3)

where ∗ =lr, hr, rh, st, mrl, icx, icv and Lt. The difference between (X −F−1(p))+ and XF−1(p) is

that the former has probability mass p on the point 0 while the latter has no mass on the point 0.

This is why we introduce the order ≤disp−lr in a separate way to the other dispersion-type orders.

The dispersion-type orders compare residual lives at quantiles. Such comparisons are of in-

terest in the following situation. Consider a system which produces units with random lifetime

X . Let the units be tested until the 100p% of the units fail, and eliminate early failures. Then

the additional residual lifetime of the remaining units is distributed as XF−1(p). Therefore the

dispersion-type orders can be used to compare two lifetimes under two different systems and under

the same policy to eliminate early failures.

A stochastic order � is said to be location-free if

X � Y =⇒ X � Y + c for any c ∈ <.

It is clear that all the orders defined in Definition 2.3 are location-free, and that

X ≤disp−mrl Y =⇒ X ≤rs Y (2.4)

and

X ≤disp−icv Y =⇒ X ≤rs Y. (2.5)

Munoz-Perez (1990) observed that

X ≤disp Y ⇐⇒ X ≤disp−st Y, (2.6)

and Belzunce (1999) showed that

X ≤rs Y ⇐⇒ X ≤disp−icx Y. (2.7)

Combining (2.5) and (2.7), we get

X ≤disp−icv Y =⇒ X ≤disp−icx Y. (2.8)

Further properties of these disperison-type variability orders are investigated in the next

sections.

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§3 Connections among the dispersion-type orders and other stochastic

orders

Throughout, Let X and Y be two random variables with distribution functions F and G,

mean residual lifetime functions mX and mY , respectively, and each with an interval support. Let

lX and uX be the left and right endpoints of the support of X ; that is,

lX = inf{x : F (x) > 0} and uX = sup{x : F (x) < 1}.

Similarly, define lY and uY . The values lX , uX , lY and uY may be infinite.

The first result states that the order ≤disp−∗ is the same as the order ≤disp−(disp−∗), where

≤∗ is any stochastic order.

Theorem 3.1. Let X and Y be two random variables with continuous distribution functions F

and G, respectively. Then

X ≤disp−∗ Y ⇐⇒(X − F−1(p)

)+≤disp−∗

(Y − G−1(p)

)+

whenever p ∈ (0, 1), (3.1)

where ≤∗ is any stochastic order. In particular,

X ≤disp Y ⇐⇒ X ≤disp−disp Y, (3.2)

X ≤rs Y ⇐⇒ X ≤disp−rs Y. (3.3)

Proof. For any p ∈ (0, 1), let F+(·; p) denote the distribution function of (X − F−1(p))+.

Similarly, define G+(·; p). It is easy to see that F+(x; p) = F (x + F−1(p)) for x ≥ 0, and 0 for

x < 0. Then the inverse function of F+(·; p) is given by

F−1+ (u; p) = (F−1(u) − F−1(p))+ for u ∈ (0, 1).

So we get that

((X − F−1(p))+ − F−1

+ (u; p))+

=

(X − F−1(p))+ for u ∈ (0, p)

(X − F−1(u))+ for u ∈ [p, 1).(3.4)

Similarly,

((Y − G−1(p))+ − G−1

+ (u; p))+

=

(Y − G−1(p))+ for u ∈ (0, p)

(Y − G−1(u))+ for u ∈ [p, 1).(3.5)

The desired result follows from (3.4) and (3.5) directly.

From the definitions in Section 2, we know that ≤disp−hr=⇒≤disp and that ≤disp−mrl=⇒≤rs.

The following corollary of Theorem 3.1 gives sufficient conditions under which the converse im-

plications above are also valid. First we recall that a random variable X is IFR [DFR] if F is

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logconcave [logconvex], and that X is DMRL [IMRL] (decreasing [increasing] mean residual life-

time) if mX(t) = E [Xt] is decreasing [increasing] in t ∈ (lX , uX).

Corollary 3.1. Let X and Y be two random variables with lX and lY finite.

(a) If X ≤disp Y and if X or Y is IFR, then X ≤disp−hr Y .

(b) If X ≤rs Y and if X or Y is DMRL, then X ≤disp−mrl Y .

Proof. (a) Suppose that X ≤disp Y . Then, by Theorem 3.1, we have

(X − F−1(p)

)+≤disp

(Y − G−1(p)

)+

for p ∈ (0, 1).

Note that X [Y ] ∈ IFR implies Xt [Yt] ∈ IFR for t < uX [t < uY ]. Using these observations, it

follows from Theorem 2.1 of Bagai and Kochar (1986) (also cf. Bartoszewicz, 1985b) that

(X − F−1(p)

)+≤hr

(Y − G−1(p)

)+

for p ∈ (0, 1);

that is, X ≤disp−hr Y .

(b) The proof is similar to that of (a) by using the fact that X [Y ] ∈ DMRL implies Xt [Yt] ∈

DMRL for each t < uX [t < uY ], and using Theorem 2.3 of Shaked and Shanthikumar (1998)

which says that for two random variables U and V with lU = lV = 0, if U ≤rs V and if U or V is

DMRL then U ≤mrl V .

Theorem 3.1 and Corollary 3.1 are applied to prove Theorems 3.9, 4.2 and 5.1 in the subsequent

sections. The next two theorems are motivated by the following well-known fact.

Fact 3.1. Let X and Y be two random variables with lY ≥ lX > −∞. Then

(a) X ≤disp−st Y =⇒ X ≤st Y (cf. Shaked and Shanthikumar, 1994, Theorem 2.B.7).

(b) X ≤disp−icx Y =⇒ X ≤icx Y (cf. Fagiuoli et al., 1999, Corollary 3.1; Belzunce, 1999,

Proposition 3).

Theorem 3.2. (a) If lY ≥ lX > −∞ and if X ≤disp−icv Y , then X ≤icv Y and X ≤icx Y .

(b) If lX = lY > −∞ and if X ≤disp−hr [≤disp−rh] Y , then X ≤hr [≤rh] Y .

(c) If lX = lY > −∞ and if X ≤disp−mrl Y , then X ≤mrl Y .

(d) If lY ≥ lX ≥ 0 and if X ≤disp−Lt Y , then X ≤Lt Y .

Proof. It is easy to prove parts (b) and (c). We only give the proofs for parts (a) and (d).

(a) Using (2.8), the result for the case X ≤icx Y follows from Fact 3.1(b). It suffices to prove

that X ≤icv Y . First, assume that lX = lY = 0. Then, X ≤disp−icv Y implies that

∫ x+F−1(p)

F−1(p)

F (u)du ≤

∫ x+G−1(p)

G−1(p)

G(u)du, ∀x ≥ 0, p ∈ (0, 1). (3.6)

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By the right continuity of F−1 and G−1, letting p → 0+ in (3.6) yields that∫ x

0

F (u)du ≤

∫ x

0

G(u)du, ∀x ≥ 0,

or, equivalently, X ≤icv Y . For the general case that lX ≤ lY , we have X − lX ≤disp−icv Y − lY

and, hence, X − lX ≤icv Y − lY . Therefore,

X ≤icv Y − (lY − lX) ≤st Y,

implying that X ≤icv Y .

(d) If lX = lY = 0, then, from (X −F−1(p))+ ≤Lt (Y −G−1(p))+, it follows that X ≤Lt Y by

letting p → 0+. If lY ≥ lX ≥ 0, then X − lX ≤Lt Y − lY and therefore X ≤Lt Y − (lY − lX) ≤st Y ,

implying that X ≤Lt Y .

The result of Theorem 3.2(a) is strengthened in Theorem 3.7 under the additional assumption

that X and Y have continuous distribution functions. Recall that a random variable X is ILR

[DLR] (increasing [decreasing] likelihood ratio) if X has a logconcave [logconvex] density function.

