Dependence, Dispersiveness, and Multivariate Hazard Rate Ordering

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1 Portland State University PDXScholar Mathematics and Statistics Faculty Publications and Presentations Fariborz Maseeh Department of Mathematics and Statistics Dependence, Dispersiveness, and Multivariate Hazard Rate rdering Baha-Eldin Khaledi Razi University Subhash C. Kochar Portland State University, Let us know how access to this document benefits you. Follow this and additional works at: Part of the ther Mathematics Commons Citation Details Khaledi, E. and Kochar, S. (2005). Dependence, dispersiveness, and multivariate hazard rate ordering. Probability in the Engineering and Informational Sciences, 19, pp This Article is brought to you for free and open access. It has been accepted for inclusion in Mathematics and Statistics Faculty Publications and Presentations by an authorized administrator of PDXScholar. For more information, please contact

2 Probability in the Engineering and Informational Sciences, 19, 2005, Printed in the U+S+A+ DEPENDENCE, DISPERSIVENESS, AND MULTIVARIATE HAZARD RATE RDERING BAHA-ELDIN KHALEDI Statistical Research Center Tehran, Iran and Department of Statistics College of Sciences Razi University Kermanshah, Iran SUBHASH KCHAR Department of Mathematics and Statistics Portland State University Portland, regon To compare two multivariate random vectors of the same dimension, we define a new stochastic order called upper orthant dispersive ordering and study its properties+ We study its relationship with positive dependence and multivariate hazard rate ordering as defined by Hu, Khaledi, and Shaked ~Journal of Multivariate Analysis, 2002!+ It is shown that if two random vectors have a common copula and if their marginal distributions are ordered according to dispersive ordering in the same direction, then the two random vectors are ordered according to this new upper orthant dispersive ordering+ Also, it is shown that the marginal distributions of two upper orthant dispersive ordered random vectors are also dispersive ordered+ Examples and applications are given+ 1. INTRDUCTIN It is of interest to compare two random variables in terms of their variability+ Although this topic has been studied extensively in the univariate case, several attempts have been made to extend it to the multivariate case+ Important contributions in this case 2005 Cambridge University Press $

3 428 B.-E. Khaledi and S. Kochar have been made by Giovagnoli and Shaked and and Fernandez-Ponce and among others+ Let X and Y be two univariate random variables with distribution functions F and G and with survival functions F and G, respectively+ A basic concept for comparing variability in distributions is that of dispersive ordering+ X is said to be less dispersed than Y ~denoted by X disp Y! if F ~b! F ~a! G ~b! G ~a! whenever 0 a b 1, (1.1) where F and G are the right continuous inverses of the distribution functions F and G, respectively+ This means that the difference between any two quantiles of X is smaller than the difference between the corresponding quantiles of Y+ In case the random variables X and Y are of continuous type with hazard rates r F and r G, respectively, then X disp Y if and only if r G ~G ~ p!! r F ~F ~ p!!, (1.2) For more details on dispersive ordering, see Shaked and Sec+ 2B#+ In analogy with the characterization ~1+2! of the univariate dispersive ordering, we introduce a new order in the multivariate case, which we call upper orthant dispersive ordering and study its properties+ According to ~1+2!, X disp Y if and only if the hazard rates of X and Y at the quantiles of the same order p are ordered for all values of To this end, we first recall the definition of hazard rate ~or hazard gradient! in the multivariate case+ Consider a random vector X ~X 1,+++,X n! with a partially differentiable survival function F~x! P$X x%+ The function R logf is called the hazard function of X, and the vector r X of partial derivatives, defined by r X ~x! ~r ~1! X ~x!,+++,r ~n! X ~x!! ] ] R~x!,+++, ]x 1 ]x n R~x!, for all x $x : F~x! 0%, is called the hazard gradient of X ~see Johnson and and Note that r! X ~x! can be interpreted as the conditional hazard rate of X i evaluated at x i, given that X j x j for all j i; that is, r X! ~x! f i x $X j x j % iji F i x $X j x j % iji where f i ~{6 ji $X j x j %! and F i ~{6 ji $X j x j %! are respectively the conditional density and the conditional survival functions of X i, given that X j x j for all j i+ For convenience, here and below we set r! X ~x! ` for all x $x : F~x! 0%+ Now, we define upper orthant dispersive ordering+,

4 DEPENDENCE, DISPERSIVENESS, MULTIVARIATE HAZARD RATE RDERING 429 Definition 1.1: Let X ~X 1,+++,X n! and Y ~Y 1,+++,Y n! be two random vectors with respective survival functions F and G. We say that X is smaller than Y according to upper orthant dispersive ordering (denoted by X uo-disp Y) if for all u j 1,+++,n, j i, X $X j F j ~u j!% disp iji Y $Y j G j ~u j!% iji, (1.3) for i 1,+++,n. In case the distributions under consideration are absolutely continuous, the upper orthant dispersive ordering can be equivalently expressed in terms of the hazard gradients at the quantiles of the same orders of the conditional distributions+ If we denote by x i ~b;u! and y i ~b;u!, the bth quantiles of the conditional distributions ~X i 6 ji $X j F j ~u j!%! and ~Y i 6 ji $Y j G j ~u j!%!, respectively, then X uo-disp Y m r! X ~F 1 ~u 1!,+++,F i ~u i!, x i ~b;u!,+++,f n ~u n!! r! Y ~G 1 ~u 1!,+++,G i ~u i!, y i ~b;u!,+++,g n ~u n!!, (1.4) for every n, and i 1,+++,n, where r X! and r Y! stand for the ith components of the hazard gradients of X and Y, respectively+ The following slightly modified version of a theorem of Saunders and provides a useful tool for establishing dispersive ordering among members of a parametric family of distributions+ Theorem 1.1: Let X a be a random variable with distribution function F a for each a R such that the following hold: Then (i) F a is supported on some interval ~x ~a!, x ~a!! ~`,`! and has density f a that does not vanish on any subinterval of ~x ~a!, x ~a!!. (ii) The derivative of F a with respect to a exists and is denoted by F a'. if and only if, X a disp X a * for a, a * R, and a a *, (1.5) F a' ~x!0f a ~x! is decreasing in x+ (1.6) In the next example we identify conditions under which two bivariate normal random vectors are ordered according to upper orthant dispersive ordering+ More examples are discussed in Section 4+

