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Preservation of Stochastic Orders for Random Minima and Maxima, with Applications Xiaohu Li,1 Ming J. Zuo2 1

2

Department of Mathematics, Lanzhou University, Lanzhou 730000, People’s Republic of China

Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, T6G 2G8, Canada

Received 28 August 2002; revised 9 October 2003; accepted 20 October 2003 DOI 10.1002/nav.10122

Abstract: It is shown, in this note, that the right spread order and the increasing convex order are both preserved under the taking of random maxima, and the total time on test transform order and the increasing concave order are preserved under the taking of random minima. Some inequalities and preservation properties in reliability and economics are given as applications. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 51: 332–344, 2004. Keywords: aging properties; increasing convex order; increasing concave order; Lorenz order; parallel system; right spread order; series system; total time on test transform order

1.

INTRODUCTION AND MOTIVATION

It is always of interest to compare variabilities of random variables in statistics, probability, and some related areas, such as reliability theory, survival analysis, economics, and actuaia science. Comparisons based on moments, for example, variance and standard deviation, are not very informative, though they are simple to use. As a result, several more refined orders, which compare variabilities of random variables based on their entire distribution functions, have been studied in the literature. Shaked and Shanthikumar [24] and Mu¨ller and Stoyan [23] present comprehensive discussions on these concepts and their properties as well. In this paper, we are interested in random variables that represent lifetimes. Given two nonnegative random variables X and Y with their distribution functions F and G, respectively, ៮ ⫽ 1 ⫺ G as their respective survival functions, and F ⫺1 and G ⫺1 denote F៮ ⫽ 1 ⫺ F and G are corresponding right continuous inverses. X is said to be smaller than Y in the increasing convex order (denoted by X ⱕ icx Y) if Correspondence to: X. Li ([email protected]). © 2003 Wiley Periodicals, Inc.

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Li and Zuo: Preservation of Stochastic Orders for Random Minima and Maxima

冕

⫹⬁

冕

F៮ 共x兲 dx ⱕ

t

⫹⬁

៮ 共x兲 dx, G

for all t ⱖ 0.

333

(1)

t

X is said to be smaller than Y in increasing concave order (denoted by X ⱕ icv Y), if

冕

t

冕

F៮ 共x兲 dx ⱕ

0

t

៮ 共x兲 dx, G

for all t ⱖ 0.

(2)

0

For more details of these two stochastic orders under more general assumptions, see Shaked and Shanthikumar [24]. Fernandez-Ponce, Kochar, and Mun˜oz-Pe´rez [12] propose a new variability ordering based upon the so-called right spread out function. A random variable X is said to be smaller than another random variable Y in the right spread order (denoted by X ⱕ RS Y) if

冕

⫹⬁

F៮ 共x兲 dx ⱕ

F⫺1共p兲

冕

⫹⬁

៮ 共x兲 dx, G

for all p 僆 共0, 1兲.

(3)

G⫺1共p兲

Completely independently, Shaked and Shanthikumar [25] define the excess wealth order through the inequality (3) and investigate some of its properties also. Recently, Kochar, Li, and Shaked [19] propose the following new order between X and Y, which are both nonnegative. X is said to be smaller than Y in the total time on test transform order (denoted by X ⱕ ttt Y) if

冕

F⫺1共p兲

0

F៮ 共x兲 dx ⱕ

冕

G⫺1共p兲

៮ 共x兲 dx, G

for all p 僆 共0, 1兲.

(4)

0

Some of its interesting properties are also studied there. Readers can refer to Klefsjo¨ [17] for more details on the total time on test transform. Suppose X 1 , . . . , X n and Y 1 , . . . , Y n are independent and identical (i.i.d.) copies of X and Y, respectively. In Kochar, Li, and Shaked [19] it is shown that the right spread order is preserved under the maxima, and the total time on test transform order is preserved under the minima. That is, 共X i ⱕRS Yi

i ⫽ 1, . . . , n兲 f max兵X1 , . . . , Xn 其 ⱕRS max兵Y1 , . . . , Yn 其

(5)

共X i ⱕttt Yi

i ⫽ 1, . . . , n兲 f min兵X1 , . . . , Xn 其 ⱕttt min兵Y1 , . . . , Yn 其.