Let X and Y be two subsets of the real line. A nonnegative function h defined on X ×Y is said to

be TP2 (totally positive of order 2) if

h(x, y)h(x′, y′) ≥ h(x, y′)h(x′, y)

whenever x ≤ x′, y ≤ y′ and x, x′ ∈ X , y, y′ ∈ Y . h is said to be RR2 (reverse regular of order 2)

if the inequality is reversed (cf. Karlin, 1968).

Theorem 3.3. If X ≤disp−lr Y with lY ≥ lX > −∞, and if X or Y is ILR, then X ≤lr Y .

Proof. We give the proof of the case Y ∈ ILR; the proof of the case X ∈ ILR is similar. Let f

and g denote the density functions of X and Y , respectively. Then g(u− v) is TP2 in (u, v) ∈ <2.

For any x ≤ x′ and t = F−1(p), p ∈ (0, 1), we have

f(x + t)g(x′ + t) = f(x + F−1(p))g(x′ + F−1(p))

≥ f(x′ + F−1(p))g(x + G−1(p))

g(x′ + G−1(p))g(x′ + F−1(p))

≥ f(x′ + t)g(x + F−1(p)) = f(x′ + t)g(x + t),

where the first inequality follows from X ≤disp−lr Y . Combining Y ∈ ILR with the fact that

F−1(p) ≤ G−1(p) (recall that ≤disp−lr=⇒≤disp−st=⇒≤st by Fact 3.1(a)), the second inequality

follows. This completes the proof.

Now we turn to consider the converse problems of Fact 3.1 and Theorems 3.2 and 3.3. First

we recall

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Fact 3.2. Let X and Y be two random variables with lX = lY > −∞.

(a) If X or Y is DFR, and if X ≤hr Y , then X ≤disp Y (cf. Bagai and Kochar, 1986; Bar-

toszewicz, 1985b).

(b) If X or Y is IMRL, and if X ≤mrl Y , then X ≤disp−icx Y (cf. Shaked and Shanthikumar,

1998, Theorem 2.4).

Theorem 3.4. (a) If X or Y is DLR, and if X ≤lr Y , then X ≤disp−lr Y .

(b) If X or Y is DFR, and if X ≤hr Y , then X ≤disp−hr Y .

(c) If X or Y is IMRL, and if X ≤mrl Y and X ≤st Y , then X ≤disp−mrl Y .

Proof. We only give the proof of part (b) for the case X ∈ DFR; the proof of the other case

and the proof of parts (a) and (c) are similar. To prove X ≤disp−hr Y , it suffices to check that

G(x + G−1(p))

F (x + F−1(p))=

G(x + G−1(p))

F (x + G−1(p))·F (x + G−1(p))

F (x + F−1(p))(3.7)

is increasing in x ≥ 0 for each p ∈ (0, 1). This is true because X ≤hr Y implies that the first term

in the right hand side of (3.7) is increasing in x ≥ 0 and that G−1(p) ≥ F−1(p) for each p, while

X ∈ DFR implies that the second term in the right hand side of (3.7) is increasing in x ≥ 0.

From Theorem 3.4(b), we can remove the assumption that X and Y have a common left

endpoint of their supports in Fact 3.2(a). It is still an open problem whether the assumption that

X ≤st Y in Theorem 3.4(c) can be removed.

For two random variables X and Y that have the support [0,∞), X is said to be smaller than

Y in the down likelihood ratio order, denoted by X ≤lr ↓ Y , if X ≤lr Yx for all x ≥ 0, and X is

said to be smaller than Y in the down hazard rate order, denoted by X ≤hr ↓ Y , if X ≤hr Yx for all

x ≥ 0, where Yx = [Y −x|Y > x] (cf. Lillo et al., 2000). Also, for X and Y with lX = lY = a finite,

X is said to be smaller than Y in the down mean residual lifetime order, denoted by X ≤mrl↓ Y , if

mX(z) ≤ mY (z + t) for all z ≥ a and t ≥ 0. This shifted mean residual lifetime order was denoted

by X ≤mrl↑ Y in Brown and Shanthikumar (1998). Here we use a different notation because of

the following observation: for X and Y with lX = lY = 0, mX(z) ≤ mY (z + t) for all z ≥ 0 and

t ≥ 0 if and only if X ≤mrl Yx for all x ≥ 0.

Theorem 3.5. Let X and Y be two nonnegative random variables.

(a) If X ≤lr ↓ Y , then X ≤disp−lr Y .

(b) If X ≤hr↓ Y , then X ≤disp−hr Y .

(c) If X ≤mrl↓ Y , then X ≤disp−icx Y .

Proof. By Theorems 6.13 and 6.28 of Lillo et al. (2000), we know that X ≤lr ↓ Y if and only if

there exists a nonnegative random variable Z ∈ DLR such that X ≤lr Z ≤lr Y , and that X ≤hr↓ Y

8


if and only if there exists a nonnegative random variable Z ∈ DFR such that X ≤hr Z ≤hr Y .

Also, by Lemma 2.3(b) of Brown and Shanthikumar (1998), X ≤mrl↓ Y if and only if there exists

an IMRL random variable Z such that X ≤mrl Z ≤mrl Y .

Using these observations, the desired results respectively follow from Theorem 3.4(a) and (b)

and Fact 3.2(b).

Theorem 3.5(b) is an extension of Theorem 6.27 of Lillo et al. (2000).

From Theorem 3.4, we obtain the following corollary.

Corollary 3.2. (a) If X or Y is DLR, and if X ≤lr Y , then Xt ≤disp−lr Yt for all t < uX .

(b) If X or Y is DFR, and if X ≤hr Y , then Xt ≤disp−hr Yt for all t < uX .

(c) If X or Y is IMRL, and if X ≤hr Y , then Xt ≤disp−mrl Yt for all t < uX .

Proof. We only give the proof of part (b); the proof of parts (a) and (c) is similar. For any

t < uX , X ≤hr Y implies Xt ≤hr Yt. Moreover, X or Y ∈ DFR implies that Xt or Yt ∈ DFR. By

Theorem 3.4(b), we have that Xt ≤disp−hr Yt.

Theorem 3.6. Let X and Y be two random variables with lY ≥ lX > −∞.

(a) If X ≤disp−lr Y , and if X or Y is ILR, then Xt ≤disp−lr Yt for all t < uX .

(b) If X ≤disp Y , and if X or Y is IFR, then Xt ≤disp−hr Yt for all t < uX .

(c) If X ≤disp−mrl Y and X ≤st Y , and if X or Y is DMRL, then Xt ≤disp−mrl Yt for all t < uX .

Proof. (a): We only give the proof of the case Y ∈ ILR; the proof of the case X ∈ ILR is

similar. By Theorem 3.3, X ≤lr Y and hence µX ≤ µY . For each fixed t < µX , let Ft denote the

distribution function of Xt, and f(·; p, t) denote the density function of [Xt−F−1t (p)|Xt > F−1

t (p)].

Similarly, define Gt and g(·; p, t). Note that

IP(Xt − F−1

t (p) > x|Xt > F−1t (p)

)=

F (x + t + F−1t (p))

qF (t)=

F (x + F−1

(qF (t))

qF (t)

for x ≥ 0, where q = 1 − p ∈ (0, 1), and the last equality follows from the identity that F −1t (p) =

F−1

(qF (t)) − t. Then

f(x; p, t) =f(x + F

−1(qF (t)))

qF (t), x ≥ 0.

Therefore, by X ≤disp−lr Y and Y ∈ ILR, we have

g(x; p, t)

f(x; p, t)=

F (t)

G(t)·g(x + G

−1(qG(t)))

f(x + F−1

(qF (t)))

=F (t)

G(t)·g(x + G

−1(qG(t)))

g(x + G−1

(qF (t)))·g(x + G

−1(qF (t)))

f(x + F−1

(qF (t)))

is increasing in x (recall that X ≤lr Y implies F (t) ≤ G(t)). This proves (a).