5 430 B.-E. Khaledi and S. Kochar Example 1.1 (Bivariate Normal Distribution): Let X and Y follow bivariate Normal distributions, each with mean vector ~0,0! and with dispersion matrices S s 2 X 1 rs 1 s 2 and s 2 1 r ' s 1 s 2 SY 2 rs 1 s 2 s 2 r ' 2 s 1 s 2 s, 2 respectively, with s i 0 for i 1,2+ For the time being, we are assuming that the marginal distributions of X and Y are identical+ The general case is considered later+ We use Theorem 1+1 to prove that in case r and r ' are of the same sign, then 6r ' 6 6r6 n X uo-disp Y+ Let us denote by G r, the distribution function of $X 1 6X 2 y% ~we are suppressing its dependence on y for the clarity of notation!+ Then G r ~x! 1 P~X 2 y! ` 1 P~X 2 y! y x ` ` y f X1, X 2 ~u,v! du dv P~X 1 x6x 2 v! f X2 ~v! dv 1 P~X 2 y! y ` F x rs 1 v s 2 s 1 ~1 r 2! s 2 f and the corresponding conditional density function is 1 g r ~x! P~X 2 y!s 1 ~1 r 2! 102y Now we compute G r' ~x! ] ]r G r~x! 1 P~X 2 y! y f x rs 1 v s 2 ` f x rs 1 v s 2 ` s 1 s 2 v~s 1 ~1 r 2! 102! s 1 ~1 r 2! s 2 f v s 2 dv, v s 2 dv s 1 ~1 r 2! s 2 f s 12 ~1 r 2! v s 2 dv+ s 1 r ~1 r 2! 102 x rs 1 s 2 v

6 DEPENDENCE, DISPERSIVENESS, MULTIVARIATE HAZARD RATE RDERING 431 which simplifies to G r' ~x! e~x 2 02s 2 1! ` P~X 2 y!m2p y exp 1 2s 22 ~1 r 2! rs 2 s 1 v M2ps 22 ~1 r 2! 302 v rxs 2 s 1 2 dv e ~x 2 02s 2 1! P~X 2 y!~2p!~1 r 2! exp s 22 ~1 r 2! Similarly, the conditional density function g r ~x! can be written as This gives e ~x 2 02s 1 2! g r ~x! P~X 2 y!m2ps 1y exp 1 2s 22 ~1 r 2! ` 1 s 2 ~1 r 2! 102 ~2p! 102 v rxs 2 s 1 h~x! G ' r~x! s 2 s 1 r Wx ~ y!, g r~x! y rxs 2 s (1.7) 2 dv+ (1.8) where r Wx ~+! denotes the hazard rate of W x, a normal random variable with mean rxs 2 0s 1 and variance s 22 ~1 r 2!+ It is known that the family of normal random variables with a fixed variance but with different means is ordered according to hazard rate order and the one with the smaller mean has the greater hazard rate+ Using this fact, it follows that if r 0, then h~x! is increasing in x+ Hence, by Theorem 1+1, 0 r ' r n X uo-disp Y+ If r 0, then h~x! is decreasing in x; hence, ~X 1, X 2! is increasing in r in the sense of upper orthant dispersive ordering+ This proves the required result+ The organization of the article is as follows+ In Section 2 we study some properties of the upper orthant dispersive ordering as defined earlier+ It is proved that if two random vectors have the same dependence structure ~copula!, then they are ordered according to upper orthant dispersive ordering if and only if their corresponding marginals are ordered according to univariate dispersive ordering+ In Section 3 we consider the special case of nonnegative random variables ~more generally, if the conditional distributions have common left end points of their supports!+ It is shown that if two random vectors have the same marginal distributions and they are ordered according to upper orthant dispersive ordering, then their bivariate copulas