(6)

and

In life-testing, if a random censoring is adopted, then the completely observed data constitute a sample of random size, say, X 1 , . . . , X N , where N ⬎ 0 is a random variable of integer value. In actuarial science, the claims received by an insurer in a certain time interval should also be

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a sample of random size, and, max{X 1 , . . . , X N }, to evaluate the one with the largest claim amount is of positive interest there. And min{X 1 , . . . , X N } arises naturally in survival analysis as the minimal survival time of a transplant operation, where N of them are defective and hence may cause a death, respectively. Another example in transportation theory may be found in Shaked and Wong [26]. Therefore, it is of special interest to conduct stochastic comparison between random maximums or random minimums in practical situations. Shaked and Wong [26] investigate the random minima and maxima of distributions. Recently, Bartoszwicz [5] studies again this topic for life distributions; it is proved there that the stochastic order and the dispersive order are both preserved under the taking of random minima and maxima. In light of the following implications, X ⱕdisp Y f X ⱕRS Y f X ⱕicx Y and X ⱕst Y f X ⱕttt Y f X ⱕicv Y, a natural question is to examine whether the above results (5) and (6) can be strengthened from maxima and minima of a finite number of elements to the random maxima and the random minima, respectively. Theorem 1 in Section 2 gives a positive answer there. Of course, it is also of interest to further investigate whether the increasing convex order and the increasing concave order are preserved under the taking of random minima and maxima, respectively. Theorem 2 in Section 2 confirms the validity of this conjecture. For sake of convenience, throughout this note, the term increasing is used for monotone nondecreasing and decreasing is used for monotone nonincreasing. It is always assumed that in the sequel that the integer-valued random variable has probability p N (n) ⫽ P(N ⫽ n) for n ⫽ 1, 2, . . . . Nonnegative random variables X i ’s and Y i ’s have F and G as their respect common ៮ ⫽ 1 ⫺ G denote their corresponding survival distribution functions and F៮ ⫽ 1 ⫺ F and G functions. 2.

MAIN RESULTS

Before stating the main results, let us recall firstly a useful conclusion which will be utilized in sequel. Denote F N:N共x兲 ⫽ P共max兵X1 , . . . , XN 其 ⱕ x兲, G N:N共x兲 ⫽ P共max兵Y1 , . . . , YN 其 ⱕ x兲, F 1:N共x兲 ⫽ P共min兵X1 , . . . , XN 其 ⱕ x兲, G 1:N共x兲 ⫽ P共min兵Y1 , . . . , YN 其 ⱕ x兲. Bartozewicz [5] shows that, for all x, ⫺1 ⫺1 G ⫺1 N:NF N:N共x兲 ⫽ G F共x兲 ⫽ G 1:NF 1:N共x兲.

(7)

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The following two theorems give our main results. THEOREM 1: Let X 1 , X 2 , . . . and Y 1 , Y 2 , . . . each be a sequence of i.i.d. random variables, and N is independent of X i ’s and Y i ’s. (i) If X i ⱕ RS Y i for i ⫽ 1, 2, . . . , then max兵X1 , . . . , XN 其 ⱕRS max兵Y1 , . . . , YN 其.

(8)

(ii) If X i ’s and Y i ’s are both nonnegative and with common left end point 0, then X i ⱕ ttt Y i for i ⫽ 1, 2, . . . , implies (9) min兵X1 , . . . , XN 其 ⱕttt min兵Y1 , . . . , YN 其. PROOF: (i) The distribution functions of the maxima of N i.i.d. copies of X and Y are, respectively,

冘 F 共x兲p 共n兲, ⫹⬁

F N:N共x兲 ⫽

冘 G 共x兲p 共n兲. ⫹⬁

G N:N共x兲 ⫽

n

N

n⫽1

n

N

n⫽1

The inequality (3) is equivalent to

冕

⫹⬁

F៮ 共x兲 d共G ⫺1F共x兲 ⫺ x兲 ⱖ 0,

t ⱖ 0.