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(b) and (c): The proof follows by a similar argument (Corollary 3.1(a) will be used in proving

(b)).

Corollary 3.2(b) and Theorem 3.6(b) respectively strengthen Theorems 3.2 and 3.4 of Belzunce

et al. (1997) in two respects. First, the results here are stronger than those of Belzunce et al.

(1997). Second, unlike in Belzunce et al. (1997), no assumption on the continuity of distribution

functions of X and Y is needed.

The relationship between the order ≤disp−icv and the order ≤st is given in the following

theorem.

Theorem 3.7. Let X and Y be two random variables with continuous distribution functions, and

with lY ≥ lX > −∞. Then

X ≤disp−icv Y =⇒ X ≤st Y. (3.8)

Proof. First suppose that X ≤disp−icv Y and that lX = lY = 0. Then, for all p ∈ (0, 1) and

x ≥ 0, ∫ x+F−1(p)

F−1(p)

F (u)du ≤

∫ x+G−1(p)

G−1(p)

G(u)du. (3.9)

Two cases arise.

(i). If F and G are not identical, and do not cross each other, then from (3.9) it is seen that

F (x) ≤ G(x) at a right neighborhood of 0, and therefore F (x) ≤ G(x) for all x; that is, X ≤st Y .

(ii). If F (x0) > G(x0) for some x0 > 0, then there exist 0 ≤ t0 < t2 such that F (t0) = G(t0),

F (u) > G(u) for all u ∈ (t0, t2), and

F (u) ≤ G(u) for all u ≤ t0. (3.10)

If F is strictly increasing in the right neighborhood of t0, then taking p = F (t0) and x = t2 − t0,

we have that p ∈ [0, 1), F−1(p) ≤ G−1(p) and t2 = x + F−1(p). Therefore,

∫ x+F−1(p)

F−1(p)

F (u)du >

∫ x+F−1(p)

F−1(p)

G(u)du ≥

∫ x+G−1(p)

G−1(p)

G(u)du,

which contradicts (3.9).

Define

t1 = inf{x : F (x) = F (t0), x ∈ <}, t′1 = inf{x : G(x) = G(t0), x ∈ <}.

If F is not strictly increasing in the right neighborhood of t0, then t1 ≤ t′1 (Otherwise, there

exists one point x0 < t0 such that F (x0) > G(x0), contradicting (3.10)). Choose 0 < p < F (t0)

10


such that G−1(p) ≥ F−1(p) when t1 < t′1, and |G−1(p) − F−1(p)| < δ/3 when t1 = t′1, and also

t′1 − G−1(p) < δ/3, where

δ =

∫ t2

t0

[F (u) − G(u)]du > 0.

Setting x = t2 − F−1(p), we have

∫ x+F−1(p)

F−1(p)

F (u)du −

∫ x+G−1(p)

G−1(p)

G(u)du

=

∫ G−1(p)

F−1(p)

F (u)du +

∫ t′1

G−1(p)

[F (u) − G(u)]du

+

∫ t2

t0

[F (u) − G(u)]du −

∫ x+G−1(p)

t2

G(u)du

> 0,

contradicting (3.9). Therefore, F and G do not cross each other, and thus X ≤st Y .

Next, if lY ≥ lX > −∞, then X − lX ≤disp−icv Y − lY and, hence, X − lX ≤st Y − lY .

Therefore, X ≤st Y − (lY − lX) ≤st Y . This completes the proof.

In Theorem 3.7, the implication (3.8) does not hold in general when lX = lY = −∞ or

lX > lY . For example, for any random variable X and a real number c > 0, X ≤disp−icv X − c,

but X 6≤st X − c.

From Theorem 3.7, we can obtain the following two interesting and somewhat surprising

results, which states that IFR(2) [DFR(2)] is equivalent to IFR [DFR], and that the orders ≤disp−icv

and ≤disp are the same. A random variable X is said to be IFR(2) [DFR(2)] if Xs ≥icv [≤icv] Xt

whenever lX < s ≤ t (cf. Deshpande et al., 1986).

Theorem 3.8. Let X be a random variable with a continuous distribution function F . Then

X ∈ IFR(2) [DFR(2)] ⇐⇒ X ∈ IFR [DFR]. (3.11)

Proof. It is well known that IFR [DFR] =⇒ IFR(2) [DFR(2)] (cf. Deshpande et al., 1986). It

suffices to prove that IFR(2) [DFR(2)] =⇒ IFR [DFR]. By Theorem 3.4 of Hu et al. (2002), we

have that

X ∈ IFR(2) [DFR(2)] ⇐⇒ Xs ≥disp−icv [≤disp−icv] Xt whenever s < t, s, t ∈ T , (3.12)

where T is the support of X . Using (3.12), it follows from Theorem 3.7 that

X ∈ IFR(2) [DFR(2)] =⇒ Xs ≥st [≤st] Xt whenever s < t, s, t ∈ T ,

where is equivalent to X ∈ IFR [DFR]. This completes the proof.

11


Theorem 3.8 gives us a new characterization of IFR [DFR] by means of the increasing concave

order of residual lives.


with lY ≥ lX > −∞. Then

X ≤disp−icv Y ⇐⇒ X ≤disp Y. (3.13)

Proof. The implication that X ≤disp Y =⇒ X ≤disp−icv Y is trivial. Now suppose that

X ≤disp−icv Y . Then, by Theorem 3.1, we have

(X − F−1(p)

)+≤disp−icv

(Y − G−1(p)

)+

for p ∈ (0, 1)

and, hence,

XF−1(p) ≤disp−icv YG−1(p) for p ∈ (0, 1).

Note that both XF−1(p) and YG−1(p) have continuous distribution functions with the origin as a

common left endpoint of their supports. By Theorem 3.7, we have

XF−1(p) ≤st YG−1(p) for p ∈ (0, 1),

which is equivalent to X ≤disp Y . This completes the proof.

The relationship between the order ≤disp−Lt and the order ≤Lt−rl is described in the following

theorem. First we recall the definitions of the order ≤Lt−rl and of the aging notions of DRLLt and

IRLLt given by Belzunce et al. (1999). Let X and Y be two random variables. X is said to be

smaller than Y in the Laplace transform order of residual lives, denoted by X ≤Lt−rl Y , if

Xt ≤Lt Yt for all t < min{uX , uY }.

X is said to have decreasing [increasing] residual lives in the Laplace order, denoted by X ∈

DRLLt[IRLLt], if

Xt ≥Lt [≤Lt] Xt′ for all t < t′ < uX .

It is well known that

X ≤Lt Y ⇐⇒ L∗X(s) ≤ L∗

Y (s) for all s > 0,

where

L∗X(s) =

∫ ∞

0

e−sxF (x)dx and L∗Y (s) =

∫ ∞

0

e−sxG(x)dx.

Theorem 3.10. Let X and Y be two random variables with continuous and strictly increasing

distribution functions F and G, respectively, and with lY ≥ lX > −∞. If X ≤disp−Lt Y and if X

or Y is DRLLt, then X ≤Lt−rl Y .

12


Proof. We give the proof of the case Y ∈DRLLt; the proof of the case X ∈DRLLt is similar.

First suppose that lX = lY = 0. Let us consider 0 ≤ t < min{uX , uY } such that F (t) ≥ G(t),

and denote p = F (t). Then

∫ ∞

te−syF (y)dy

e−stF (t)≤

∫ ∞

G−1(p) e−syG(y)dy

e−sG−1(p)G(G−1(p))≤

∫ ∞

te−syG(y)dy

e−stG(t)(3.14)

for every s > 0, where the first inequality follows from X ≤disp−Lt Y , and the second one follows

from Y ∈ DRLLt and t = F−1(p) ≤ G−1(p).