7 432 B.-E. Khaledi and S. Kochar are ordered, implying that one random vector is more dependent in the sense of positive quadrant dependence than the other+ We also study the connection between upper orthant dispersive ordering and multivariate hazard rate ordering as introduced by Hu, Khaledi, and The last section is devoted to some examples and applications+ It is shown that if two univariate distributions are ordered according to dispersive ordering, then the corresponding vectors of order statistics from them are ordered according to upper orthant dispersive ordering+ 2. PRPERTIES F UPPER RTHANT DISPERSIVE RDERING In this section we establish an interesting property of the upper orthant dispersive ordering that if two n-dimensional random vectors X and Y have the same dependence structure in the sense that they have the same copula, then dispersive ordering among the marginal distributions implies upper orthant dispersive ordering and vice versa+ The notion of copula has been introduced by and studied by, among others, Kimeldrof and under the name of uniform representation and by under the name of dependence function+ A copula C is a cumulative distribution function with uniform margins Given a copula C, if one defines F~x! C~F 1 ~x 1!, F 2 ~x 2!,+++F n ~x n!!, x IR n, (2.1) then F is a multivariate distribution function with margins as F 1, F 2,+++,F n + For any multivariate distribution function F with margins as F 1, F 2,+++,F n, there exists a copula C such that ~2+1! holds+ If F is continuous, then C is unique and can be constructed as follows: C~u! 1 ~u 1!, F 2~u 2!,+++,F n ~u n!#, n + (2.2) It follows that if X and Y are two n-dimensional random vectors with margins as ~F 1, F 2,+++,F n! and ~G 1,G 2,+++,G n!, respectively, and if they have the same copula, then ~F 1 ~X 1!, F 2 ~X 2!,+++F n ~X n!! st ~G 1 ~Y 1!,G 2 ~Y 2!,+++G n ~Y n!!+ (2.3) For i 1,+++,n, let us denote by H X i, u the cumulative distribution function ~cdf! of the conditional distribution ~X i 6 ji $X j F j ~u j!%! and by H Y i, u that of ~Y i 6 ji $Y j G j ~u j!%!+ To prove the next theorem, we first prove the following lemma, which may be of independent interest+ Lemma 2.1: If two n-dimensional random vectors X and Y have the same copula, then, for i 1,+++n, X F i H i, u Y ~b! G i H i, u ~b!, n + (2.4)

8 DEPENDENCE, DISPERSIVENESS, MULTIVARIATE HAZARD RATE RDERING 433 Proof: Proving ~2+4! is equivalent to proving that for i,+++n and n, HP X i, u F i ~v! HP Y i, u G i ~v!, m P X i F i ~v! $X j F j ~u j!% ji P Y i G i ~v! ji m P F i~x i! v $F j ~X j! u j % ji P G i~y i! v $G j ~Y j! u j % ji $Y j G j ~u j!%#, which is true because of ~2+3!, since X and Y have the same copula+ Theorem 2.1: Let X and Y be two n-dimensional random vectors with the same copula. Then X uo-disp Y if and only if X i disp Y i,i,+++,n. Proof: By definition, X uo-disp Y Y m H X i, u ~b! H i, u ~b! is increasing in 1#, n, for i 1, +++, n+ (2.5) It follows from ~2+4! that if X and Y have the same copula, then for i 1,+++n and n, Y H X i, u ~b! H X i, u ~b! G i F i ~H X i, u ~b!! H i, u ~b!, (2.6) for i 1,+++n and for every It is easy to see that the right-hand side of ~2+6! is increasing in b if and only if G i F i ~x! x is increasing in x +e+, if and only if X i disp Y i, i 1,+++,n!+ This proves the desired result+ Recently, Müller and have investigated some other multivariate stochastic orders for which results parallel to Theorem 2+1 hold for those orders+ The following interesting property of the upper orthant dispersive ordering immediately follows from Theorem 2+1+ Corollary 2.1: Let Y ~a 1 X 1 b 1, a 2 X 2 b 2,+++,a n X n b n!. Then for a i 1, b i IR, X uo-disp Y. Proof: Since X and Y have the same copula, the required result follows immediately from Theorem 2+1+

9 434 B.-E. Khaledi and S. Kochar Example 2.1 (Multivariate Normal Distributions): Let X follow the p-variate multivariate Normal distribution with mean vector m and dispersion matrix S ~~s ij!!, with s ij r ij s i s j, r ii 1, s i 0, and i, j 1,+++,p+ Let Y follow the p-variate multivariate Normal distribution with mean vector m ' and dispersion matrix S ' ~~s ij'!!, with s ij' r ij s i' s j', s i' 0, and i, j,+++,p+ It is known that X and Y have the same copula+ It follows from Theorem 2+1 that X uo-disp ' Y if and only if s i s i for i 1,+++,p+ This result in conjunction with Example 1+1 leads us to the following result for comparing two bivariate Normal distributions+ Let X and Y follow bivariate Normal distributions with dispersion matrices S s 2 1 rs 1 s 2 2 rs 1 s 2 s, s '2 1 r ' s 1' s 2 ' S' 2 r ' ' '2 s 1' s 2 s, 2 respectively+ If 0 s i s i' for i 1,2 and 6r ' 6 6r6 1, then X uo-disp Y+ It will be interesting to find necessary and sufficient conditions under which two multivariate normal random vectors will be ordered according to upper orthant dispersive ordering in the general case+ It follows immediately that if two random vectors are ordered according to upper orthant dispersive ordering, then so are their corresponding subsets+ In particular, their marginal distributions will be then ordered according to univariate dispersive ordering+ Theorem 2.2: Let X and Y be two n-dimensional random vectors such that X uo-disp Y. Then X uo-disp I Y I, where I $i 1, i 2,+++,i k % $1,2,+++,n%, X I ~X i1,+++,x ik!, Y I ~Y i1,+++,y ik!, and k 1,+++,n. The proof of the next result is also immediate+ Theorem 2.3: Let X 1,+++,X m be a set of independent random vectors for which the dimension of X i is k i,i 1,+++,m. Let Y 1,+++,Y m be another set of independent random variables for which the dimension of Y i is k i,i 1,+++,m. Then ~X uo-disp i Y i, i 1,+++,m! n ~X 1,+++,X m! uo-disp ~Y 1,+++,Y m!+ (2.7) Remark 2.1: A consequence of ~2+7! is that if X 1,+++,X n is a collection of independent univariate random variables and Y 1,+++,Y n is another set of independent random variables, then X i disp Y i, i 1,+++,n implies X uo-disp Y+ In general, there does not seem to be any direct connection between upper orthant dispersive ordering and the multivariate dispersive ordering as introduced by Fernandez-Ponce and According to their definition, X disp Y may not imply that X i disp Y i for i 1,+++,n+ Also, the multivariate dispersive ordering as defined by them may not be preserved under permutations of the vari-