t

Since the function

F៮ N:N共x兲 ⫽ 1 ⫺

冘 F 共x兲p 共n兲 ⫽ 冘 关1 ⫺ F 共x兲兴p 共n兲 ⫹⬁

⫹⬁

n

n

N

n⫽1

N

n⫽1

and

冋 册

冘 冘 F 共x兲 p 共n兲

⫹⬁ ⫹⬁ 关1 ⫺ F n共x兲兴p N共n兲 F៮ N:N共x兲 ¥ n⫽1 ⫽ ⫽ 1 ⫺ F共x兲 F៮ 共x兲 n⫽1

n⫺1

k

N

k⫽0

is increasing and positive for all x ⱖ 0, it follows from (7) and Lemma 7.1(a) (Barlow and Proschan [3], p. 121) that, for all t ⱖ 0,

冕

⫹⬁

F៮ N:N共x兲 d共G ⫺1 N:NF N:N共x兲 ⫺ x兲 ⫽

t

冕

⫹⬁

F៮ N:N共x兲 d共G ⫺1F共x兲 ⫺ x兲

t

⫽

冕

t

⫹⬁

F៮ N:N共x兲 F៮ 共x兲 ៮ d共G ⫺1F共x兲 ⫺ x兲 ⱖ 0. F 共x兲

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That is to say, max兵X1 , . . . , XN 其 ⱕRS max兵Y1 , . . . , YN 其. (ii) The survival functions of the minima of N i.i.d. copies are, respectively,

冘 F៮ 共x兲p 共n兲, ⫹⬁

F៮ 1:N共x兲 ⫽

n

N

៮ 1:N共x兲 ⫽ G

n⫽1

冘 G៮ 共x兲p 共n兲. ⫹⬁

n

N

n⫽1

The inequality (4) is equivalent to

冕

t

F៮ 共x兲 d共G ⫺1F共x兲 ⫺ x兲 ⱖ 0,

t ⱖ 0.

0

៮ n⫺1 ( x) p N (n) is decreasing and positive for all x ⱖ 0, it follows from Since the function ¥ ⫹⬁ n⫽1 F (7) and Lemma 7.1(b) (Barlow and Proschan [3], p. 121) that, for all t ⱖ 0,

冕

t

⫺1 F៮ 1:N共x兲 d共G 1:N F 1:N共x兲 ⫺ x兲 ⫽

0

冕

t

F៮ 1:N共x兲 d共G ⫺1F共x兲 ⫺ x兲

0

冕 冋 t

⫽

0

F៮ 共x兲

冘 F៮ ⫹⬁

册

共x兲p N共n兲 d共G ⫺1F共x兲 ⫺ x兲 ⱖ 0.

n⫺1

n⫽1

That is to say, min兵X1 , . . . , XN 其 ⱕttt min兵Y1 , . . . , YN 其. Now, the proof is complete.

䊐

THEOREM 2: Let X 1 , X 2 , . . . , and Y 1 , Y 2 , . . . , each be a sequence of i.i.d. random variables, and N is independent of X i ’s and Y i ’s. (i) If X i ⱕ icx Y i for i ⫽ 1, 2, . . . , then max兵X1 , . . . , XN 其 ⱕicx max兵Y1 , . . . , YN 其.

(10)

(ii) If X i ’s and Y i ’s are both nonnegative and with common left end point, then X i ⱕ icv Y i for i ⫽ 1, 2, . . . implies min兵X1 , . . . , XN 其 ⱕicv min兵Y1 , . . . , YN 其. (11) PROOF: (i) By (1), the order X 1 ⱕ icx Y 1 states that

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冕

⫹⬁

៮ 共x兲 ⫺ F៮ 共x兲兴 dx ⫽ 关G

t

冕

337

⫹⬁

关F共x兲 ⫺ G共x兲兴 dx ⱖ 0,

t ⱖ 0.

t

Since the function F n⫺1共x兲 ⫹ F n⫺2共x兲G共x兲 ⫹ · · · ⫹ F共x兲G n⫺2共x兲 ⫹ G n⫺1共x兲 is increasing and positive for all x ⱖ 0, it follows from Lemma 7.1(a) (Barlow and Proschan [3], p. 121) that, for all t ⱖ 0,

