Likelihood ratio order of sample minimum from heterogeneous Weibull random variables

Statistics and Probability Letters 97 (2015) 46–53

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Likelihood ratio order of sample minimum from heterogeneous Weibull random variables Chen Li a , Xiaohu Li b,∗ a

School of Mathematical Sciences, Xiamen University, Xiamen Fujian 361005, China

b

Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ 07030, USA

article

info

Article history: Received 21 June 2014 Received in revised form 29 October 2014 Accepted 31 October 2014 Available online 13 November 2014

abstract For heterogeneous Weibull samples with a common shape parameter and weakly majorized scale parameters, we study the likelihood ratio order and the stochastic order between minimums of independent and dependent samples, respectively. Also, an application in fabric strength is presented. © 2014 Elsevier B.V. All rights reserved.

Keywords: Archimedean copula Log-convex Majorization Schur-convex Weakest link

1. Introduction Order statistics and sample spacings are rather useful in reliability theory, life testing, auction theory and other related areas. For example, the kth order statistic of a sample of size n characterizes the lifetime of a (n − k + 1)-out-of-n system in reliability, and the sample minimum represents the winner’s price in the first-price Dutch auction. Ever since (Pledger and Proschan, 1971) first compared order statistics from heterogeneous exponential samples, many researchers followed them to consider order statistics from various samples, for example Weibull samples, proportional hazard samples, gamma samples etc. For more details, one may refer to Navarro and Lai (2007), Zhao et al. (2009), Mao and Hu (2010), Balakrishnan and Zhao (2013) and references therein. As for stochastic comparisons of sample spacings, Pledger and Proschan (1971), Kochar and Korwar (1996), Kochar and Rojo (1996) and Kochar and Xu (2011) studied independent heterogeneous and homogeneous exponential samples, and Torrado and Lillo (2013) considered two independent heterogeneous exponential samples. For more on this line of research, we refer readers to Kochar (2012) and references therein. We will handle stochastic comparisons of sample minimums from independent heterogeneous and homogeneous Weibull samples and those from two independent heterogeneous Weibull samples in this paper. Also this paper will study sample minimums of Weibull random variables with a common Archimedean survival copula based on the log-convex or log-concave generator. Since the sample minimum of nonnegative random variables is just the first sample spacing, all results to be presented in this note also extend the corresponding literature on stochastic comparison of the first sample spacings. α The Weibull distribution W (α, λ) has the probability density f (x; α, λ) = α xα−1 λα e−(λx) for x > 0, α > 0 and λ > 0. Denote X1:n the smallest order statistic of X1 , . . . , Xn . For Xi ∼ W (α, λi ), i = 1, . . . , n and Yi ∼ W (α, µi ), i = 1, . . . , n, both mutually independent, it is of both theoretical and practical interest to investigate how the majorization of scale parameters

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (X. Li).

http://dx.doi.org/10.1016/j.spl.2014.10.019 0167-7152/© 2014 Elsevier B.V. All rights reserved.

C. Li, X. Li / Statistics and Probability Letters 97 (2015) 46–53

47

λ = (λ1 , . . . , λn ) and µ = (µ1 , . . . , µn ) impacts the distribution of the sample minimums. Khaledi and Kochar (2006, Theorem 2.3) firstly proved that m

λ ≼ µ H⇒



Y1:n ≤hr X1:n ,

for α ≥ 1,

X1:n ≤hr Y1:n ,

for α ≤ 1,

(1.1)

m

where ‘≼’ and ‘≤hr ’ respectively denote the majorization order and the hazard rate order (see Section 2 for their definitions). Lately, Fang and Zhang (2013, Theorem 3.2) proved that m

λ ≼ µ H⇒



Y1:n ≤rh X1:n ,

for α ≥ 1,

X1:n ≤rh Y1:n ,

for α ≤ 1,

(1.2)

where ‘≤rh ’ denotes the reversed hazard rate order (see Section 2 for the definition). Further suppose mutually independent Zi ∼ W (α, λ), i = 1, . . . , n. Fang and Zhang (2013, Theorem 3.1) also showed that

λn =

n 

λk H⇒ X1:n ≤rh Z1:n ,

(1.3)

k=1

λα ≥

n 1 α λk H⇒ Z1:n ≤disp X1:n , n k=1

(1.4)

where ‘≤disp ’ denotes the dispersive order (see Section 2 for the definition). Recently, for Xi ∼ W (α, λi ), i = 1, . . . , n and Yi ∼ W (α, µi ), i = 1, . . . , n, both mutually independent, Torrado (in press, Theorem 3.1) further showed that X1:n ≤hr Y1:n is equivalent to X1:n ≤lr Y1:n , here ‘≤lr ’ denotes the likelihood ratio order (see Section 2 for the definition), and thus Torrado (in press, Theorem 3.2) pointed out that (1.1) and (1.2) can be strengthened to m

λ ≼ µ H⇒



Y1:n ≤lr X1:n ,

for α ≥ 1,

X1:n ≤lr Y1:n ,

for α ≤ 1.