10 DEPENDENCE, DISPERSIVENESS, MULTIVARIATE HAZARD RATE RDERING 435 ables+ n the other hand, the upper orthant dispersive ordering is invariant under the same permutation of the two vectors and their marginals are also ordered according to univariate dispersive ordering+ bviously, if X uo-disp Y, then tr S X tr S Y, where S X and S Y denote the dispersion matrices of X and Y, respectively+ 3. THE CASE F NNNEGATIVE RANDM VARIABLES In this section, we will restrict our attention to the case in which the random vectors under consideration are nonnegative or, more generally, they have a finite common left end point of their supports+ We will see that certain results hold in this case that may not hold in the general case+ The following assumption will be made at some places in this article+ Assumption A: The random variables $X i 6 ji $X j F j ~u j!%% and $Y i 6 ji $Y j G j ~u j!%% have afinite common left endpoint of their supports for all u and for i 1,+++,n. In the univariate case, for nonnegative random variables, there is an intimate connection between hazard rate ordering and dispersive ordering and which is made more explicit in the following result of Bagai and We use this theorem to prove some of the results of this section+ Theorem 3.1: Let X and Y be two univariate random variables with distribution functions F and G, respectively, such that F~0! G~0! 0. Then the following hold: (a) If Y hr X and either F or G is DFR (decreasing failure rate), then Y disp X. (b) If Y disp X and either F or G is IFR (increasing failure rate), then Y hr X. For a bivariate random vector ~S, T!, we say that T is right tail increasing in S if t 6S s# is increasing in s for all t, and we denote this relationship by RTI~T 6S!+ If S and T are continuous lifetimes, then T is right tail increasing in S if and only if r~s6t t! r~s6t 0! r S ~s! for all s 0 and for each fixed t+ The RTI property is weaker than the RCSI ~right corner set increasing! property, but stronger than PQD ~positive quadrant dependence!+ In the next theorem, we study the effect of positive dependence on upper orthant dispersive ordering for nonnegative random vectors+ Theorem 3.2: Let X ~X 1, X 2! be a bivariate random vector such that the left end point of the support of $X i 6X j F j ~u!% is finite and independent of for i, j 1,2. Let X I ~X 1I, X 2I! be a random vector of independent random variables such that X i st X ii, i 1,2. (a) If X i is RTI in X j,i j, and X i is DFR for i, j 1,2, then ~X 1, X 2! uo-disp ~X 1I, X 2I!+ (3.1)

11 436 B.-E. Khaledi and S. Kochar (b) If ~X 1, X 2! uo-disp ~X 1I, X 2I! and X i is IFR for i 1,2, then X i is RTI in X j, i j, i, j 1,2. Proof: ~a! Note that RTI~X i 6X j! if and only if, for all u 0, $X i 6X j F j ~u!% hr X i + (3.2) It follows from Theorem 3+1~a! that if, in addition, X i is DFR, then $X i 6X j F j ~u!% disp X i st X ii + (3.3) Since this holds for i, j 1,2, the required result follows+ ~b! ~X 1, X 2! uo-disp ~X 1I, X 2I! implies for all u 0, $X i 6X j F j ~u!% disp X ii st X i, i j, i, j 1,2+ This together with the assumption that X 1 and X 2 are IFR implies ~3+2! by Theorem 3+1~b!+ This proves that RTI~X i 6X j!, i j, i, j 1,2+ Theorem 3.3: Let X and Y be two n-dimensional random vectors satisfying Assumption A and such that X i st Y i,i,+++,n. Then X uo-disp Y implies that C X Y i, j C i, j for i, j 1, +++, n, i j, (3.4) where C X i, j ~C Y i, j! denotes the copula of ~X i, X j!~~y i,y j!!. Proof: X uo-disp Y n ~X i, X j! uo-disp ~Y i,y j!, i j, i, j $1,+++, n%, (3.5) from which it follows that $X i 6X j F j ~u j!% disp $Y i Y j G j ~u j!%, u and which, in turn, implies $X i 6X j F j ~u j!% st $Y i 6Y j G j ~u j!%, u (3.6) under Assumption A since dispersive ordering implies stochastic ordering when the random variables have a finite common left end point of their supports+ If we denote by F i, j ~ G i, j! the joint survival function of ~X i, X j!~~y i,y j!!, then ~3+6! can be written as

