On usual multivariate stochastic ordering of order statistics from heterogeneous beta variables

Journal of Multivariate Analysis 127 (2014) 147–150

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Journal of Multivariate Analysis journal homepage: www.elsevier.com/locate/jmva

On usual multivariate stochastic ordering of order statistics from heterogeneous beta variables Narayanaswamy Balakrishnan a,c,∗ , Ghobad Barmalzan b , Abedin Haidari b a

Department of Mathematics and Statistics, McMaster University, Hamilton, Canada

b

Department of Statistics, Zabol University, Sistan and Baluchestan, Iran

c

Department of Statistics, King Abdulaziz University, Jeddah, Saudi Arabia

article

info

Article history: Received 15 July 2013 Available online 1 March 2014 AMS subject classifications: 60E15 62E10

abstract Let Xi ∼ beta(αi , 1) and Yi ∼ beta(γi , 1), i = 1, 2, be all independent. We show that m

(α1 , α2 ) ≽ (γ1 , γ2 ) implies (Y1:2 , Y2:2 ) ≥st (X1:2 , X2:2 ). We then extend this result to the general case of the proportional reversed hazard rates (PRHR) model. © 2014 Elsevier Inc. All rights reserved.

Keywords: Usual multivariate stochastic order Order statistics Proportional reversed hazard rates model Beta distribution Exponentiated Weibull distribution

1. Introduction Order statistics have been studied quite extensively in the literature due to the key role they play in many areas of statistics. They especially play a special role in reliability theory wherein they correspond to k-out-of-n systems. Such a system, consisting of n components, works as long as at least k components work. Let X1 , . . . , Xn denote the lifetimes of the components and X1:n ≤ · · · ≤ Xn:n be the corresponding order statistics. Then, Xn−k+1:n corresponds to the lifetime of such a k-out-of-n system. So, many properties of k-out-of-n systems have been established in the literature by using the theory of order statistics; see [1] for some recent results in this direction. Interested readers may refer to Balakrishnan and Rao [2,3] for elaborate discussions on theory and applications of order statistics. Let X1 , . . . , Xn denote the lifetimes of n components of a system with distribution functions F1 , . . . , Fn , respectively. Then, X1 , . . . , Xn are said to follow the PRHR model if there exist positive constants α1 , . . . , αn and a distribution function F (x) with corresponding density function f (x) such that Fi (x) = F αi (x) for i = 1, . . . , n. In this case, F (x) and r˜ (x) = f (x)/F (x) are called the baseline distribution and baseline reversed hazard functions, respectively, and α1 , . . . , αn are the proportional reversed hazard rate parameters. Distributions such as power, generalized exponential and exponentiated Weibull are all special cases of this model. One may refer to Chapter 7 of Marshall and Olkin [5] for a discussion on this model. We now review briefly some common notions of stochastic orders and majorization order. Throughout, the terms increasing and decreasing are used for non-decreasing and non-increasing, respectively. Let X and Y be two random variables ¯ = 1 − G, and density functions f and g, respectively. with distribution functions F and G, survival functions F¯ = 1 − F and G

∗

Corresponding author at: Department of Mathematics and Statistics, McMaster University, Hamilton, Canada. E-mail addresses: [email protected], [email protected] (N. Balakrishnan), [email protected] (G. Barmalzan), [email protected] (A. Haidari). http://dx.doi.org/10.1016/j.jmva.2014.02.008 0047-259X/© 2014 Elsevier Inc. All rights reserved.

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¯ (x). A multivariate version of X is said to be larger than Y in the usual stochastic order (denoted by X ≥st Y ) if F¯ (x) ≥ G the usual stochastic order is as follows. Let X = (X1 , . . . , Xn ) and Y = (Y1 , . . . , Yn ) be two random vectors. Then, X is said to be larger than Y in the usual multivariate stochastic order (denoted by X ≥st Y) if E [φ(X)] ≥ E [φ(Y)] for all increasing functions φ : Rn → R. It is easy to see that the multivariate stochastic ordering implies componentwise usual stochastic ordering. Interested readers may refer to Müller and Stoyan [9] and Shaked and Shanthikumar [11] for detailed discussions on univariate and multivariate stochastic orderings. For two vectors x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ), let {x(1) , . . . , x(n) } and {y(1) , . . . , y(n) } denote the increasing m