(1.5)

This paper is devoted to developing criteria to verify the existence of the likelihood ratio order between sample minimums in the context that the majorization in (1.5) is relaxed to the weak majorizations. Moreover, the dispersive order in (1.4) is also generalized to samples sharing a common Archimedean survival copula. The rest of this paper is organized as follows: As a preliminary Section 2 recalls related stochastic orders of random variables, majorization of real vectors and two weak versions, and some useful lemmas to be used in the sequel. In Section 3, we obtain the likelihood ratio order between two sample minimums of independent Weibull samples with a common shape parameter and weakly majorized scale parameters. Section 4 obtains sufficient conditions for both the stochastic order and the dispersive order on sample minimums of Weibull samples sharing a common Archimedean survival copula. Finally, for the likelihood ratio order between sample minimums we present in Section 5 an application in the blended fibre strength. Throughout this note, for convenience, we use the notations R = (−∞, +∞), R+ = [0, +∞) and R++ = (0, +∞), ¯ = 1 − G survival functions of distribution functions F and G, respectively, and the terms we also denote F¯ = 1 − F and G increasing and decreasing mean nondecreasing and nonincreasing, respectively. 2. Preliminaries For ease of reference, we recall some related concepts and useful lemmas that play a role in developing our main theories in the sequel. For two random variables X and Y with distribution functions F and G, density functions f and g, respectively, denote F −1 and G−1 the right continuous inverses1 of F and G, X is said to be smaller than Y in the (i) likelihood ratio order (denoted as ¯ (t )/F¯ (t ) is increasing in t, (iii) reversed hazard X ≤lr Y ) if g (t )/f (t ) increases in t, (ii) hazard rate order (denoted as X ≤hr Y ) if G ¯ (t ) rate order (denoted as X ≤rh Y ) if G(t )/F (t ) is increasing in t, (iv) usual stochastic order (denoted as X ≤st Y ) if F¯ (t ) ≤ G for all t, and (v) dispersive order (denoted as X ≤disp Y ) if F −1 (β) − F −1 (α) ≤ G−1 (β) − G−1 (α) for all 0 < α ≤ β < 1. For more on stochastic orders please refer to Shaked and Shanthikumar (2007) and Li and Li (2013). Let x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) be two real vectors, denote x(1) ≤ · · · ≤ x(n) the increasing arrangement of x1 , . . . , xn . x is said to be

n n i=j x(i) ≤ i=j y(i) for all j = 1, . . . , n, j  j (ii) weakly supermajorized by y (denoted as x ≼w y) if i=1 x(i) ≥ i=1 y(i) for all j = 1, . . . , n, m n n j j (iii) majorized by y (denoted as x ≼ y) if i=1 xi = i=1 x(i) ≥ i=1 y(i) for all j = 1, . . . , n − 1. i=1 yi and (i) weakly submajorized by y (denoted as x ≼w y) if

1 The right continuous inverse of an increasing function h is defined as h−1 (u) = sup{x ∈ R : h(x) ≤ u}.

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C. Li, X. Li / Statistics and Probability Letters 97 (2015) 46–53 p

n Also in R+ , y is said to be p-larger than x (denoted as x ≼ y) if p

m

x ≼ y ⇐H x ≼w y ⇐H x ≼ y H⇒ x ≼w y ,

j

i=1

x(i) ≥

j

i=1

y(i) for j = 1, . . . , n. It is well-known that

n for x, y ∈ R+ .

Lemma 2.1 (Marshall et al.,2011, Theorem 5.A.2). If an increasing function g is convex (concave), then x ≼w (≼w ) y implies    w g (x1 ), . . . , g (xn ) ≼w (≼ ) g (y1 ), . . . , g (yn ) . m

A function ℓ defined on A ⊂ Rn is said to be Schur-convex (Schur-concave) on A if x ≼ y on A implies ℓ(x) ≤ (≥) ℓ(y ). For more details on the above partial orders of real vectors, Schur-convexity and Schur-concavity we refer readers to Marshall et al. (2011) and Bon and Pˇaltˇanea (1999). In the following we list some useful lemmas. Lemma 2.2 (Marshall et al., 2011, Theorem 3.A.4). For an open I  ⊂ R, a continuously differentiable ℓ : I n → R is  ∂ℓ(xinterval ∂ℓ(x) ) n Schur-convex if and only if it is symmetric on I and (xi − xj ) ∂ x − ∂ x ≥ 0 for all i ̸= j and x ∈ I n . i

j

Lemma 2.3 (Marshall et al., 2011, Proposition 3.C.1). If I ⊂ R is an interval and g : I → R is convex, then ℓ(x) = is Schur-convex on I n .