12 DEPENDENCE, DISPERSIVENESS, MULTIVARIATE HAZARD RATE RDERING 437 F i, j ~x, F j ~u j!! G i+j ~x,g j ~u j!! for x 0, u m F i, j ~F i~u i!, F j ~u j!! G i+j ~F i~u i!,g j ~u j!! m F i, j ~F i~u i!, F j ~u j!! G i+j ~G i~u i!,g j ~u j!! for all u i, u for all u i, u ~since X i st Y i, i 1, +++, n! m C X i, j ~u i, u j! C Y i, j ~u i, u j! m C X i, j ~u i, u j! C Y i, j ~u i, u j! for all u i, u for all u i, u where C~u,v! 1 u v C~u+v!+ If ~3+4! holds and the margins of ~X i, X j! and ~Y i,y j! are equal, then we say that ~Y i,y j! is more PQD than ~X i, X p+ 36#!+ Note that C X i, j ~u i, u j![u i u j in the case X i and X j are independent and C X i, j ~u i, u j! u i u j for all u i, u in the case X i and X j are PQD+ Thus, according to Theorem 3+3, if Assumption A holds and if X and Y have the same margins and X uo-disp Y, then the Y i s are more dependent than the X i s according to PQD ordering+ We obtain the following result as a special case+ Corollary 3.1: Let X ~X 1, X 2! be a bivariate random vector such that the left end point of the support of $X i 6X j F j ~u!% is finite and independent of for i j and i, j 1,2. Let X I ~X 1I, X 2I! be a random vector of independent random variables such that X i st X ii,i 1,2+ Then ~X 1, X 2! uo-disp ~X 1I, X 2I! n X is PQD+ (3.7) Contrast this result with Theorem 3+2~b!, which is a stronger one since RTI implies PQD+ However, here no assumption on the monotonicity of the hazard rates is made in the second case+ Remark 3.1: Assumption A is very crucial for Theorem 3+3 and Corollary 3+1 to hold+ As a counterexample, let Y 1 and Y 2 be two independent U~0,1! random variables+ Let X 1 X 2 be uniformly distributed over ~0,1! also+ Note that X 1 and X 2 are strongly positively dependent, as they satisfy the Frechet upper bound+ Let us compare ~X 1, X 2! with ~Y 1,Y 2! according to upper orthant dispersive ordering+ The relevant conditional distributions to compare 1 6X 1 u# 1 6Y 2 u#, 0 u 1+ The left-hand conditional distribution is U~u,1! and the right-hand conditional distribution is U~0,1!+ Hence, ~X 1, X 2! uo-disp ~Y 1,Y 2!, but C X 1,2 C Y 1,2, contradictory to ~3+4!+ The reason for this contradiction is that unless Assumption A is satisfied, dispersive ordering may not imply stochastic ordering+

13 P P P 438 B.-E. Khaledi and S. Kochar Theorem 3.4: Let X and Y be two n-dimensional random vectors satisfying Assumption A and such that X uo-disp Y. Then implies uv C X i, j C Y i, j, i, j 1,+++,n, i j, (3.8) Cov~h 1 ~X i!, h 2 ~X j!! Cov~h 1 ~Y i!, h 2 ~Y j!! (3.9) for all increasing convex functions h 1 and h 2 for which the above covariances exist. Proof: Without loss of generality, let i 1 and j 2+ The survival functions of ~h 1 ~X 1!, h 2 ~X 2!!, h 1 ~X 1!, and h 2 ~X 2! are, respectively, H~x P 1, x 2! F~h 1 ~x 1!, h 2 ~x 2!!, HP 1 ~x 1! F 1 ~h 1 ~x 1!!, and HP 2 ~x 2! F 2 ~h 2 ~x 2!!+ Similarly, the survival functions of ~h 1 ~Y 1!, h 2 ~Y 2!!, h 1 ~Y 1!, and h 2 ~Y 2! are, respectively, K~x 1, x 2! G~h 1 ~x 1!, h 2 ~x 2!!, K 1 ~x 1! G 1 ~h 1 ~x 1!!, and K 2 ~x 2! G 2 ~h 2 ~x 2!!+ Covariance between h 1 ~X 1! and h 2 ~X 2!, if it exists, can be expressed as Cov~h 1 ~X 1!, h 2 ~X 2!! ~ H~x P 1, x 2! HP 1 ~x 1! HP 2 ~x 2!! dx 1 dx 2 ~ F~h 1 ~x 1!, h 2 ~x 2!! F 1 ~h 1 ~x 1!! F 2 ~h 2 ~x 2!!! dx 1 dx 2 ~ F~F 1 ~u!, F 2 ~v!! ~1 u!~1 v!! h 1 ' ~F 1 ~u!! 0 1 f 1 ~F 1 ~u!! h ' 2~F 2 f 2 ~F 2 ~v!! 1 du dv ~ C X 1,2 ~u,v! ~1 u!~1 v!! 0 h 1 ' ~F 1 f 1 ~F 1 ~u!! h ' 2~F 2 du dv, (3.10) f 2 ~F 2 ~v!! uo-disp where h 1 ~x 1! F 1 ~u! and h 2 ~x 2! F 2 ~v!+ The assumption X Y implies that X i disp Y i, from which it follows that f i ~F i ~u!! g i ~G i ~u!! and F i ~u! G i ~u!, i 1,2, under Assumption A+ Now h i' ~x! is increasing in x since h~x! is convex+ Combining these facts, the required result follows from ~3+8!+ Theorem 3.5: Let X and Y be two n-dimensional random vectors satisfying the Assumption A and let f 1,+++,f n be increasing convex functions on IR. Then X uo-disp Y n ~f 1 ~X 1!,+++,f n ~X n!! uo-disp ~f 1 ~Y 1!,+++,f n ~Y n!!+