Now consider 0 < t < min{uX , uY } such that F (t) < G(t). Then there exists 0 ≤ t1 < t such

that F (t1) = G(t1) and F (u) < G(u) for all u ∈ (t1, t). From X ≤disp−Lt Y , we have

∫ ∞

t1e−syF (y)dy

e−st1≤

∫ ∞

t1e−syG(y)dy

e−st1for s > 0

or, equivalently, ∫ ∞

t1

e−syF (y)dy ≤

∫ ∞

t1

e−syG(y)dy for s > 0. (3.15)

Since ∫ t

t1

e−syF (y)dy ≥

∫ t

t1

e−syG(y)dy for s > 0,

it follows from (3.15) that

∫ ∞

t

e−syF (y)dy ≤

∫ ∞

t

e−syG(y)dy for s > 0.

Then ∫ ∞

t e−syF (y)dy

F (t)≤

∫ ∞

t e−syG(y)dy

F (t)≤

∫ ∞

t e−syG(y)dy

G(t)(3.16)

for every s > 0. Therefore, we conclude from (3.14) and (3.16) that X ≤Lt−rl Y for the case

lX = lY = 0.

Next, suppose that lY ≥ lX > −∞. Note that the order ≤disp−Lt is location-free, and that

Y ∈ DRLLt implies Y − lY ∈ DRLLt. Then X − lX ≤Lt−rl Y − lY . On the other hand, it is easy

to check that

Y ∈ DRLLt ⇐⇒ Y + t ≤Lt−rl Y + t′ whenever t < t′

[It should be noted that this property does not hold when Y ∈ IRLLt], and that for any random

variables U and V ,

U ≤Lt−rl V =⇒ U + c ≤Lt−rl V + c for all c ∈ <.

Therefore,

X ≤Lt−rl Y − (lY − lX) ≤Lt−rl Y,

implying that X ≤Lt−rl Y . This completes the proof.

13


In the end of this subsection, we give one result concerning the comparison between a non-

negative random variable with an exponential random variable.

Theorem 3.11. Let X be a random variable with continuous distribution function and with lX = 0,

and Y be an exponential random variable. Then

X ≤disp−∗ [≥disp−∗] Y ⇐⇒ Xt ≤∗ [≥∗] Y whenever t < uX , (3.17)

where ∗ = lr, hr, st, icv, icx, mrl and Lt.

Proof. We only give the proof of the necessity; the proof of the sufficiency is trivial. Suppose

that X ≤disp−∗ Y .

(a) ∗ =lr or hr: It follows from Theorem 3.6(a) and (b).

(b) ∗ =st: From Theorem 2.B.13(b) of Shaked and Shanthikumar (1994), it follows that X ≤hr Y ,

which is equivalent to Xt ≤st Y for all t < uX .

(c) ∗ =icv: It follows from Theorem 3.9 and (b).

(d) ∗ =icx: From Theorem 2.3 of Shaked and Shanthikumar (1998), it follows that X ≤mrl Y ,

which is equivalent to Xt ≤icx Y for all t < uX .

(e) ∗ =mrl: It follows easily from the implications that ≤disp−mrl=⇒≤disp−icx and that Xt ≤icx

Y ⇐⇒ Xt ≤mrl Y whenever Y is exponential.

(f) ∗ =Lt: It follows from Theorem 3.10.

In the literature of reliability, several aging notions are defined or characterized in terms

of some stochastic orders between the underlying random variable and an exponential random

variable with the equal means.

Let X be a nonnegative random variable, and Y be an exponential random variable with mean

E X . X is said to be exponentially better [worse] than used in the ∗-order, denoted by X ∈ EBU∗

[X ∈ EWU∗] if

Xt ≤∗ [≥∗] Y whenever 0 ≤ t < uX ,

where ∗ = lr, hr, st, mrl, icx, icv and Lt. For these aging notions, the following implications are easy

to prove:

EBUlr =

EBUicv

‖

EBUhr

‖

EBUst

= EBULt = {Exponential distribution} =⇒

EBUicx

m

NBUE

m

EBUmrl

The implication EBUst ⇐⇒ EBUicv follows from Theorems 3.9 and 3.11. The classes EBUlr,

EBUhr and EBUst only contain the exponential distribution by Theorem 1.A.7 of Shaked and

14


Shanthikumar (1994), which states that if U ≤st V and E U = E V then Ust= V . To prove that

EBULt contains only the exponential distribution, we first recall the aging notions L and L from

Klefsjo (1983). A random variable X ∈ L [L] if X ≥Lt [≤Lt] Y , where Y is exponentially distributed

with mean E X . From this definition, it is seen that EBULt =⇒ L. Also, EBULt =⇒ NBUE =⇒ L.

So EBULt = L ∩ L = {Exponential distribution}.

The equivalence of NBUE, EBUicx and EBUmrl follows from Theorem 2.6 of Belzunce et al.

(1997) and the fact that ≤mrl=⇒≤icx for nonnegative random variables. Similar implications also

hold for the above dual aging notions. EBULt and EWULt were introduced by Belzunce et al.

(1999).

§4 Some preservation properties

§4.1 Preservation under order statistics and spacings

Let X1:n ≤ X2:n ≤ · · · ≤ Xn:n denote the order statistics from a random sample X1, X2, . . . , Xn

of independent and identically distributed (i.i.d.) random variables that have the same distribution

function F as X . Similarly, let Y1:n ≤ Y2:n ≤ · · · ≤ Yn:n denote the order statistics from another

random sample Y1, Y2, . . . , Yn of i.i.d. random variables that have the same distribution function

G as Y .

Let Bα,β(·) denote the distribution function of a beta distribution with parameters α > 0 and

β > 0 with density function given by

bα,β(u) =Γ(α + β)

Γ(α)Γ(β)uα−1(1 − u)β−1, 0 < u < 1.

Then the distribution and survival functions of Xi:n can be respectively written as

IP(Xi:n ≤ x) = Bi,n−i+1(F (x)), ∀x, (4.1)

and

IP(Xi:n ≥ x) = Bn−i+1,i(F (x)), ∀x. (4.2)

Lemma 4.1. (Nanda and Shaked, 2001) If X ≤hr [≤rh] Y , then Xi:n ≤hr [≤rh] Yj:m whenever

j ≥ i and j − i ≥ m − n.

It should be noted that Nanda and Shaked (2001) established the above lemma under the

assumption that X and Y are absolutely continuous. In fact, this assumption can be dropped off

by Lemma 1.B.5 of Shaked and Shanthikumar (1994) and the fact that the orders ≤hr and ≤rh are

closed under weak convergence.

15


From (4.1) and (4.2), a particular case of Lemma 4.1 can be restated as follows: for each

i = 1, . . . , n,

F ≤hr G =⇒Bn−i+1,i(G(x))

Bn−i+1,i(F (x))is increasing in x < uX , (4.3)

and

F ≤rh G =⇒Bi,n−i+1(G(x))

Bi,n−i+1(F (x))is increasing in x > lX , (4.4)

where (lX , uX) is the support of X .

Bartoszewicz (1986) proved that if X ≤disp Y then Xi:n ≤disp Yi:n for all i = 1, . . . , n.

Recently, Kochar et al. (2001) proved that if X ≤disp−icx Y then Xn:n ≤disp−icx Yn:n. We

establish the following analogous results for some other dispersion-type orders.

Theorem 4.1. (a) If X ≤disp−hr Y , then Xi:n ≤disp−hr Yi:n for each i = 1, . . . , n.

(b) If X ≤disp−rh Y , then Xi:n ≤disp−rh Yi:n for each i = 1, . . . , n.

(c) If X ≤disp−lr Y , then X1:n ≤disp−lr Y1:n and Xn:n ≤disp−lr Yn:n.