n

i=1

g (xi )

Lemma 2.4 (Marshall et al., 2011, Theorem 3.A.8). For a function ℓ on A ⊂ Rn , x ≼w (≼w ) y implies ℓ(x) ≤ ℓ(y ) if and only if it is increasing (decreasing) and Schur-convex on A. For a n-monotone2 (n ≥ 2) function ψ : [0, +∞) → [0, 1] with ψ(0) = 1 and limx→+∞ ψ(x) = 0, let φ = ψ −1 , the right continuous inverse,3 then C (u1 , . . . , un ) = ψ φ(u1 ) + · · · + φ(un ) ,





for all ui ∈ [0, 1], i = 1, . . . , n,

is called an Archimedean copula with generator ψ . Evidently, for such ψ and φ , the function J (λ, α; x) = ψ

n    α φ e−λk x ,

for x > 0 and α > 0,

(2.1)

k=1 n is symmetric with respect to λ ∈ R++ . For more on Archimedean copulas, readers may refer to Nelsen (2006) and McNeil and Nešlehová (2009). Next, let us present a lemma on J (λ, α; x), which will be used to complete the proofs in Section 4.

Lemma 2.5. J (λ, α; x) is decreasing in λi for i = 1, . . . , n, and the log-convex (log-concave) ψ implies that J (λ, α; x) is Schurconcave (Schur-convex) with respect to λ. Proof. Let η(λi , x) = by (2.1) we have

α e−λi x

α

ψ ′ (φ(e−λi x ))

for i = 1, . . . , n. Since ψ is n-monotone, it holds that ψ ′ (x) ≤ 0 for all x ≥ 0 and hence

n    ∂ J (λ, α; x) α = −xα η(λi , x)ψ ′ φ e−λk x ≤ 0, ∂λi k=1

for all x, α > 0.

That is, J (λ, α; x) is decreasing in λi for i = 1, . . . , n. Furthermore, for i ̸= j, n      ∂ J (λ, α; x) ∂ J (λ, α; x) α − = −xα η(λi , x) − η(λj , x) ψ ′ φ e−λk x . ∂λi ∂λj k=1

Note that, for i = 1, . . . , n,

   ′   −λ xα 2 ∂η(λi , x)        α α −1 α  α α 2 ψ φ e i = xα e−λi x ψ ′ φ e−λi x ψ (2) φ e−λi x e−λi x − ψ ′ φ e−λi x . ∂λi Since ψ is log-convex (log-concave), it holds that

 2 ψ (2) (x)ψ(x) − ψ ′ (x) ∂ 2 ln ψ(x) = ≥ (≤) 0, ∂ x2 ψ 2 (x)

for x ≥ 0.

2 A real function h is n-monotone on (a, b) ⊆ [−∞, +∞] if (−1)k h(k) (x) ≥ 0 for all x ∈ (a, b), k = 0, 1, . . . , n − 2, and (−1)n−2 h(n−2) is decreasing and convex in (a, b). 3 The right continuous inverse of a decreasing function h is defined as h−1 (u) = sup{x ∈ R : h(x) > u}.

C. Li, X. Li / Statistics and Probability Letters 97 (2015) 46–53

49

As a result, we have

          α  α α 2 α    α  α 2 ψ (2) φ e−λi x e−λi x − ψ ′ φ e−λi x = ψ (2) φ e−λi x ψ φ e−λi x − ψ ′ φ e−λi x ≥ (≤) 0, and thus, i ̸= j,

∂η(λi ,x) ∂λi

≤ (≥) 0. That is, η(λi , x) is decreasing (increasing) in λi for i = 1, . . . , n. Consequently, it holds that, for

 ∂ J (λ, α; x) ∂ J (λ, α; x) (λi − λj ) − ≤ 0, ∂λi ∂λj   ∂ J (λ, α; x) ∂ J (λ, α; x) (λi − λj ) − ≥ 0, ∂λi ∂λj 

for log-convex ψ , and for log-concave ψ.

Now, the desired results follow immediately from Lemma 2.2.