14 DEPENDENCE, DISPERSIVENESS, MULTIVARIATE HAZARD RATE RDERING 439 Proof: Note that the cdf of f j ~X j! is F fj ~x! F j ~f j ~x!!, with its inverse as F fj ~u! f j ~F j ~u!!+ We have to prove that for i,+++,n, u j ~0,1!, j,+++,n, j i, f i~x i! $f j ~X j! F fj ~u j!% ji disp f i~y i! $f j ~Y j! G fj ~u j!% ; ji that is, f i~x i! $f j ~X j! f j ~F j ~u j!!% ji disp f i~y i! $f j ~Y j! f j ~G j ~u j!!%, ji which is equivalent to f i~x i! $X j F j ~u j!% disp f i~y i! $Y j G j ~u j!% + ji ji Using Assumption A, for i 1,+++,n, u j ~0,1!, j 1,+++,n, j i, X $X j F j ~u j!% disp iji Y $Y j G j ~u j!% iji implies ~X i 6$X j F j ~u j!%! st ~Y i 6$Y j G j ~u j!%!+ Now the required result follows from Theorem 2+2 of Rojo and since f i s are increasing convex functions+ Hu et gave the following definition of multivariate ~weak! hazard rate ordering+ Definition 3.1: Let X and Y be n-dimensional random vectors with hazard gradients r X and r Y, respectively. We say that X is smaller than Y according to weak hazard rate ordering (written as X whr Y)if X $X j x j % hr iji Y $Y j x j % iji, for i 1,2,+++,n, x IR n ; that is, if r X! ~x! r Y! ~x!, i 1,2,+++,n, x IR+ In the next theorem we establish results analogous to Theorem 3+1 between upper orthant dispersive ordering and multivariate weak hazard rate ordering under Assumption A+

15 440 B.-E. Khaledi and S. Kochar Theorem 3.6: Let X and Y be two n-dimensional random vectors satisfying Assumption A. Proof: (a) If Y whr X and either r! X ~x! or r! Y ~x! is decreasing in x, for i 1,+++,n, then Y uo-disp X. (b) If Y uo-disp X and either r! X ~x! or r! Y ~x! is increasing in x, for i,+++,n, then Y whr X. ~a! Under Assumption A, Y whr X implies that Y i st X i, i,+++,n, which is equivalent to G i ~u! F i ~u!, for i,+++,n+ Again, Y whr X implies that for i 1,+++,n, ~Y i 6 ji $Y j x j %! st ~X i 6 ji $X j x j %!+ Taking x j F j ~u j!, j 1,+++,n, j i, we find that this implies y i' ~b;u! x i ~b;u!, where y i' ~b;u! is the bth quantiles of the conditional distribution ~Y i 6 ji $Y j F j ~u j!%! and x i ~b;u! is as defined earlier+ n the other hand, r Y! ~x! decreasing in x implies that Y iji $Y j x j % st Y iji $Y j x j' %, for x j x j', j 1 +++,n, j i+ This, along with G i ~u i! F i ~u i!, implies that y i ~b;u! y i' ~b;u! x i ~b;u!, i 1,+++,n+ Using these observations, we obtain r! X ~F 1 ~u 1!,+++,F i ~u i!, x i ~b;u!,+++,f n ~u n!! r! Y ~F 1 ~u 1!,+++,F i ~u i!, x i ~b;u!,+++,f n ~u n!!, since Y whr X+ The right-hand side of this inequality is less than or equal to r! Y ~G 1 ~u 1!,+++,G i ~u i!, y i ~b;u!,+++,g n ~u n!! since r! Y ~x! is decreasing in x+ This completes the proof of ~a!+ ~b! From the assumption Y uo-disp X, it follows that r! X ~F 1 ~u 1!,+++,F i ~u i!, x i ~b;u!,+++,f n ~u n!! r! Y ~G 1 ~u 1!,+++,G i ~u i!, y i ~b;u!,+++,g n ~u n!! (3.11)

16 DEPENDENCE, DISPERSIVENESS, MULTIVARIATE HAZARD RATE RDERING 441 and X $X j F j ~u j!% st iji Y $Y j G j ~u j!% iji, which implies that x i ~b;u! y i ~b;u! and F j ~u j! G j ~u j!, j,+++,n, j i+ Using these facts and the assumption that r Y! ~x! is increasing in x, it follows that, for i 1,+++,n, the right-hand side of ~3+11! is less than or equal to r! Y ~F 1 ~u 1!,+++,F i ~u i!, x i ~b;u!,+++,f n ~u n!! r! Y ~x!; that is, we have shown that for i 1,+++,n, and hence the required result+ r X! ~x! r Y! ~x! Remark 3.2: If r X! ~x 1,+++,x i,+++,x n! increases in x i for i 1,+++,n, then we say that the random vector X has a multivariate increasing hazard rate distribution ~cf+ Johnson and The condition r X! ~x 1,+++,x i,+++,x n! increasing in x j, j 1,+++,n, j i, i 1,+++,n describes a condition of positive dependence that is equivalent to saying that the random vector X has RCSI; that is, increases in x i', i 1,+++,n+ 1 x 1,+++,X n x n 6X 1 x 1',+++,X n x n' # We will now study some preservation properties of the upper orthant dispersive order under random compositions+ Such results are often referred to as preservations under random mapping ~see Shaked and or preservations of stochastic convexity ~see Shaked and Chap+ 6# and Denuit, Lefèvre, and and references therein!+ Let $ F u, u X % be a family of n-dimensional survival functions, where X is a subset of the real line+ Let X~u! denote a random vector with survival function F u + For any random variable Q with support in X and with distribution function H, let us denote by X~Q! a random vector with survival function G given by G~x! F u ~x! dh~u!, x IR X n + Theorem 3.7: Consider a family of n-dimensional survival functions $ F u, u X % as above. Let Q 1 and Q 2 be two random variables with supports in X and distribution functions H 1 and H 2, respectively. Let Y 1 and Y 2 be two random vectors such that Y i st X~Q i!, i 1,2; that is, suppose that the survival function of Y i is given by G i ~x! F u ~x! dh i ~u!, x IR, i 1,2+ X