Proof. (a) Let Fi:n and Gi:n denote the distribution functions of Xi:n and Yi:n, respectively.

For each p ∈ (0, 1), denote p = B−1i,n−i+1(p). Then p ∈ (0, 1). From (4.1) and (4.2), it follows that

F−1i:n (p) = F−1(B−1

i,n−i+1(p)) = F−1(p)

and

G−1i:n(p) = G−1(p).

To prove that Xi:n ≤disp−hr Yi:n for each i, it suffices to show that

Gi:n(x + G−1i:n(p))

F i:n(x + F−1i:n (p))

=Bn−i+1,i(G(x + G−1(p)))

Bn−i+1,i(F (x + F−1(p)))is increasing in x ∈ [0, uX − F−1(p)) (4.5)

for any p ∈ (0, 1). Since X ≤disp−hr Y implies that (X − F−1(p))+ ≤hr (Y − G−1(p))+, (4.5)

follows from the observation (4.3).

(b) The proof is similar to that of part (a) by using the observation (4.4).

(c) We only give the proof of Xn:n ≤disp−lr Yn:n; the proof of X1:n ≤disp−lr Y1:n is similar.

Let fn:n and gn:n denote the densities of Xn:n and Yn:n, respectively. It suffices to prove that

gn:n(x + G−1n:n(p))

fn:n(x + F−1n:n(p))

=g(x + G−1(p1/n))

f(x + F−1(p1/n))·

[G(x + G−1(p1/n))

F (x + F−1(p1/n))

]n−1

is increasing in x > lX − F−1(p1/n) for each p ∈ (0, 1). However, this follows from the assumption

that X ≤disp−lr Y and the fact that X ≤disp−lr Y implies X ≤disp−rh Y .

The next counterexample illustrates that Theorem 4.1 does not hold in general with respect

to the order ≤disp−mrl for series systems of i.i.d. components.

16


Counterexample 4.1 (X ≤disp−mrl Y 6=⇒ X1:n ≤disp−mrl Y1:n for n ≥ 2) Let X and Y be two

nonnegative random variables with corresponding survival functions given by

F (x) = e−x2

, G(x) = e−x, ∀x ≥ 0.

Singh and Vijayasree (1991) showed that X ≤mrl Y and X1:2 6≤mrl Y1:2. Note that both Y

and Y1:2 have exponential distributions. It follows from Theorem 3.11 that X ≤disp−mrl Y and

X1:2 6≤disp−mrl Y1:2. /

Theorem 4.2. If X is an exponential random variable, then Xi:n ≤disp−hr Xj:m for all i ≤ j and

n − i ≥ m − j.

Proof. By Lemma 2.1 of Khaledi and Kochar (2000a), Xi:n ≤disp Xj:m for i ≤ j and n − i ≥

m − j. Since X has exponential distribution, Xi:n ∈ IFR for each i. Therefore, the desired result

follows from Corollary 3.1(a).

A stronger result will be given in Theorem 4.6. An immediate consequence of Theorems 4.1

and 4.2 is

Corollary 4.1. Let X and Y be two exponential random variables with hazard rates λ1 and λ2,

respectively. If λ1 ≥ λ2, then Xi:n ≤disp−hr Yj:m for all i ≤ j and n − i ≥ m − j.

We now turn to present two results concerning stochastic comparisons of the normalized

spacings, Ui:n and Vj:m, of order statistics from X and Y -samples, where Ui:n = (n− i+1)(Xi:n −

Xi−1:n) and Vj:m = (m − j + 1)(Yj:m − Yj−1:m) for i = 1, . . . , n and j = 1, . . . , m. Here X0:n =

Y0:n ≡ 0.

Theorem 4.3. Let X and Y be two nonnegative random variables such that X ≤lr Y . If either

X or Y is DFR, then

Ui:n ≤disp−hr Vj:m whenever i ≤ j and n − i ≥ m − j.

Proof. It follows from Theorem 2.2 of Khaledi and Kochar (1999) that

Ui:n ≤hr Vj:m whenever i ≤ j and n − i ≥ m − j.

On the other hand, the normalized spacings from DFR distributions are also DFR (cf. Barlow and

Proschan, 1966). Combining these, the required result follows from Theorem 3.4(b).

Theorem 4.4. Let X and Y be two nonnegative random variables such that X ≤lr Y . If either

X or Y is DLR, then

Ui:n ≤disp−lr Vj:m whenever i ≤ j and n − i = m − j.

17


Proof. By Theorem 2.3 of Khaledi and Kochar (1999), we get that

Ui:n ≤lr Vj:m whenever i ≤ j and n − i = m − j.

Moreover, it can be checked that spacings from DLR distributions are also DLR by using the

closure property of DLR under mixtures (cf. Barlow and Proschan, 1975, p.102). Therefore, the

desired result follows from Theorem 3.4(a).

Theorems 4.3 and 4.4 are extensions of Corollary 2.2 of Khaledi and Kochar (1999) from the

disp-st order to the disp-hr and disp-lr orders.

§4.2 Preservation under transformations

The following facts state that both the order ≤disp and the order ≤rs are closed under increas-

ing and convex transformations.

Fact 4.1. Let X and Y be two random variables with continuous distribution functions, and with

lY ≥ lX > −∞ . Then for any increasing and convex function φ, we have

(a) X ≤disp Y =⇒ φ(X) ≤disp φ(Y ) (cf. Rojo and He, 1991, Theorem 2.2; Bartoszewicz, 1985a,

p. 389).

(b) X ≤rs Y =⇒ φ(X) ≤rs φ(Y ), provided that X and Y have finite means (cf. Kochar et al.,

2001, Theorem 4.2).

In the next theorem, it is first shown that the orders ≤disp−hr and ≤disp−rh are closed under

increasing and convex transformations.


with a common finite left endpoint of their supports. Then, for any increasing and convex function

φ,

X ≤disp−hr [≤disp−rh] Y =⇒ φ(X) ≤disp−hr [≤disp−rh] φ(Y ). (4.6)

Proof. We give the proof of the order ≤disp−hr; the proof of the order ≤disp−rh is similar.

Assume that X ≤disp−hr Y . Let φ be any increasing and convex function, and let F [resp. G, Fφ

and Gφ] denote the distribution function of X [resp. Y , φ(X) and φ(Y )]. Then

Fφ(x) = F (φ−1(x)), Gφ(x) = G(φ−1(x)), ∀x,

F−1φ (p) = φ(F−1(p)), G−1

φ (p) = φ(G−1(p)), ∀p ∈ (0, 1).

Note that X ≤disp−hr Y if and only if

G(x + G−1(p))

F (x + F−1(p))is increasing in x ≥ 0 for each p ∈ (0, 1)

18


or, equivalently,

G(y + G−1(p) − F−1(p))

F (y)is increasing in y ≥ F−1(p) for each p ∈ (0, 1). (4.7)

Thus, in order to prove the theorem we need to show that for each p ∈ (0, 1),

Gφ(x + G−1φ (p))

F φ(x + F−1φ (p))

=G

(φ−1

(x + φ(G−1(p))

))

F (φ−1 (x + φ(F−1(p))))is increasing in x ≥ 0,

or thatG(y + ∆(x, p))

F (y)is increasing in y ≥ F−1(p) and in x ≥ 0, (4.8)

where

∆(x, p) ≡ φ−1(x + φ(G−1(p))

)− φ−1

(x + φ(F−1(p))

). (4.9)

Since φ is increasing and convex, it follows that φ−1 is increasing and concave. By Fact 3.1(a)

and the implication ≤disp−hr=⇒≤disp, we get that G−1(p) ≥ F−1(p) for each p ∈ (0, 1). Therefore,

∆(x, p) is decreasing in x ≥ 0 for each p, which implies that G(y + ∆(x, p))/F (y) is increasing in

x ≥ 0 for each y and that

0 ≤ ∆(x, p) ≤ ∆(0, p) = G−1(p) − F−1(p), x ≥ 0, p ∈ (0, 1).