¯ where G¯ is strictly decreasing on For exchangeable (X1 , . . . , Xn ) with joint and marginal survival functions F¯ and G, [0, +∞] with G¯ (0) = 1 and G¯ (+∞) = 0, the multivariate ageing function  −1   ¯ F¯ (− ln u1 , . . . , − ln un ) , u ∈ [0, 1]n . BF¯ (u) = exp −G According to Bassan and Spizzichino (2005, Lemma 4.2) and Durante et al. (2010, Proposition 3.6(c)), F¯ is Schur-concave if and only if, for any pair {i, j} ⊂ {1, . . . , n}, BF¯ (u1 , . . . , sui , . . . , uj , . . . , un ) ≥ BF¯ (u1 , . . . , ui , . . . , suj , . . . , un ), Set F¯ (λ) = ψ

−λk xα

ui ≥ uj , s ∈ (0, 1). (2.2)  α n − tx ) = J (λ, α; x) for λ ∈ R++ and some x > 0, α > 0. Then G¯ (t ) = e and BF¯ (u) =

k=1 φ(e  n x−α xα ψ for t ≥ 0 and u ∈ [0, 1]n . As a consequence, the log-convexity of ψ implies (2.2) and hence J (λ, α; x) k=1 φ(uk ) is Schur-concave with respect to λ, and this coincides with the Schur-concavity in Lemma 2.5.

n

3. Mutually independent variables 3.1. Heterogeneous samples α Assume that both X1 , . . . , Xn and Y1 , . . . , Yn are mutually independent. For convenience, we denote ϕ(λ, α) = k =1 λ k n for α > 0 and λ ∈ R++ . Evidently, ϕ(λ, α) increases in λi for i = 1, . . . , n. It is easy to verify that xα is convex for α ≥ 1 and −xα is convex for α ∈ (0, 1]. According to Lemma 2.3, (i) ϕ(λ, α) is Schur-convex with respect to λ for α ≥ 1 and (ii) −ϕ(λ, α) is Schur-convex with respect to λ for α ≤ 1.

n

Theorem 3.1. Suppose both Xi ∼ W (α, λi ) for i = 1, . . . , n and Yi ∼ W (α, µi ) for i = 1, . . . , n are mutually independent. Then, (i) for α ≥ 1, λ ≼w µ implies X1:n ≥lr Y1:n , and (ii) for α ≤ 1, λ ≼w µ implies X1:n ≤lr Y1:n . α

α

Proof. X1:n has the distribution function Fn (x) = 1 − e−ϕ(λ,α)x and Y1:n has the distribution function Gn (x) = 1 − e−ϕ(µ,α)x , their respective density functions are α

fn (x) = α xα−1 ϕ(λ, α)e−ϕ(λ,α)x , Thus, h¯ n (x) =

fn (x) gn (x)

α

gn (x) = α xα−1 ϕ(µ, α)e−ϕ(µ,α)x ,

x > 0.

  ϕ(λ,α) = ϕ(µ,α) exp −[ϕ(λ, α) − ϕ(µ, α)]xα satisfies

  ∂ h¯ n (x) ϕ(λ, α) α−1 = [ϕ(µ, α) − ϕ(λ, α)] αx exp −[ϕ(λ, α) − ϕ(µ, α)]xα . ∂x ϕ(µ, α) (i) For α ≥ 1, ϕ(λ, α) is Schur-convex with respect to λ and increasing in λi for i = 1, . . . , n. According to Lemma 2.4, λ ≼w µ implies ϕ(λ, α) ≤ ϕ(µ, α). Then, it follows that ∂ h¯∂nx(x) ≥ 0 for all x > 0 and thus fn (x)/gn (x) increases in x > 0. That is, Y1:n ≤lr X1:n . (ii) For α ≤ 1, −ϕ(λ, α) is Schur-convex with respect to λ and decreasing in λi for i = 1, . . . , n. Again by Lemma 2.4, λ ≼w µ implies −ϕ(λ, α) ≤ −ϕ(µ, α). Then, ∂ h¯∂nx(x) ≤ 0 for all x > 0 and thus gn (x)/fn (x) increases in x > 0. That is, X1:n ≤lr Y1:n .



m

Note that (i) λ ≼ µ implies both λ ≼w µ and λ ≼w µ and (ii) X1:n ≤lr Y1:n implies both X1:n ≤hr Y1:n and X1:n ≤rh Y1:n , Theorem 3.1 substantially improves (1.1), (1.2) and (1.5). Naturally, one may wonder whether the following two statements p

are actually also true: (i) For α ≥ 1, λ ≼w µ gives rise to the likelihood ratio order between X1:n and Y1:n ; (ii) For α ≤ 1, λ ≼ µ or λ ≼w µ yields the likelihood ratio order between X1:n and Y1:n . Example 3.2 provides with negative answers to these two conjectures.