17 442 B.-E. Khaledi and S. Kochar If (a) X~u! whr X~u '! whenever u u ', (3.12) (b) Q 1 and Q 2 are ordered in the univariate hazard rate order; that is, Q 1 hr Q 2, (3.13)! (c) r X~u! ~x 1,+++,x n! is decreasing in x j,j,+++,n, i 1,+++,n, then Y uo-disp 1 Y 2 + (3.14) Proof: Hu et proved that assumptions ~a! and ~b! imply that Y 1 whr Y 2 + Now, we show that for i 1,+++,n, r! Y1 ~x 1,+++,x n! is decreasing in x j, j 1,+++,n, then the required result will follow from Theorem 3+6~a!+ Assumption ~a! is equivalent to F u ~x! being TP 2 ~recall from that a function f : R 2 r R is said to be totally positive of order 2 ~TP 2! if f ~x 1, y 1! f ~x 2, y 2! f ~x 1, y 2! f ~x 2, y 1!,! for x 2 x 1 and y 2 y 1! in ~u, x j!, j 1,+++,n+ r X~u! ~x 1,+++,x n! decreasing in x j, j 1,+++,n, j i is equivalent to F u ~x 1,+++,x n! being TP 2 in ~x i, x j!, i, j 1,+++,n, j i+ Using these observations, it follows that G 1 ~x! is TP 2 in ~x i, x j!, i, j 1,+++,n ~cf+ which is equivalent to r! Y1 ~x 1,+++,x n! decreasing in! x j, j 1,+++,n, j i+ It is worth noting that r X~u! ~x 1,+++,x n! decreasing in x i is equivalent to the fact that $X i 6 ji $X j x j %% is a DFR random variable, i 1,+++,n+ Now, G 1 ~x! can be written as G 1 ~x! P u X i x i $X j x j % ji P uji $X j x j %, j 1,+++,n, j i dh 1~u!; that is, ~Y 1,+++,Y n! is a sort of mixture of DFR random variables; therefore, ] log G j ~x!0]x i is decreasing in x i, which is equivalent to r! Y1 ~x 1,+++,x n! decreasing in x i + This completes the proof+ 4. EXAMPLES AND APPLICATINS Example 4.1 (Multivariate Pareto Distributions): For a 0, let X a ~X a,1,+++,x a, n! have the survival function F a given by F a ~x 1,+++,x n! ( n i x i n 1a, x i 1, i 1,2,+++,n; see, for example, Kotz, Balakrishnan, and p+ 600#+ The corresponding density function is given by f a ~x 1,+++,x n! a~a 1!+++~a n 1!( x i n 1an, n i x i 1, i 1,2,+++,n+

18 DEPENDENCE, DISPERSIVENESS, MULTIVARIATE HAZARD RATE RDERING 443 Hu et showed that X a1 whr X a2 whenever a 1 a 2 + n the other hand, r! n X ~x! a0~( j x j n 1!, is decreasing in x j, j 1,+++,n+ Then from Theorem 3+6~a! it follows that X uo-disp a1 X a2 whenever a 1 a 2 + Example 4.2 (Bivariate Farlie Gumbel Morgenstern Distributions): For a ~, 1!, let X a ~X a,1, X a,2! have the survival function F a given by F a ~x 1, x 2! F 1 ~x 1! F 2 ~x 2!@1 a~1 F 1 ~x 1!!~1 F 2 ~x 1!!# and Y a ~Y a,1,y a,2! have the survival function G a given by G a ~x 1, x 2! G 1 ~x 1! G 2 ~x 2!@1 a~1 G 1 ~x 1!!~1 G 2 ~x 1!!#, where F 1, F 2, G 1, and G 2 are arbitrary univariate survival functions ~which happen to be the marginal survival functions of X a,1, X a,2, Y a,1, and Y a,2, respectively, independently of a!+ Assume that X a, i disp Y a, i, i 1,2+ It is easy to see that C X a ~u,v! C Y a~u,v!+ Then, from Theorem 2+1, it follows that uo-disp Xa Y a + Example 4.3 (Multivariate Gumbel Exponential Distributions): For positive parameters l $l I : I $1,2,+++,n%, I %, let X l ~X 1, X 2,+++,X n! have the survival function F l given by F l ~x 1, x 2,+++,x n! exp ( I l I ) x i, ~x 1, x 2,+++,x n! ~0,0,+++,0!; ii see Kotz et p+ 406#+ For another set of positive parameters l * $l I* : I $1,2,+++,n%, I %, let Y l * ~Y 1,Y 2,+++,Y n! have the survival function G l *+ Let X i st Y i, i 1,+++,n; that is, l i l i* + We show that if l l *, then for i 1,+++,n, X $X j F j ~u j!% disp iji Y $Y j G j ~u j!% iji, u (4.1) Since X i st Y i, i 1,+++,n, ~4+1! is equivalent to X $X j x j % disp iji $Y j x j %, iji i 1,+++,n+ (4.2) Let x i ~x 1,+++,x i, x i,+++,n! + The survival function of ~X i 6 ji $X j x j %!, denoted by F i ~x i ;x i!, is F i ~x i ;x i! exp x i l i ( j l ij x j ( jk l ijk x j x k {{{ l 12+++n ) x j + (4.3) ji Similarly, the survival function of $Y i 6Y j x j, j i %, denoted by G i ~x i ;x i!, is G i ~x i ;x i! exp x i l i ( j l ij * x j ( jk l * ijk x j x k {{{ l 12+++n * ) ji x j + (4.4)