Now fix x ≥ 0 and p ∈ (0, 1). If ∆(x, p) > 0, then there exists q ∈ (0, p] such that

∆(x, p) = G−1(q) − F−1(q)

because G−1(u) − F−1(u) is continuous and

[0, G−1(p) − F−1(p)

]⊂

{G−1(u) − F−1(u) : u ∈ [0, p]

}.

Thus, (4.8) follows from (4.7). If ∆(x, p) = 0, then (4.8) follows from Theorem 3.2(b) trivially.

This completes the proof.

From Theorem 4.5, we can obtain the following two results, which respectively strengthen

Theorems 2.1 and 2.2 of Khaledi and Kochar (2000a).

Theorem 4.6. Let X be a DFR random variable with continuous distribution function F . Then

Xi:n ≤disp−hr Xj:m for all i ≤ j and n − i ≥ m − j.

Proof. Let {Z, Z1, . . . , Zn} be a set of i.i.d. exponential random variables with mean 1, and

denote R(x) = − log F (x). Then R(x) is strictly increasing and concave, and Xst= φ(Z), where

st= means equal in distribution, and φ(z) = R−1(z) is increasing and convex. By Theorem 4.2, we

have

Zi:n ≤disp−hr Zj:m for all i ≤ j and n − i ≥ m − j,

19


where Zi:n is the ith order statistic of random variables Z1, . . . , Zn. Then, by Theorem 4.5, we get

Xi:nst= φ(Zi:n) ≤disp−hr φ(Zj:m)

st= Xj:m

for all i ≤ j and n − i ≥ m − j. This completes the proof.

Theorem 4.7. If either X or Y is DFR, then

X ≤disp−hr Y =⇒ Xi:n ≤disp−hr Yj:m for all i ≤ j and n − i ≥ m − j.

Proof. We give the proof of the case X ∈ DFR; the proof of the case Y ∈ DFR is similar.

By Theorem 4.6, Xi:n ≤disp−hr Xj:m for all i ≤ j and n − i ≥ m − j. By Theorem 4.1(a),

Xj:m ≤disp−hr Yj:m for all 1 ≤ j ≤ m. Combining these we get the required result.

Using the method in the proof of Theorem 4.5, we can also derive the following closure property

of the order ≤disp−lr under increasing and convex transformations.

Theorem 4.8. Let X and Y be two absolutely continuous random variables with a common fi-

nite left endpoint of their supports, and φ be a strictly increasing and twice differentiable convex

functions defined on the union of the supports of X and Y such that φ′′(x)/(φ′(x))2 is decreasing,

where φ′ and φ′′ are the first and second derivatives of φ, respectively. If either X or Y has a

decreasing density function, then

X ≤disp−lr Y =⇒ φ(X) ≤disp−lr φ(Y ).

Proof. We give the proof of the case that Y has a decreasing density; the proof of the other

case is similar. We will use the same notations as in the proof of Theorem 4.5. Again, let f [resp.

g, fφ and gφ] denote the density of X [resp. Y , φ(X) and φ(Y )].

To prove the theorem, we need to show that for each p ∈ (0, 1),

gφ

(x + G−1

φ (p))

fφ

(x + F−1

φ (p)) =

g(φ−1

(x + φ(G−1(p))

))

f (φ−1 (x + φ(F−1(p))))·(d/dx)φ−1(x + φ(G−1(p)))

(d/dx)φ−1(x + φ(F−1(p)))

=g(y + ∆(x, p))

f(y)·(d/dx)φ−1(x + φ(G−1(p)))

(d/dx)φ−1(x + φ(F−1(p)))(4.10)

is increasing in y ≥ F−1(p) and in x ≥ 0, where y = φ−1(x + φ(F−1(p))) and ∆(x, p) is defined by

(4.9).

Note that X ≤disp−lr Y if and only if

g(y + G−1(p) − F−1(p)

)

f(y)is increasing in y ≥ F−1(p) for each p ∈ (0, 1). (4.11)

20


Similar argument to that in the proof of Theorem 4.5 shows that the first term in (4.10) is in-

creasing in y and x by using (4.11) and the decreasingness of g. The decreasingness property of

φ′′(x)/(φ′(x))2 ensures that the second term in (4.10) is also increasing in x (cf. Theorem 6.16 of

Lillo et al., 2000). This completes the proof.

An immediate consequence of Theorem 4.8 is

Corollary 4.2. Let X and Y be two absolutely continuous random variables with lX = lY ≥ 0. If

either X or Y has a decreasing density function, then

X ≤disp−lr Y =⇒ Xr ≤disp−lr Y r whenever r > 1.

It is still an open problem whether the order ≤disp−mrl is closed under increasing and convex

transforms.

§4.3 Preservation under mixtures of distributions

Consider a family of distribution functions {Hθ, θ ∈ X}, where X is a subset of the real

line. Let Θ1 and Θ2 be two random variables with supports in X and with respective distribution

functions F1 and F2. Let Y1 and Y2 be two random variables such that Yi =st X(Θi) for i = 1, 2;

that is, the distribution functions of Yi is given by

Gi(y) =

∫

X

Hθ(y) dFi(θ), y ∈ <, i = 1, 2. (4.12)

Theorem 4.9. Let Y1 and Y2 be as described above.

(a) If X(θ) is DLR for each θ ∈ X , and if X(θ) ≤lr X(θ′) whenever θ < θ′, then

Θ1 ≤lr Θ2 =⇒ Y1 ≤disp−lr Y2. (4.13)

(b) If X(θ) is DFR for each θ ∈ X , and if X(θ) ≤hr X(θ′) whenever θ < θ′, then

Θ1 ≤hr Θ2 =⇒ Y1 ≤disp−hr Y2. (4.14)

(c) If X(θ) is IMRL for each θ ∈ X , and if X(θ) ≤mrl X(θ′) whenever θ < θ′, then

Θ1 ≤hr Θ2 =⇒ Y1 ≤disp−icx Y2. (4.15)

Additionally, if X(θ) ≤st X(θ′) whenever θ < θ′, then

Θ1 ≤hr Θ2 =⇒ Y1 ≤disp−mrl Y2. (4.16)

Proof. (a) Since DLR is closed under mixtures (cf. Barlow and Proschan, 1975, p. 102), it

follows from (4.12) that Yi is DLR for each i. By Theorem 1.C.11 of Shaked and Shanthikumar

21


(1994), Θ1 ≤lr Θ2 implies Y1 ≤lr Y2 under the assumption that X(θ) ≤lr X(θ′) whenever θ < θ′.

Combining these observations, the implication (4.13) follows from Theorem 3.4(a).

(b) Since DFR is closed under mixtures (cf. Barlow and Proschan, 1975, p. 103), it follows

from (4.12) that Yi is DFR for each i. By Corollary 38(a) of Hu et al. (2001a) (see also Shaked and

Wong, 1995), Θ1 ≤hr Θ2 implies Y1 ≤hr Y2 under the assumption that X(θ) ≤hr X(θ′) whenever

θ < θ′. Combining these observations, the implication (4.14) follows from Theorem 3.4(b).

(c) Since IMRL is closed under mixtures (cf. Brown, 1981), it follows from (4.12) that Yi is

IMRL for each i. By Corollary 38(b) of Hu et al. (2001a), Θ1 ≤hr Θ2 implies Y1 ≤mrl Y2 under

the assumption that X(θ) ≤mrl X(θ′) whenever θ < θ′. On the other hand, by Theorem 1.A.6

of Shaked and Shanthikumar (1994), Θ1 ≤st Θ2 implies Y1 ≤st Y2 if X(θ) ≤st X(θ′) whenever

θ < θ′. Combining these observations, the implications (4.15) and (4.16) follows from Fact 3.2(b)

and Theorem 3.4(c), respectively.