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C. Li, X. Li / Statistics and Probability Letters 97 (2015) 46–53

Example 3.2. Let both Xi ∼ W (α, λi ) (i = 1, 2) and Yi ∼ W (α, µi ) (i = 1, 2) be independent. (i) Set α = 2. Clearly, for λ = (1.5, 2.5) ≼w (1, 2) = µ,

(1.4, 2.1) = µ,

f2 (x) g2 (x)

f2 (x) g2 (x)

is decreasing in x > 0; however, for λ = (1.5, 2.01) ≼w

is increasing in x > 0. So, λ ≼w µ implies neither X1:n ≤lr Y1:n nor X1:n ≥lr Y1:n for α ≥ 1. p

(ii) Set α = 0.5. For λ = (1.1, 2.14) ≼(1, 2.25) = µ, p

f2 (x) g2 (x)

p

decreases in x > 0; however, for λ = (1, 2) ≼(0.6, 3) = µ,

f2 (x) g2 (x)

increases in x > 0. So, λ ≼ µ implies neither X1:n ≤lr Y1:n nor X1:n ≥lr Y1:n for α ≤ 1. f (x) (iii) Set α = 0.5. It is plain that λ = (1.18, 1.8) ≼w (1, 2) = µ and g2 (x) decreases in x > 0; however, for λ = (1, 1.5) ≼w

(1.2, 2) = µ,

f2 (x) g2 (x)

2

increases in x > 0. So, λ ≼w µ implies neither X1:n ≤lr Y1:n nor X1:n ≥lr Y1:n for α ≤ 1.



3.2. Heterogeneous and homogeneous samples For X1 , . . . , Xn and Z1 , . . . , Zn both mutually independent, we obtain the likelihood ratio order between sample minimums X1:n and Z1:n . Theorem 3.3. n and Zi ∼ W (α, λ) for i = 1, . . . , n are mutually independent. Then,  Suppose both Xi ∼ W (α, λi ) for i = 1, . . . , (i) λα ≥ 1n nk=1 λαk implies X1:n ≥lr Z1:n , and (ii) λα ≤ 1n nk=1 λαk implies X1:n ≤lr Z1:n . α

α xα

α−1 −ϕ(λ,α)x Proof. X1:n has the and Z1:n has the density hn (x) = α xα−1 nλα e−nλ  density fn (x) = αα x α  ϕ(λ, α)e ϕ(λ,α) fn (x) = nλα exp −[ϕ(λ, α) − nλ ]x satisfies h (x)

. Then, ℓn (x) =

n

 ϕ(λ, α)   ∂ℓn (x)  α 1 = λ − ϕ(λ, α) α xα−1 exp −[ϕ(λ, α) − nλα ]xα . ∂x n λα n (i) If λα ≥ 1n λα , then ∂ℓ∂nx(x) ≥ 0 for all x > 0 and hence fn (x)/hn (x) increases in x > 0. That is, Z1:n ≤lr X1:n . kn=1 kα ∂ℓn (x) 1 α (ii) If λ ≤ n ≤ 0 for all x > 0 and hence hn (x)/fn (x) increases in x > 0. That is, X1:n ≤lr Z1:n .  k=1 λk , then ∂x 1   1 1 n  1n  1 n  n n α α α n×α ≥ = , Theorem 3.3(ii) articulates that λn = k=1 λk implies X1:n ≤lr Due to n k=1 λk k=1 λk k=1 λk Z1:n . Since X1:n ≤lr Z1:n implies X1:n ≤rh Z1:n , Theorem 3.3(ii) serves as one improved version of (1.3). While both the weak majorization of scale parameters and the size of the common shape parameter play an individual role in deriving the likelihood ratio order of sample minimum in Theorem 3.1, Theorem 3.3 imposes no individual requirement on the shape parameter due to the homogeneous scale parameters of one sample. 4. Interdependent variables with Archimedean copulas Although in the literature most research on stochastic comparison on sample minimums assume the mutual independence among observations, recently some authors deal with the sample with dependent random variables. For example, Navarro and Spizzichino (2010) studied stochastic comparisons of series system with component lifetimes sharing a common copula. Here, we consider samples of Weibull random variables with a common Archimedean survival copula. 4.1. Heterogeneous samples For X1 , . . . , Xn and Y1 , . . . , Yn both sharing a common Archimedean survival copula, we verify the usual stochastic order between the sample minimums. Denote λα = (λα1 , . . . , λαn ). Theorem 4.1. Suppose Xi ∼ W (α, λi ) (i = 1, . . . , n) and Yi ∼ W (α, µi ) (i = 1, . . . , n) share a common Archimedean survival copula with generator ψ . Then, (i) Y1:n ≤st X1:n if λα ≼w µα and ψ is log-convex, and (ii) X1:n ≤st Y1:n if λα ≼w µα and ψ is log-concave. Proof. X1:n and Y1:n have their respective survival functions, for all x ≥ 0, F¯n (x) = P(Xk > x, 1 ≤ k ≤ n) = ψ

n n      α α  φ P(Xk > x) = ψ φ e−λk x = J (λα , α; x), k=1

k=1

n n      α α  ¯ n (x) = P(Yk > x, 1 ≤ k ≤ n) = ψ G φ P(Yk > x) = ψ φ e−µk x = J (µα , α; x). k=1