19 444 B.-E. Khaledi and S. Kochar Now, the ratio G i ~x i ;x i! F i ~x i ;x i! (4.5) is increasing in x i ; that is, X $X j x j % hr iji Y iji $Y j, x j %, i 1,+++,n+ (4.6) n the other hand, the random variable $X i 6X j x j, j i % has an exponential distribution that is DFR+ Combining this observation with ~4+6!, it follows from Theorem 3+1~a! that ~4+2! holds+ Now, applying Theorem 3+3 to this example, we get that C l i, j is decreasing in l, where C l i, j denotes the copula of ~X i, X j! Application 4.1 (rder Statistics): Let X 1,+++,X n ~Y 1,+++,Y n! be a random sample from a univariate distribution with strictly increasing distribution function F ~G!+ has shown that F disp G implies X i:n disp Y i:n, i,+++,n, where X i:n ~Y i:n! is the ith-order statistic of the X sample ~Y sample!+ We will strengthen this result to prove that F disp G implies ~X 1:n,+++,X n:n! uo-disp ~Y 1:n,+++,Y n:n!+ (4.7) We first show that in the case of random samples from continuous distributions, the copulas of order statistics are independent of the parent distributions+ Note that Y i:n st G F~X i:n!, i,+++n+ Since the function G F is strictly increasing, it follows from Theorem of that C~X i:n, X j:n! C~Y i:n,y j:n!, for i, j 1,+++,n+ It now immediately follows from Theorem 2+1 that F disp G implies ~4+7!+ Since the order statistics from a random sample are positively associated ~cf+ Boland, Hollander, Joag-Dev, and since ~X 1:n,+++,X n:n! and ~Y 1:n,+++, Y n:n! have the same copula, the conditions of Theorem 3+4 are satisfied+ Hence, for i, j $1,+++,n%, Cov~h 1 ~X i:n!, h 2 ~X j:n!! Cov~h 1 ~Y i:n!, h 2 ~Y j:n!!, for all increasing convex functions h 1 and h 2 for which the above covariances exist+ This result was originally proved by using a different method+ A similar result can be established for record values+ Application 4.2 (Record Values): Let X 1,+++,X n,+++ ~Y 1,+++,Y n,+++! be a sequence of random variables from a univariate distribution F ~G!+ It is known that F disp G implies R mx disp R my, where R mx ~R my! is the mth record value of the X sequence ~Y sequence!+ We first show that in the case of random sequences from continuous distributions, the copulas of record values are independent of the parent distributions+ Then it will immediately follow from Theorem 2+1 that F disp G implies

20 DEPENDENCE, DISPERSIVENESS, MULTIVARIATE HAZARD RATE RDERING 445 ~R X m1,+++,r X mn! uo-disp ~R Y m1,+++,r Y mn!+ (4.8) Let M~x! log F~x!; then the distribution function of R mx can be expressed as F Rm X~x! G m ~M~x!!, where G m ~x! is the distribution function of a Gamma random variable with scale parameter one and shape parameter m+ Similarly, let N~x! log G~x!; then the distribution function of R my is F Y Rm ~x! G m ~N~x!!+ Now, both M~X 1!,+++,M~X n!,+++ and N~Y 1!,+++,N~Y n!,+++ are sequences of independent and identically distributed +i+d+! exponential random variables with mean one+ Using this observation, it follows that and ~R X m1,+++,r X mn! st ~M ~R * m1!,+++,m ~R * mn!! ~R Y m1,+++,r Y mn! st ~N ~R * m1!,+++,n ~R * mn!!, where R m* is the mth record value of a sequence of i+i+d+ exponential random variables with mean one+ C RX ~u 1,+++,u n! F R X~F X Rm1 ~u 1!,+++,F X Rmn ~u n!! P~R X m1 F X Rm1 ~u 1!,+++,R X mn F X Rmn ~u n!! P~M ~R * m1! M G m 1 ~u 1!,+++,M ~R * mn! M G mn ~u n!! * * P~R m1 G m1 ~u 1!,+++,R mn G mn ~u n!! P~N ~R * m1! N G m1 ~u 1!,+++,N ~R * mn! N G mn ~u n!! P~R Y m1 F Y Rm1 C RY ~u 1,+++,u n!+ This proves the desired result+ Acknowledgments ~u 1!,+++,R Y mn F Y Rmn ~u n!! Research by the first author was partially supported by the Statistical Research Center, Tehran, Iran+ The authors are grateful to the referees for their helpful comments and suggestions, which have greatly improved the presentation of the article+ References 1+ Bagai, I+ & Kochar, S+C+ ~1986!+ n tail ordering and comparison of failure rates+ Communications in Statistics Theory and Methods 15: Bartoszewicz, J+ ~1985!+ Moment inequalities for order statistics from ordered families of distributions+ Metrika 32:

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