It should be noted that, in Theorem 4.9, the corresponding order relationship between Y1 and

Y2 is reversed if the order relationship between X(θ) and X(θ′) is reversed. Some applications of

Theorem 4.9 are given in next section.

§5 Applications

In this section, we give various applications of the results that were developed in previous

sections. These applications are not exhaustive.

§5.1 New proofs of the characterizations of IFR and DMRL aging notions

Hu et al. (2002) characterized several aging notions in terms of the dispersion-type and

dilation-type orders by using the mean residual lifetime of the underlying random variable. To

illustrate the usefulness of Theorem 3.1, we give a simple proof of Theorems 3.1 and 3.3 of Hu et

al. (2002) for the IFR and DMRL cases.

Theorem 5.1. (Hu et al., 2002) Let X be a random variable with a continuous distribution F

and support of the form T = (a,∞), where a ≥ −∞. Then

X ∈ IFR ⇐⇒ Xs ≥disp−hr Xt whenever a < s < t (5.1)

⇐⇒ X ≥disp−hr Xt, ∀ t > a, (5.2)

and

X ∈ DMRL ⇐⇒ Xs ≥disp−mrl Xt whenever a < s < t (5.3)

⇐⇒ X ≥disp−mrl Xt, ∀ t > a. (5.4)

22


Proof. (i). It suffices to prove that X ∈ IFR implies the right hand sides of both (5.1) and

(5.2). Suppose that X ∈ IFR. By Theorem 2.2 of Pellerey and Shaked (1997), we have

X ∈ IFR ⇐⇒ Xs ≥disp Xt, ∀ a < s < t ⇐⇒ X ≥disp Xt, ∀ t > a.

Observing that X ∈ IFR implies Xt ∈ IFR, by Corollary 3.1(a), we get that Xs ≥disp−hr Xt for

all a < s < t, and that X ≥disp−hr Xt for all t > a.

(ii). It suffices to prove that X ∈ DMRL implies the right hand sides of both (5.3) and (5.4).

Suppose that X ∈ DMRL. By Theorem 2 of Belzunce (1999), we have

X ∈ DMRL ⇐⇒ Xs ≥rs Xt, ∀ a < s < t ⇐⇒ X ≥rs Xt, ∀ t > a.

Also observe that X ∈ DMRL implies Xt ∈ DMRL. Therefore, the desired result follows from

Corollary 3.1(b) immediately. This completes the proof.

§5.2 Stochastic comparisons of epoch times of non-homogeneous Poisson processes

Let T1,n, n ≥ 1, be the epoch times of a non-homogeneous Poisson process (NHPP) with

intensity function λ such that∫ ∞

tλ(u)du = ∞ for all t ≥ 0, and denote by X a nonnegative

random variable with hazard rate λ, and with distribution function F . Similarly, Let T2,n, n ≥ 1,

be the epoch times of another NHPP with intensity function γ such that∫ ∞

t γ(u)du = ∞ for all

t ≥ 0, and denote by Y a nonnegative random variable with hazard rate γ, and with distribution

function G.

Recently, Belzunce et al. (2001), Belzunce and Shaked (2001) and Belzunce and Ruiz (2002)

investigated sufficient conditions which enable one to compare epoch times of two NHPP’s in some

multivariate and univariate orders. In this subsection, we consider the same comparison problem

with respect to the orders ≤disp−icx, ≤disp−hr and ≤disp−lr.

The density functions of T1,n and T2,n are given by

fn(t) = f(t)(− log F (t))n−1

(n − 1)!, t ≥ 0, (5.5)

and

gn(t) = g(t)(− logG(t))n−1

(n − 1)!, t ≥ 0, (5.6)

respectively, where f and g denote the density functions of X and Y , respectively. The survival

functions of T1,n and T2,n can be written as

F n(t) = Ψn(F (t)) and Gn(t) = Ψn(G(t)), (5.7)

respectively, where Ψn(p) = 1 − Γn(− ln p) for p ∈ (0, 1), and Γn is the distribution function of

a gamma distribution with scale parameter 1 and shape parameter n. Similarly, the distribution

23


functions of T1,n and T2,n can be written as

Fn(t) = Φn(F (t)) and Gn(t) = Φn(G(t)), (5.8)

respectively, where Φn(p) = Γn(− ln(1 − p)) for p ∈ (0, 1). Eqs. (5.5)–(5.8) can be found in

Belzunce et al. (2001), Gupta and Kirmani (1988) or Kochar (1996).

Fact 5.1. (Belzunce et al., 2001) If X ≤hr Y , then T1,n ≤hr T2,n for all n ≥ 1

From Fact 5.1 and (5.7), we get that

F ≤hr G =⇒Ψn(G(x))

Ψn(F (x))is increasing in x < uX . (5.9)

To prove Theorem 5.2, we first recall one notion of relative aging of two life distributions from

Sengupta and Deshpande (1994) and Rowell and Siegrist (1998), and then give one useful lemma.

For two nonnegative random variables U and V with respective distribution functions H and K, U

is said to be aging faster than V in average (denoted by U ≤afa V ) if ln H(t)/ ln K(t) is increasing

in t ≥ 0.

Lemma 5.1. (Barlow and Proschan, 1975, p. 120) Let W 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∫ b

t dW (x) ≥ 0 for all t ∈ (a, b), and if h is increasing, then∫ b

a h(x)dW (x) ≥ 0.

Let X and Y be two random variables associated with two NHPP’s as described earlier.

Belzunce et al. (2001, Theorem 3.10) proved that if X ≤disp Y then T1,n ≤disp T2,n for all n ≥ 1.

In the next theorem we establish analogous results for the orders ≤disp−icx, ≤disp−hr and ≤disp−lr.

Theorem 5.2. (a) If X ≤disp−icx Y , then T1,n ≤disp−icx T2,n for all n ≥ 1.

(b) If X ≤disp−hr Y , then T1,n ≤disp−hr T2,n for all n ≥ 1.

(c) If X ≤disp−lr Y and YG−1(p) ≤afa XF−1(p) for all p ∈ (0, 1), then T1,n ≤disp−lr T2,n for all

n ≥ 1.

Proof. (a). Fix an n ≥ 1 and suppose that X ≤disp−icx Y . Observe that

X ≤disp−icx Y ⇐⇒

∫ ∞

F−1(p)

F (x)dx ≤

∫ ∞

G−1(p)

G(x)dx

⇐⇒

∫ 1

p

(1 − u)d(G−1(u) − F−1(u)

)≥ 0 for all p ∈ (0, 1)

⇐⇒

∫ ∞

t

F (x)d(G−1(F (x)) − x

)≥ 0 for all t ≥ 0, (5.10)

and that

G−1n (Fn(x)) = G−1(F (x)), x ≥ 0.

24


Then, for all t ≥ 0,∫ ∞

t

F n(x) d(G−1

n (Fn(x)) − x)

=

∫ ∞

t

F n(x) d(G−1(F (x)) − x

)

=

n−1∑

j=0

∫ ∞

t

(− lnF (x))j

j!F (x) d

(G−1(F (x)) − x

)≥ 0,

where the inequality follows from (5.10), Lemma 5.1 and the fact that (− lnF (x))j/j! is nonnegative

and increasing in x ≥ 0 for each j ≥ 0. By (5.10) agian, we get T1,n ≤disp−icx T2,n.

(b). Fix an n ≥ 1 and suppose that X ≤disp−hr Y . It suffices to show that

Gn(x + G−1n (p))

F n(x + F−1n (p))

is increasing in x ≥ 0 for all p ∈ (0, 1). (5.11)

Observing that F−1n (p) = F−1(p) and G−1

n (p) = G−1(p), where p = Φ−1n (p) ∈ (0, 1), we can rewrite

(5.11) asΨn(G(x + G−1(p)))

Ψn(F (x + F−1(p)))is increasing in x ≥ 0 for all p ∈ (0, 1),

which follows from (5.9) and the assumption (X − F−1(p))+ ≤hr (Y − G−1(p))+. Therefore,

T1,n ≤disp−hr T2,n.