k=1

(i) If ψ is log-convex, due to Lemma 2.5, −J (λ, α; x) is Schur-convex with respect to λ and increasing in λi for i = 1, . . . , n. ¯ n (x) ≤ F¯n (x) for x ≥ 0. That According to Lemma 2.4, λα ≼w µα implies −J (λα , α; x) ≤ −J (µα , α; x). Then, it follows that G is, Y1:n ≤st X1:n . (ii) If ψ is log-concave, due to Lemma 2.5, J (λ, α; x) is Schur-convex with respect to λ and decreasing in λi for i = 1, . . . , n. ¯ n (x) for x ≥ 0. That is, Again due to Lemma 2.4, λα ≼w µα implies J (λα , α; x) ≤ J (µα , α; x). Then, it follows that F¯n (x) ≤ G X1:n ≤st Y1:n . 

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51

In accordance with (You and Li, 2014), for an Archimedean survival copula with generator ψ , log-convexity (logconcavity) of ψ , characterizing RTIS (RTDS),4 plays a role in verifying the stochastic orders in Theorem 4.1. For more details on log-convexity and log-concavity of generators of Archimedean copulas, one may refer to Bassan and Spizzichino (2005) and Müller and Scarsini (2005). The next example shows that the log-convexity and log-concavity of ψ in Theorem 4.1 cannot be dropped. Example 4.2. Assume Xi ∼ W (α, λi ) (i = 1, 2) and Yi ∼ W (α, µi ) (i = 1, 2) share a common Archimedean survival copula with generator ψ . (i) Let α = 4, λ = (1.4, 2.1) and µ = (1.19, 2.15), it is plain that λα ≼w µα . For φ(x) = (1 − ln x)0.1 − 1 with x ∈ [0, 1], we have ψ(x) = e1−(1+x)

10

∂ 2 ln ψ(x) ∂ x2

= −90(1 + x)8 < 0 tells that ψ is not log-convex. On the other ¯ 2 of sample minimums satisfy F¯2 (0.45) − G¯ 2 (0.45) = −0.0151 < 0, then X1:2 ̸≥st Y1:2 . hand, survival functions F¯2 and G (ii) Set α = 2, it is evident that λα = (0.82 , 1.22 ) ≼w (0.72 ,1.252 ) = µα . For φ(x) = − ln(1 − (1 − x)3 ) with x ∈ [0, 1],  2 1 x)  ψ(x) = 1 − (1 − e−x ) 3 for x ∈ [0, +∞) and ∂ ln∂ xψ( = 0.8717 > 0 invalidates the log-concavity of ψ . Also, 2  for x ∈ [0, +∞) and

x=0.5

¯ 2 of sample minimums satisfy G¯ 2 (0.6) − F¯2 (0.6) = −0.0166 < 0, i.e., X1:2 ̸≤st Y1:2 . survival functions F¯2 and G



As a direct consequence of Theorem 4.1, we present the following corollary. Corollary 4.3. Assume Xi ∼ W (α, λi ) (i = 1, . . . , n) and Yi ∼ W (α, µi ) (i = 1, . . . , n) share a common Archimedean survival copula with generator ψ . (i) For α ≥ 1, the convex ln ψ implies Y1:n ≤st X1:n if λ ≼w µ; (ii) For α ≤ 1, the concave ln ψ implies X1:n ≤st Y1:n if λ ≼w µ. Proof. Note that xα is increasing and convex for α ≥ 1 and is increasing and concave for α ∈ (0, 1]. According to Lemma 2.1, λ ≼w µ implies λα ≼w µα for α ≥ 1 and λ ≼w µ implies λα ≼w µα for α ∈ (0, 1]. Then, the desired result follows immediately from Theorem 4.1.



4.2. Heterogeneous and homogeneous samples For X1 , . . . , Xn and Z1 , . . . , Zn sharing a common Archimedean survival copula, we provide sufficient conditions for the dispersive order between sample minimums. Theorem 4.4. Suppose Xi ∼ W (α, λi ) (i = 1, . . . , n) and Zi ∼ W (α, λ) (i = 1, . . . , n) share a common Archimedean survival ψ ln ψ copula with generator ψ . Then, log-convex (log-concave) ψ and convex (concave) ψ ′ imply X1:n ≤disp (≥disp )Z1:n whenever

λα ≤ (≥) 1n

n

k=1

λαk .