冕

⫹⬁

៮ N:N共x兲 ⫺ F៮ N:N共x兲兴 dx ⫽ 关G

t

冕

⫹⬁

关F N:N共x兲 ⫺ G N:N共x兲兴 dx

t

⫽

冕

冘 p 共n兲关F 共x兲 ⫺ G 共x兲兴 dx

⫹⬁ ⫹⬁

n

n⫽1

t

冕 冋 ⫹⬁

⫽

n

N

t

冘 p 共n兲 冘 F ⫹⬁

册

n⫺1

n⫽1

共x兲G k共x兲 共F共x兲 ⫺ G共x兲兲 dx ⱖ 0.

n⫺k⫺1

N

k⫽0

That is to say, max兵X1 , . . . , XN 其 ⱕicx max兵Y1 , . . . , YN 其. (ii) The order X 1 ⱕ icv Y 1 states that

冕

t

៮ 共x兲 ⫺ F៮ 共x兲兴 dx ⱖ 0, 关G

t ⱖ 0.

0

Since the function ៮ 共x兲 ⫹ · · · ⫹ F៮ 共x兲G ៮ n⫺2共x兲 ⫹ G ៮ n⫺1共x兲 F៮ n⫺1共x兲 ⫹ F៮ n⫺2共x兲G is decreasing and positive for all x ⱖ 0, it follows from Lemma 7.1(b) (Barlow and Proschan [3], p. 121) that, for all t ⱖ 0,

冕

t

៮ 1:N共x兲 ⫺ F៮ 1:N共x兲兴 dx ⫽ 关G

冕冘 t ⫹⬁

៮ n共x兲 ⫺ F៮ n共x兲兴 dx p N共n兲关G

0 n⫽1

0

⫽

冕 冋冘 t

⫹⬁

0

n⫽1

冘 F៮

n⫺1

p N共n兲

k⫽0

册

៮ k共x兲 共G ៮ 共x兲 ⫺ F៮ 共x兲兲 dx ⱖ 0. 共x兲G

n⫺k⫺1

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That is to say, min兵X1 , . . . , XN 其 ⱕicv min兵Y1 , . . . , YN 其. The proof is now complete.

䊐 3.

3.1.

SOME APPLICATIONS

Location Independent Riskier Order and Lorenz Order

In economics, many stochastic orders are built to compare the risk of two random assets. To keep the comparison independent of locations, Jewitt [16] proposes the following conception: A random asset X is said to be location independent riskier than another random asset Y, denoted by X ⱕ lir Y, if

冕

F⫺1共p兲

F共x兲 dx ⱕ

⫺⬁

冕

G⫺1共p兲

G共x兲 dx,

for all p 僆 共0, 1兲.

⫺⬁

Fagiuoly, Pellerey, and Shaked [11] establish the following equivalence between right spread order and location independent riskier order, X ⱕlir Y N ⫺X ⱕRS ⫺Y. Note that max{⫺X 1 , . . . , ⫺X n } ⫽ ⫺min{X 1 , . . . , X n }, an immediate application of Theorem 1(i) can be obtained as follows. COROLLARY 3: The location independent riskier order is preserved under the random minima of i.i.d. copies. That is, X ⱕlir Y f min兵X1 , . . . , XN 其 ⱕlir min兵Y1 , . . . , YN 其,

(12)

where X i ’s and Y i ’s are i.i.d. copies of X and Y, respectively, and N is independent of X i ’s and Y i ’s. It is always of interests to compare variabilities of two random assets in portfolio theory. Convex ordering is a natural alternative. However it depends on locations involved. To overcome this problem, the Lorenz order is suggested for nonnegative random variables (see, e.g., Arnold [2]). X is said to be smaller than Y in Lorenz order, denoted by X ⱕ lorenz Y, if X/EX ⱕ icx Y/EY. By Theorem 2, we have the following: COROLLARY 4: Let X i ’s and Y i ’s each be i.i.d. copies of X and Y, respectively, and N is independent of X i ’s and Y i ’s. If X ⱕ lorenz Y, then E max兵X1 , . . . , XN 其/EX ⱕ E max兵Y1 , . . . , YN 其/EY, E min兵X1 , . . . , XN 其/EX ⱖ E min兵Y1 , . . . , YN 其/EY.

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3.2.