arrangements of their components, respectively. A vector x is said to majorize another vector y (written as x ≥ y) if i i n n n j=1 x(j) ≤ j=1 y(j) for i = 1, . . . , n − 1, and j=1 x(j) = j=1 y(j) . A real-valued function φ defined on a set A ⊆ R is m

said to be Schur-convex (Schur-concave) on A if x ≽ y implies φ(x) ≥ (≤)φ(y ) for any x, y ∈ A. One may refer to Marshall et al. [6] for a detailed discussion on majorization and Schur-type functions. In this work, we obtain some new results about stochastic comparison of vectors of order statistics. Specifically, taking Xi ∼ beta(αi , 1) and Yi ∼ beta(γi , 1), i = 1, 2, all being independent, we prove that

    m (α1 , α2 ) ≽ (γ1 , γ2 ) H⇒ Y1:2 , Y2:2 ≥st X1:2 , X2:2 .

(1.1)

We further extend the result in (1.1) to the general PRHR model. It is useful to mention here that the beta distributions considered here are also special cases of the Kumaraswamy distributions [4]. 2. Main results In this section, we consider stochastic comparison of vectors of order statistics in the PRHR model. In this case, the beta distribution mentioned above is the simplest element of the PRHR model. Also, some other known distributions belonging to the PRHR model can be derived by a simple transform on the beta distribution. For this reason, we first focus on this case and present a result concerning some properties of order statistics arising from heterogeneous beta random variables. The following theorem gives necessary and sufficient conditions for characterizing Schur-convex and Schur-concave functions. Theorem 2.1 ([6, p. 84]). Let I ⊂ R be an open interval and φ : I n → R be continuously differentiable. Then, the necessary and sufficient conditions for φ to be Schur-convex on I n are φ is symmetric on I n and for all i ̸= j,

 ∂φ ∂φ (zi − zj ) (z) − (z) ≥ 0 for all z ∈ I n , ∂ zi ∂ zj 

∂φ

where ∂ z (z) denotes the partial derivative of φ with respect to its ith argument. Function φ is Schur-concave if and only if it is i symmetric and the reversed inequality sign holds in the above inequality. Lemma 2.1. Let Xi ∼ beta(αi , 1), i = 1, 2, be independent. Then, (i) X2:2 ∼ beta(α1 + α2 , 1); (ii) X1:2 /X2:2 and X2:2 are independent; (iii) the distribution function of X1:2 /X2:2 is Schur-convex in (α1 , α2 ). Proof.

(i) The distribution function of X2:2 , for x ∈ (0, 1), is FX2:2 (x) = FX1 (x) FX2 (x) = xα1 +α2 ,

x ∈ (0, 1),

and so Part (i) is immediate. (ii) The joint density function of (X1:2 , X2:2 ) is given by f (x1 , x2 ) = fX1 (x1 )fX2 (x2 ) + fX1 (x2 )fX2 (x1 )

  α −1 α −1 α −1 α −1 = α1 α2 x1 1 x2 2 + x1 2 x2 1 I(0,1) (x1 ) I(x1 ,1) (x2 ), where IA (x) = 1 if and only if x ∈ A. Let U1 = X1:2 /X2:2 and U2 = X2:2 . Now, to prove the required result, we must show that fU1 ,U2 (x1 , x2 ) = fU1 (x2 ) fU2 (x2 )

for all x1 , x2 ,

(2.2)

where fU1 ,U2 (., .) is the joint density function of (U1 , U2 ), and fUi (.) are the density functions of Ui , i = 1, 2. It is easy to see that fU1 ,U2 (x1 , x2 ) = x2 f (x1 x2 , x2 )

 =

    α1 α2  α1 −1 α 2 −1 α +α −1 x1 + x1 I(0,1) (x1 ) (α1 + α2 ) x2 1 2 I(0,1) (x2 ) α1 + α2

= fU1 (x1 ) fU2 (x2 ), where the last equality follows from Part (i). Thus, Part (ii) follows.