α α

−λk x Proof. X1:n has distribution function Fn (x) = 1 − ψ k =1 φ e   −λα xα  ψ nφ e . Then their respective densities are, for x > 0,

n

fn (x) = α xα−1 ψ ′

n n   α α   φ e−λk x k=1

k=1





and Z1:n has distribution function Hn (x) = 1 −

α α

e−λk x λα   −λα kxα  , ψ′ φ e k α α

 ne−λ x λα    . hn (x) = α x ψ nφ e ψ ′ φ e−λα xα  n  −λα xα   n  −λα xα    k k Denote L(λ, α; x) = ψ 1n and L′ (λ, α; x) = ψ ′ 1n . Then, it holds that Hn−1 Fn (x) = k=1 φ e k =1 φ e  1 1 − ln L(λ, α; x) α and λ α−1

hn Hn−1 Fn (x)



(i) Since



ψ ln ψ ψ′



′





−λα xα

n  α α    α−1 L(λ, α; x) ′  = nαλ − ln L(λ, α; x) α ′ ψ φ e−λk x . L (λ, α; x) k=1

is convex, we have

L(λ, α; x) ln L(λ, α; x) L′ (λ, α; x)

  α α    α α  α α n n  ψ φ e−λk x ln ψ φ e−λk x 1  λαk xα e−λk x =− ≤   −λα xα    −λα xα  . n k=1 ψ ′ φ e k nψ ′ φ e k k=1

(4.1)

4 A random vector (X , . . . , X ) is right tail increasing (decreasing) in sequence, abbreviated as RTIS (RTDS) if, for i = 2, . . . , n, P(X > x | X > x , j = 1 n i i j j 1, . . . , i − 1) is increasing (decreasing) in xj , j = 1, . . . , i − 1. For more on RTIS and RTDS please refer to Ebrahimi and Ghosh (1981).

52

C. Li, X. Li / Statistics and Probability Letters 97 (2015) 46–53

ψ(x) ∈ [0, 1] implies − ln ψ(x) ≥ 0 for x ≥ 0. Log-convex ψ together with λα ≤

1 n

n

k=1

λαk implies

 1   α−1 − ln L(λ, α; x) = − ln L(λ, α; x) α − ln L(λ, α; x) α n  1   α α  α1   α−1 ≥ − ln ψ φ e−λk x − ln L(λ, α; x) α n k=1

=x

n 1 

n k=1

λαk

 α1   α−1   α−1 − ln L(λ, α; x) α ≥ xλ − ln L(λ, α; x) α .

(4.2)

Since ψ is n-monotone, we have ψ ′ (x) ≤ 0 for x ≥ 0. Thus, it holds that, for x > 0, α α n n   α α   λαk xα e−λk x φ e−λk x   −λα xα  ′ k x k=1 k=1 ψ φ e n     α α nα φ e−λk x [L(λ, α; x) ln L(λ, α; x)]/L′ (λ, α; x) ≥ − ψ′

fn (x) =

α

ψ′

x

=

nα  x

k=1 n    α α  − ln L(λ, α; x) ψ ′ φ e−λk x L(λ, α; x)/L′ (λ, α; x) k=1

n   α−1   α α  ≥ nαλ − ln L(λ, α; x) α ψ ′ φ e−λk x L(λ, α; x)/L′ (λ, α; x) k=1

   = hn Hn−1 Fn (x) , where the first and the second inequalities follow from (4.1) and (4.2), respectively. Consequently, we have fn Fn−1 (x) ≥





hn Hn−1 (x) for all x ∈ (0, 1). By (3.B.11) in Shaked and Shanthikumar (2007), this invokes X1:n ≤disp Z1:n .





(ii) Similarly, fn (x) ≤ hn Hn−1 Fn (x) for x > 0 and then fn Fn−1 (x) ≤ hn Hn−1 (x) for x ∈ (0, 1). Again by (3.B.11) of Shaked and Shanthikumar (2007), we have Z1:n ≤disp X1:n . 















As for the convexity and concavity in Theorem 4.4 we provide the following examples.

• For n ≥ 2, Gumbel survival copula with φ(x) = (− ln x)θ (θ ≥ 1, x ∈ [0, 1]) and the one with φ(x) = (1 − ln x)θ − 1 (θ ≥ ψ ln ψ 1, x ∈ [0, 1]) both have log-convex ψ and convex ψ ′ . √ • Gumbel–Hougaard survival copula (n = 3) with φ(x) = ln(1 − θ ln x) (θ ∈ (0, 0.5(3 − 5)], x ∈ [0, 1]) and the one ψ ln ψ (n = 2) with φ(x) = (1 − ln x)θ − 1 (θ ∈ (0, 1], x ∈ [0, 1]) both have log-concave ψ and concave ψ ′ . To close this section, we present an example revealing that the log-convexity (log-concavity) of ψ and the convexity ψ ln ψ (concavity) of ψ ′ in Theorem 4.4 are not trivial. Example 4.5. Assume Xi ∼ W (α, λi ) (i = 1, 2) and Yi ∼ W (α, µi ) (i = 1, 2) share a common Archimedean survival copula with generator ψ . (i) Evidently, λα ≤ (λα1 + λα2 )/2 for (α, λ1 , λ2 , λ) = (0.8, 1, 3, 1.9). For φ(x) = ln(2x−0.5 − 1) with x ∈ [0, 1], ψ(x) =