339

Preservation Property of Some Aging Classes

The IFR (increasing failure rate), IFRA (increasing failure rate average), NBU (new better than used), IFR(2) (increasing failure rate in second order stochastic dominance(2)), NBU(2) (new better than used in second order stochastic dominance(2)), DMRL (decreasing mean residual life), NBUC (new better than used in convex order), and NBUE (new better than used in expectation) classes of life distributions as well as their duals have been proved to have some preservation properties under series and parallel systems of i.i.d. components. For more on these aging notions, please see Barlow and Proschan [3], Deshpande, Kochar, and Singh [10], and Cao and Wang [8, 9]. Let X 1 , X 2 , . . . , be i.i.d. and N be independent of X i ’s. 1. If X 1 is IFR, then max{X 1 , . . . , X n } is IFR (Grosh [14]). 2. If X 1 is IFRA (NBU), then both max{X 1 , . . . , X n } and min{X 1 , . . . , X n } are IFRA (NBU) (Barlow and Proschan [3]). 3. If X 1 is DMRL (NBUE), then max{X 1 , . . . , X n } is DMRL (NBUE) (Abouammoh and El-Neweihi [1]). 4. If X 1 is NBUC, then max{X 1 , . . . , X n } is NBUC (Hendi, Mashhour and Montasser [15]). 5. If X 1 is NBU(2) (IFR(2)), then min{X 1 , . . . , X n } is NBU(2) (IFR(2)) (Li and Kochar [22] and Franco, Ruiz, and Ruiz [13]. Let N be a positive integer-valued random variable which is independent of X i ’s and Y i ’s. 1. If X 1 is IFR (DFR), then both max{X 1 , . . . , X N } and min{X 1 , . . . , X N } are IFR (DFR) (Shaked, [27]). 2. If X 1 is IFRA (DFRA), then both max{X 1 , . . . , X N } and min{X 1 , . . . , X N } are IFRA (DFRA) (Bartozewicz [5]). 3. If X 1 is NBU (NWU) then both max{X 1 , . . . , X N } and min{X 1 , . . . , X N } are NBU (NWU) (Bartozewicz [5]). As our second application, we will present respectively the preservation results of NBUC and NBU(2) under parallel systems and under series systems which are composed of a random number of i.i.d. components. COROLLARY 5: Let X 1 , X 2 , . . . be a sequence of i.i.d. random lives, and N be independent of X i ’s. (i) If X 1 is NBUC, then max{X 1 , . . . , X N } is also of NBUC property. (ii) If X 1 is NBU(2), then min{X 1 , . . . , X N } is also of NBU(2) property. PROOF: (i) X 1 is NBUC; thus, for all t ⱖ 0, 共X i兲 t ⱕicx Xi ,

i ⫽ 1, 2, . . . ,

where X t ⫽ (X ⫺ t兩X ⬎ t) is the residual life length of X at time t ⱖ 0. By Theorem 2(i), we have, for all t ⱖ 0, max兵共X1 兲t , . . . , 共XN 兲t 其 ⱕicx max兵X1 , . . . , XN 其.

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According to Li and Lu [21], it holds that, for any positive integer n, max兵共X1 兲t , . . . , 共Xn 兲t 其 ⱖst 共max兵X1 , . . . , Xn 其兲t ,

t ⱖ 0.

By a straightforward evaluation, it can be deduced that max兵共X1 兲t , . . . , 共XN 兲t 其 ⱖst 共max兵X1 , . . . , XN 其兲t ,

t ⱖ 0.

From transitivity, it follows that 共max兵X1 , . . . , XN 其兲t ⱕicx max兵X1 , . . . , XN 其,

t ⱖ 0.

That is, max{X 1 , . . . , X N } is NBUC. (ii) X 1 is NBU(2), then, for all t ⱖ 0, 共X i兲 t ⱕicv Xi ,

i ⫽ 1, 2, . . . .

By Theorem 2(ii), we have, for all t ⱖ 0, min兵共X1 兲t , . . . , 共XN 兲t 其 ⱕicv min兵X1 , . . . , XN 其. Since, for any positive integer n, st

min兵共X1 兲t , . . . , 共Xn 兲t 其 ⫽ 共min兵X1 , . . . , Xn 其兲t ,

t ⱖ 0,

it holds also that st

min兵共X1 兲t , . . . , 共XN 兲t 其 ⫽ 共min兵X1 , . . . , XN 其兲t ,

t ⱖ 0,

and hence 共min兵X1 , . . . , XN 其兲t ⱕicv min兵X1 , . . . , XN 其, That is to say, min{X 1 , . . . , X N } is NBU(2) too.

t ⱖ 0.