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2

(iii) For fixed 0 < x < 1, let us define the symmetrical differentiable function φ : R+ → R+ as follows:

φ(α1 , α2 ) = α2 xα1 + α1 xα2 . Under this setting and according to Part (ii), the distribution function of U1 , for 0 < x < 1, can be expressed as FU1 (x) =

1

α1 + α2

φ(α1 , α2 ).

Since the function α1 + α2 is Schur-constant, in order to obtain the desired result, it is sufficient to show that φ(α1 , α2 ) is Schur-convex in (α1 , α2 ). The partial derivatives of φ(α1 , α2 ) with respect to α1 and α2 are

∂φ(α1 , α2 ) = xα2 + α2 xα1 ln x, ∂α1

∂φ(α1 , α2 ) = xα1 + α1 xα2 ln x. ∂α2

Hence, we have

∂φ(α1 , α2 ) ∂φ(α1 , α2 ) − = (1 − α1 ln x) xα2 + (α2 ln x − 1) xα1 . ∂α1 ∂α2 Now, let us assume that α1 > α2 . Then, xα2 > xα1 , which along with the fact that ln x < 0, yields   ∂φ(α1 , α2 ) ∂φ(α1 , α2 ) − > xα1 (1 − α1 ln x) + (α2 ln x − 1) > 0. ∂α1 ∂α2 For the case α2 > α1 , the above inequality is reversed. Thus, we have  ∂φ(α , α ) ∂φ(α , α )  1 2 1 2 − (α1 − α2 ) > 0, ∂α1 ∂α2 which, according to Theorem 2.1, implies that φ(α1 , α2 ) is Schur-convex in (α1 , α2 ). Hence, the lemma.



Theorem 2.2. Let X = (X1 , . . . , Xn ) be a set of independent random variables, and Y = (Y1 , . . . , Yn ) be another set of independent random variables. If Xi ≥st Yi for i = 1, . . . , n, then X ≥st Y. Proof. This result is a special case of Theorem 6.B.16(b) of Shaked and Shanthikumar [11].



Theorem 2.3 ([11, p. 273]). Let X and Y be two n-dimensional random vectors. If X ≥st Y and h : Rn → Rk is any k-dimensional increasing function, then for any positive integer k, the k-dimensional vectors h(X) and h(Y) satisfy the multivariate ordering h(X) ≥st h(X). A sufficient condition for the usual multivariate stochastic ordering between vectors of order statistics from two sets of heterogeneous beta random variables is established in the following theorem. Theorem 2.4. Let Xi ∼ beta(αi , 1) and Yi ∼ beta(γi , 1), i = 1, 2, be all independent. Then,

    m (α1 , α2 ) ≽ (γ1 , γ2 ) H⇒ Y1:2 , Y2:2 ≥st X1:2 , X2:2 . Proof. According to Parts (i) and (iii) of Lemma 2.1, and under the majorization order between the two parameter vectors, it follows that Y1:2 Y2:2

≥st

X1:2 X2:2

and

st

Y2:2 = X2:2 .

So, based on Theorem 2.2 and Part (ii) of Lemma 2.1, we obtain

Y

1:2

Y2:2

 X  1:2 , Y2:2 ≥st , X2:2 .

(2.3)

X2:2

Now, let us consider the function h(x1 , x2 ) = (x1 x2 , x2 ) for 0 < x1 , x2 < 1. It is clear that h is a 2-dimensional increasing function. Therefore, from Theorem 2.3 and the ordering in (2.3), we obtain



Y1:2 , Y2:2

as required.



Y  X    1:2 1:2 =h , Y2:2 ≥st h , X2:2 = X1:2 , X2:2 , Y2:2

X2:2



In the following theorem, we extend Theorem 2.4 to the PRHR model.