x) = (1 + e−x ) ln[0.5(ex + 1)] for x ∈ [0, +∞). Also it can be verified that ∂ ln∂ xψ( = 2   2 ′ − x x)/ψ (x)]  −2e ≤ 0 and ∂ [ψ(x) ln∂ψ( = −0.2071 < 0, the log-convexity of ψ and convexity of ψ ψln′ ψ in Theorem 4.4(i)  (1+e−x )2 x2 x=0.5    are violated. On the other hand, f2 (0.3) − h2 H2−1 F2 (0.3) = −0.0035 < 0 tends to X1:2 ̸≤disp Z1:2 . (ii) Plainly, λα ≥ 21 (λα1 + λα2 ) for (α, λ1 , λ2 , λ) = (2.5, 3, 4, 3.9). For φ(x) = (x−1 − 1)2 with x ∈ [0, 1], ψ(x) = √x1+1 and  √  √ √ 2+1/ x ∂ 2 [ψ(x) ln ψ(x)/ψ ′ (x)]  ψ(x) ln ψ(x) ∂ 2 ln ψ(x) √ = ( x + x ) ln ( x + 1 ) for x ≥ 0. Again, = > 0 and = 0.77 > 0  2ψ ′ (x) ∂ x2 4( x+x)2 ∂ x2 x=0.4    ψ ln ψ invalidate log-concavity of ψ and concavity of ψ ′ . Also, h2 H2−1 F2 (0.35) − f2 (0.35) = −0.076 < 0 yields X1:2 ̸≥disp

[0.5(ex + 1)]−2 and

Z1:2 .

ψ(x) ln ψ(x) ψ ′ (x)

2



5. An application in weakest link theory The Weibull distribution is rather flexible in describing the lifetime distributions and thus very commonly used in reliability theory. As one natural application, Theorem 3.1 guarantees that, for series systems of components having independent Weibull distributed lifetimes with a common shape parameter, the weakly submajorized scale parameter vector leads to a

C. Li, X. Li / Statistics and Probability Letters 97 (2015) 46–53

53

larger system’s lifetime in the sense of the likelihood ratio order when the shape parameter is larger than 1, and the weakly supermajorized scale parameter vector leads to smaller system’s lifetime in the sense of the likelihood ratio order when the shape parameter is smaller than 1. Here we present one application in material strength. Based on the weakest-link theory the well-known Weibull distribution is widely used to describe the tensile strength of brittle materials such as carbon fibres and glass fibres. Recently, the Weibull distribution was also applied to analyse the tensile properties of bast fibres such as jute (Doan et al., 2006), hemp (Pickering et al., 2007), and flax (Zafeiropoulos and Baillie, 2007). In material science, the fibre strength is usually described by a two-parameter Weibull distribution F (x; α, λ) = 1 − exp{−λα xα } for x > 0, where the shape parameter α > 0 is the Weibull modulus, the scale parameter λ > 0 and hence the characteristic strength λ−1 is related to the fibre diameter and gauge length. Due to modern technologies, textile fabrics are usually made of blended yarns, for example, jute/cotton and polyester. Consider a yarn blended with two types of fibres, and assume that their strengths are of Weibull distributions with a common shape parameter α but different scale parameters λ1 and λ2 . Say, the total number of fibres in this yarn is n, let Xi ’s be i.i.d. fibre strengths with scale parameter λ1 and Yi ’s be i.i.d. fibre strengths with scale parameter λ2 . For convenience set λ1 < λ2 . Denote, for k = 0, 1, . . . , n − 1,

λk = (λ1 , . . . , λ1 , λ2 , . . . , λ2 ),       k

n −k

λk+1 = (λ1 , . . . , λ1 , λ2 , . . . , λ2 ).       k+1

n−k−1

Then, it is easy to verify that λk ≼w λk+1 and λk+1 ≼w λk . As a direct application of Theorem 3.1, we have, for any Weibull modulus α > 0, min{X1 , . . . , Xk+1 , Y1 , . . . , Yn−k−1 } ≥lr min{X1 , . . . , Xk , Y1 , . . . , Yn−k }. According to the well-known weakest-link theory, this actually tells us that the yarn blended with stronger fibres (with smaller scale parameter λ1 ) has a larger strength than the yarn blended with less strong fibres in the sense of the likelihood ratio order, and this solid theory helps fabric material engineers to adjust the percentages of two types of fibres and obtain a stronger blended yarn. Acknowledgements The authors would like to thank the two reviewers for their valuable comments, which have greatly improved the earlier version of this manuscript. In particular, one reviewer directed us to the interesting relation between Lemma 2.5 and Bassan and Spizzichino (2005). 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