䊐

Belzunce, Candel, and Ruiz [7] propose the NBUD (new better than used in dispersion) class of life distributions. A random life X is NBUD if X [t,⫹⬁) is smaller than X in dispersive order (denoted by X [t,⫹⬁) ⱕ disp X) for all t ⱖ 0, where X 关t,⫹⬁兲 ⫽ 共X兩X ⬎ t兲 is the additional residual life at time t ⱖ 0. In light of the fact that X [t,⫹⬁) ⫽ t ⫹ X t , it can be verified that if X 1 , X 2 , . . . , are i.i.d., then, for all t ⱖ 0, st

共min兵X1 , . . . , Xn 其兲关t,⫹⬁兲 ⫽ min兵共X1 兲关t,⫹⬁兲 , . . . , 共Xn 兲关t,⫹⬁兲 其.

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Through an similar argument in Corollary 5(ii) above, it stems from the preservation property of dispersive order under random minima (Theorem 1(b); Bartoszewicz [5]) that NBUD is preserved under the taking of random minima too. Since X is IFR(2) if X t is decreasing in increasing concave order in t ⱖ 0 (Deshpande, Kochar, and Singh [10]), and X is DMRL if X t is decreasing in increasing convex order in t ⱖ 0 (Cao and Wang [8]), as another application of Theorem 1, an argument similar to the proof of Corollary 5(ii) will give the preservation of IFR(2) under random minima. Based upon the preservation properties of IFR(2) (Franco, Ruiz, and Ruiz [13]), DMRL and NBUE under random maxima, straightforward calculations can tend to corresponding properties under random maxima. COROLLARY 6: Let X 1 , X 2 , . . . , is a sequence of i.i.d. random lives, and N is independent of X i ’s. (i) If X 1 is DMRL (NBUE), then max{X 1 , . . . , X N } is also of DMRL (NBUE) property. (ii) If X 1 is IFR(2), then max{X 1 , . . . , X N }(min{X 1 , . . . , X N }) is also of IFR(2) property. 3.3.

Bounds of Expected Life Lengths of Series and Parallel Systems

Our next application result presents an upper bound for the mean and the variance of a parallel system with a random number of i.i.d. components as well as a lower one for a series system with those elements also. These bounds rely only on the mean of the common life and distribution of the number of its components, and hence are quite convenient in practical situations. Kochar and Wiens [20] propose the bivariate extension of NBUE. A random life X is smaller than another one Y in NBUE ordering, denoted by X ⱕ nbue Y, if

冕

⫹⬁

F៮ 共x兲 dx/EX ⱕ

F⫺1共p兲

冕

⫹⬁

៮ 共x兲 dx/EY, G

p 僆 共0, 1兲.

G⫺1共p兲

It is pointed out in Kochar, Li, and Shaked [19] that X ⱕnbue Y N X/EX ⱖttt Y/EY N X/EX ⱕRS Y/EY. According to Theorem 1, Corollary 7 can be deduced as below. COROLLARY 7: Let X i ’s and Y i ’s each be i.i.d. copies of X and Y, respectively, and N is independent of X i ’s and Y i ’s. If X ⱕ nbue Y, then E关max兵X1 , . . . , XN 其兴 E关max兵Y1 , . . . , YN 其兴 ⱕ , EX EY

(13)

Var关max兵X1 , . . . , XN 其兴 Var关max兵Y1 , . . . , YN 其兴 ⱕ , E2 X E2 Y

(14)

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E关min兵X1 , . . . , XN 其兴 E关min兵Y1 , . . . , YN 其兴 ⱖ , EX EY

(15)