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Theorem 2.5. Let Xi ∼ F αi and Yi ∼ F γi , i = 1, 2, be all independent. Then,

    m (α1 , α2 ) ≽ (γ1 , γ2 ) H⇒ Y1:2 , Y2:2 ≥st X1:2 , X2:2 . Proof. Let Xi′ ∼ beta(αi , 1) and Yi′ ∼ beta(γi , 1), i = 1, 2, be all independent. Then, according to Theorem 2.4, we have

    m (α1 , α2 ) ≽ (γ1 , γ2 ) H⇒ Y1′ :2 , Y2′ :2 ≥st X1′ :2 , X2′ :2 . Since the function K (x) = F (x) is increasing in x, K −1 (x) is also an increasing function, and so from Theorem 2.3, we obtain

    m (α1 , α2 ) ≽ (γ1 , γ2 ) H⇒ K −1 (Y1′ :2 ), K −1 (Y2′ :2 ) ≥st K −1 (X1′ :2 ), K −1 (X2′ :2 ) ,       st st and the required result follows immediately from the fact that Y1:2 , Y2:2 = K −1 (Y1′ :2 ), K −1 (Y2′ :2 ) and X1:2 , X2:2 =   K −1 (X1′ :2 ), K −1 (X2′ :2 ) .  From Theorems 2.3 and 2.5, we get the following result on the comparison of convolutions of independent heterogeneous PRHR variables with respect to the usual stochastic ordering. Corollary 2.1. Let Xi ∼ F αi and Yi ∼ F γi , i = 1, 2, be all independent. Then, m

(α1 , α2 ) ≽ (γ1 , γ2 ) H⇒ Y1 + Y2 ≥st X1 + X2 .  The PRHR α model includes the exponentiated Weibull distribution with cumulative distribution function F (x; α, β) = −x β 1−e , x > 0, α > 0, β > 0, denoted by X ∼ EW (α, β); see [8]. This distribution, in addition to including exponential, Weibull, generalized exponential, Rayleigh and Burr type X distributions all as special cases, has its hazard rate admitting increasing, decreasing, bathtub and upside-down bathtub shapes for different choices of α and β ; see Mudholkar and Hutson [7]. The following corollary is a direct consequence of Theorem 2.5 with regard to this exponentiated Weibull distribution. Corollary 2.2. Let Xi ∼ EW (αi , β) and Yi ∼ EW (γi , β), i = 1, 2, be all independent. Then, for any β > 0, we have

    m (α1 , α2 ) ≽ (γ1 , γ2 ) H⇒ Y1:2 , Y2:2 ≥st X1:2 , X2:2 . Navarro and Lai [10] have studied some ordering properties of marginal order statistics in systems with two dependent components. Thus, a generalization of the present work to the case of systems with dependent components will be of interest. We are currently looking into this problem and hope to report the findings in a future paper, Acknowledgments Our sincere thanks go to the Associate Editor and two anonymous reviewers for their useful comments and suggestions on an earlier version of this manuscript which led to this improved one. References [1] N. Balakrishnan, G. Barmalzan, A. Haidari, Stochastic orderings and ageing properties of residual lifelengths of live components in (n − k + 1)-out-of-n systems, J. Appl. Probab. (2014) in press. [2] N. Balakrishnan, C.R. Rao (Eds.), Handbook of Statistics 16—Order Statistics: Theory and Methods, North-Holland, Amsterdam, 1998. [3] N. Balakrishnan, C.R. Rao (Eds.), Handbook of Statistics 17—Order Statistics: Applications, North-Holland, Amsterdam, 1998. [4] P. Kumaraswamy, A generalized probability density function for double-bounded random processes, J. Hydrol. 46 (1980) 79–88. [5] A.W. Marshall, I. Olkin, Life Distributions, Springer, New York, 2007. [6] A.W. Marshall, I. Olkin, B.C. Arnold, Inequalities: Theory of Majorization and its Applications, Springer, New York, 2011. [7] G.S. Mudholkar, A.D. Hutson, The exponentiated Weibull family: some properties and a flood data application, Commun. Stat.—Theory Methods 25 (1996) 3059–3083. [8] G.S. Mudholkar, D.K. Srivastava, Exponential Weibull family for analyzing bathtub failure rate data, IEEE Trans. Reliab. 42 (1993) 299–302. [9] A. Müller, D. Stoyan, Comparison Methods for Stochastic Models and Risks, John Wiley & Sons, New York, 2002. [10] J. Navarro, C-D. Lai, Ordering properties of systems with two dependent components, Commun. Stat.—Theory Methods 36 (2007) 645–655. [11] M. Shaked, J.G. Shanthikumar, Stochastic Orders, Springer, New York, 2007.