(c). Fix an n ≥ 1 and suppose that X ≤disp−lr Y . Using (5.5) and (5.6), we have to prove

thatgn(x + G−1

n (p))

fn(x + F−1n (p))

=g(x + G−1(p)

f(x + F−1(p))·

[ln G(x + G−1(p))

ln F (x + F−1(p))

]n−1

is increasing in x ≥ 0 for every p ∈ (0, 1), where p is the same as above. This follows from the

assumptions that X ≤disp−lr Y and XF−1(p) ≥afa YG−1(p). This completes the proof.

In the end of this subsection, we give one result concerning comparisons of interepoch intervals

of a NHPP. Let T1 ≤ T2 ≤ · · · be the epoch times of a NHPP with intensity function λ such that∫ ∞

t λ(u)du = ∞ for all t ≥ 0, and denote by X a nonnegative random variable with hazard rate

λ. Let S1 = T1 and Sn = Tn − Tn−1, n ≥ 2, be the interepoch intervals of the NHPP.

Theorem 5.3. Let X and Sn be as described above. Then

(a) If X is DLR, then Sn ≤disp−lr Sn+1 for all n ≥ 1.

(b) If X is DFR, then Sn ≤disp−hr Sn+1 for all n ≥ 1.

(c) If X is IMRL, then Sn ≤disp−icx Sn+1 for all n ≥ 1.

Proof. We shall prove the results by applying Theorem 4.9. Define X(θ) = [X − θ|X > θ] for

θ ∈ <+, and X(θ) = X(0) for θ ∈ (−1, 0). Then

Snst= X(Tn−1) for n ≥ 1,

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where T0 is uniform on (−1, 0). Observe that if X is DLR [DFR, IMRL] then X(θ) is DLR [DFR,

IMRL] for each θ and X(θ) ≤lr [≤hr,≤mrl] X(θ′) whenever θ ≤ θ′. Also, it can be checked from

(5.5) and (5.7) that Tn−1 ≤hr Tn for n ≥ 1 in the general case and Tn−1 ≤lr Tn for n ≥ 1 in the

absolutely continuous case. Therefore, the desired result follows from Theorem 4.9.

§5.3 Stochastic comparisons of concomitants of order statistics

Let (X1, Y1), . . . , (Xn, Yn) be a random sample of size n from a continuous bivariate distri-

bution. If we arrange the X ’s in ascending order as X1:n ≤ X2:n ≤ · · · ≤ Xn:n, then the Y ’s

associated with these order statistics are denoted by Y[1], Y[2], . . . , Y[n] and are called concomitants

of order statistics. They are also known as induced order statistics in the literature. Let H(y|x)

denote the conditional distribution function of Y given X = x. Then the distribution function of

Y[r] is given by

G[r](y) =

∫ +∞

−∞

H(y|x) dFr:n(x), (5.12)

where Fr:n is the density function of Xr:n. For a comprehensive review of this topic, see David

and Nagaraja (1998).

Khaledi and Kochar (2000b) investigated sufficient conditions which enable one to compare

concomitants of order statistics in the sense of the order ≤st [resp. ≤hr, ≤lr, ≤mrl and ≤disp]. In

this subsection, we shall compare concomitants of order statistics in the sense of the order ≤disp−hr

[resp. ≤disp−lr and ≤disp−mrl].

Observe Lemma 4.1 and the fact that Xi:n ≤lr Xj:n for i < j if X is absolutely continuous.

Applying Theorem 4.9 to model (5.12), we immediately obtain the following three theorems, which

are possible extensions of Theorem 3.7 in Khaledi and Kochar (2000b).

Theorem 5.4. Suppose that r(y|x), the conditional hazard rate of Y given X = x, is decreasing

in x and y. Then

Y[i] ≤disp−hr Y[j] for 1 ≤ i < j ≤ n. (5.13)

The inequality in (5.13) is reversed in case r(y|x) is increasing in x for each fixed y.

To state the next theorem, recall that two random variables X and Y are said to be TP2

[RR2] dependent if their joint density function exists and is TP2 [RR2].

Theorem 5.5. Let h(y|x) denote the conditional density function of Y given X = x. Suppose that

h(y|x) is logconvex in y for each fixed x, and that X and Y are TP2 dependent. Then

Y[i] ≤disp−lr Y[j] for 1 ≤ i < j ≤ n. (5.14)

The inequality in (5.14) is reversed in case X and Y are RR2 dependent.

26


Theorem 5.6. Let r(y|x) and m(y|x) denote the conditional hazard rate and the conditional mean

residual life of Y given X = x, respectively. Suppose that r(y|x) is decreasing in x for each fixed

y, and that m(y|x) is increasing in y for every fixed x. Then

Y[i] ≤disp−mrl Y[j] for 1 ≤ i < j ≤ n. (5.15)

The inequality in (5.15) is reversed in case r(y|x) is increasing in x for every fixed y.

§5.4 Stochastic comparisons of concomitants of record values

Let {(Xi, Yi), i ≥ 1} be a sequence of i.i.d. random variables with a continuous bivariate distri-

bution. If we denote by Tn the corresponding record values (or epoch times of a non-homogeneous

Poisson process) of the sequence {Xi, i ≥ 1}, then the Y -variate associated with Tn is called the

concomitant of the nth record value and is denoted by Y[n].

From Khaledi and Kochar (2001), we know that the distribution function of Y[n] is given by

G[n](y) =

∫ +∞

−∞

H(y|x) dFn(x), (5.16)

where H(·|x) and Fn are respective distribution functions of [Y |X = x] and Tn.

From (5.5) and (5.7), it is easy to check that Ti ≤hr Tj whenever i < j and that Ti ≤lr Tj

whenever i < j if X is absolutely continuous. Therefore, noting that (5.12) and 5.16) have the

same representation, we conclude that Theorems 5.4, 5.5 and 5.6 also hold for concomitants of

record values.

§5.5 Dispersion comparisons of Bayesian imperfect repair

Consider a unit whose lifetime has the distribution function H . For i = 1, 2, let Θi be a

random probability (that is, Θi takes on values in [0,1]), where the realized value of Θi is the

probability of minimal repair of the unit (cf. Brown and Proschan, 1983). Then the waiting time

Yi for the first perfect repair has the survival function Gi given by

Gi(y) =

∫ 1

0

H1−θ

(y)dFi(θ), (5.17)

where Fi is the prior distribution of Θi, i = 1, 2.

Hu et al. (2001b) and Belzunce and Shaked (2001) obtained some results involving stochastic

comparisons of Y1 and Y2 in the sense of the orders ≤lr, ≤hr, ≤rh, ≤st and ≤icx.

Applying Theorem 4.9 to model (5.17), we state the following theorem, which compares Y1

and Y2 according to the orders ≤disp−hr and ≤disp−icx.

Theorem 5.7. (a) If H is DFR, and if Θ1 ≤hr Θ2, then Y1 ≤disp−hr Y2.

(b) If H is IMRL, and if Θ1 ≤hr Θ2, then Y1 ≤disp−mrl Y2.

27


Proof. First note that model (5.17) is the special case of (4.12) with the survival function of

X(θ) being H1−θ

. It is seen that X(θ) ≤hr X(θ′) for all 0 < θ < θ′ ≤ 1. On the other hand,

by Theorem 2.5 of Block et al. (1985), it follows that H1−θ

is DFR [IMRL] for each θ ∈ (0, 1)

whenever H is. Combining these observations, the desired results follow from Theorem 4.9 (b) and

(c). This completes the proof.

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