EV n EW n ⱕ , EX EY

(16)

where V n ⫽ max兵X1 , . . . , XN 其 ⫺ min兵X1 , . . . , XN 其, W n ⫽ max兵Y1 , . . . , YN 其 ⫺ min兵Y1 , . . . , YN 其. For the case of minima and maxima of finite elements, Bartoszewicz [4, Corollary 2] presented inequalities (13), (15), and (16) under the assumption of the star-shaped ordering between X i and Y i , i ⫽ 1, . . . , n. As star-shaped ordering implies the NBUE order, they are in fact special cases of Corollary 7. Since X is NBUE if and only if X ⱕ nbue Y, where Y is exponentially distributed with mean EX, Corollary 8 can be deduced as follows. COROLLARY 8: Let X 1 , . . . , X n are i.i.d. NBUE random variables with finite mean ␮ and second moments, and N is independent of X i ’s. Then n

N

n⫽1

冋冘

冘

⫹⬁

冉 冊 冉 冊 冉冘

冘 p 共n兲 冘 1k ,

⫹⬁

E关max共X1 , . . . , XN 兲兴 ⱕ ␮

共⫺1兲k⫹1 pN 共n兲 Var关max共X1 , . . . , XN 兲兴 ⱕ ␮2 2 k2 n⫽1 k⫽1 n

(17)

k⫽1

n k

冘

⫹⬁

n

冊册

1 ⫺ pN 共n兲 k n⫽1 k⫽1

E关min共X1 , . . . , XN 兲兴 ⱖ ␮E共N⫺1 兲.

2

,

(18)

(19)

PROOF: By the NBUE property, there exist Y 1 , . . . , Y n , which are mutually independent and exponentially distributed with parameter ␭, such that (13) and (14) and (15) hold. Notice that

E max共Y1 , . . . , YN 兲 ⫽

冕

冘 关1 ⫺ 共1 ⫺ e

⫹⬁ ⫹⬁

0

冘 p 共n兲 冕 冘 e ⫹⬁

⫽

⫺␭x n

⫹⬁ n⫺1

N

n⫽1

兲 兴pN 共n兲 dx

n⫽1

0

k⫽0

⫺␭x

共1 ⫺ e

1 兲 dx ⫽ ␭

⫺␭x k

冘 p 共n兲 冘 1k , ⫹⬁

n

N

n⫽1

k⫽1

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343

and

E共max共Y1 , . . . , YN 兲兲2 ⫽ 2

冕

冘 共1 ⫺ 共1 ⫺ e

⫹⬁

⫹⬁

⫺␭x n

x

冘 ⫹⬁

⫽2

兲 兲pN 共n兲 dx

n⫽1

0

1 p 共n兲 ␭ 2 n⫽1 N

E min共Y1 , . . . , YN 兲 ⫽

冕

冘 共⫺1兲k 冉 nk 冊, n

2

k⫽1

冘e

⫹⬁ ⫹⬁

0

k⫹1

n⫽1

冘

⫹⬁

⫺n ␭ x

1 1 1 p 共n兲 ⫽ E共N⫺1 兲; pN 共n兲 dx ⫽ ␭ n⫽1 n N ␭

bounds (17), (18), and (19) follow immediately.

䊐

In fact, one can prove that both (17) and (19) hold for parallel systems and series systems with i.i.d. HNBUE (harmonic new better than used in expectation) components, respectively. Recall that X is HNBUE if, for all t ⱖ 0, X ⱕ icx (ⱖ icv ) Y, where Y is exponential with mean ␮. By Theorem 2, we will have max共X1 , . . . , XN 兲 ⱕicx max共Y1 , . . . , YN 兲, min共X1 , . . . , XN 兲 ⱖicv min共Y1 , . . . , YN 兲. Taking expectation on both sides of the above inequalities will yield (17) and (19). It is obvious that NBUE class is a subclass of HNBUE class. In Corollary 8, we get lower bound (18) for the variance of a system with a random number of components but at the cost of restricting ourselves to NBUE distributions. This is probably not a great restriction from a practical point of view. In addition, since right spread order may not be preserved under the minima of finite i.i.d. components (Kochar, Li, and Shaked [19]), neither does it under the random minima. ACKNOWLEDGMENTS This research was supported by the National Natural Science Foundations of China (10201010), Action Programming Funding of Lanzhou University, and the Natural Sciences and Engineering Research Council of Canada. The authors would like to thank the editor and the referee for their valuable suggestions, which have greatly improved the presentation of this paper. REFERENCES [1] A. Abouammoh and E. El-Neweihi, Closure of the NBUE and DMRL classes under formation of parallel systems, Statist Probab Lett 4 (1986), 223–225. [2] B.B. Arnold, Majorization and Lorenz order: A brief introduction, Springer-Verlag, New York, 1987.

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[3] R.E. Barlow and F. Proschan, Statistical theory of reliability and life testing, To Begin with, Silver Spring, Maryland, 1981. [4] J. Bartoszewicz, Applications of a general composition theorem to the star order of distributions, Statist Probab Lett 38 (1998), 1–9. [5] J. Bartoszewicz, Stochastic comparisons of random minima and maxima from life distributions, Statist Probab Lett 55 (2001), 107–112. [6] F. Belzunce, On a characterization of right spread order by the increasing convex order, Statist Probab Lett 45 (1999), 103–110. [7] F. Belzunce, J. Candel, and J.M. Ruiz, Dispersive orderings and characterization of ageing classes, Statist Probab Lett 28 (1996), 321–327. [8] J. Cao and Y. Wang, Correction for ‘The NBUC and NWUC classes of life distributions,’ J Appl Probab 29 (1992), 753. [9] J. Cao and Y. Wang, The NBUC and NWUC classes of life distributions, J Appl Probab 28 (1991), 473– 479. [10] J.V. Deshpande, S.C. Kochar, and H. Singh, Aspects of positive ageing, J Appl Probab 23 (1986), 748 –758. [11] E. Fagiuoly, F. Pellerey, and M. Shaked, A characterization of the dilation order and its applications, Statist Papers 40 (1999), 393– 406. [12] J.M. Fernandez-Ponce, S.C. Kochar, and J. Mun˜oz-Pe´rez, Partial orderings of distributions based upon right spread function, J Appl Probab 35 (1998), 221–228. [13] M. Franco, J.M. Ruiz, and M.C. Ruiz, On closure of the IFR(2) and NBU(2) classes, J Appl Probab 38 (2001), 236 –242. [14] D.L. Grosh, A parallel system of IFR units is IFR, IEEE Trans Reliability 31 (1982), 403. [15] M.I. Hends, A.F. Mashhour, and M.A. Montasser, Closurer of the NBud class of life distribution under formation of Parallel system, J Appl Probob 30 (1993), 925–978. [16] I. Jewitt, Choosing between risky prospects: The characterization of comparative statics results, and location independent risk, Management Sci 35 (1989), 60 –70. [17] B. Klefsjo¨, TTT-plotting—a tool for both theoretical and practical problems, J Statist Plann Inference 29 (1991), 111–124. [18] S.C. Kochar and K.C. Carrie´re, Connections among various variability orderings, Statist Probab Lett 35 (1997), 327–333. [19] S.C. Kochar, X. Li, and M. Shaked, The total time on test transform and the excess wealth stochastic order of distributions, Adv Appl Probab 34 (2002), 826 – 845. [20] S.C. Kochar and D.P. Wiens, Partial ordering of life distributions with respect to their aging properties, Naval Res Logist Quart 34 (1987), 823– 829. [21] X. Li and J. Lu, Stochastic comparisons on residual life and inactivity time of series and parallel systems, Probab Engrg Inform Sci 17 (2003), 267–276. [22] X. Li and S.C. Kochar, Some new results involving the NBU(2) classes of life distributions, J Appl Probab 38 (2001), 242–247. [23] A. Mu¨ller and D. Stoyan, Comparison methods for stochastic models and risks, Wiley, New York, 2002. [24] M. Shaked and J.G. Shanthikumar, Stochastic orders and their applications, Academic Press, San Diego, 1994. [25] M. Shaked and J.G. Shanthikumar, Two variability orders, Probab Engrg Inform Sci 12 (1998), 1–23. [26] M. Shaked and T. Wong, Stochastic comparisons of random minima and maxima, J Appl Probab 34 (1997), 420 – 425. [27] M. Shaked, On the distribution of the minimum and of the maximum of a random number of i.i.d. random variables. In: G.P. Patil, S. Kotz and J. Ord (Editors), Statistical Distribution in Scientific Work, vol. 1. Reidel, Dordrecht, (1975), 363–380.