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Stochastic comparisons of parallel and series systems with heterogeneous Birnbaum–Saunders components
Accepted Manuscript Stochastic comparisons of parallel and series systems with heterogeneous Birnbaum-Saunders components Longxiang Fang, Xiaojun Zhu, N. Balakrishnan PII: DOI: Reference:
S0167-7152(16)00014-6 http://dx.doi.org/10.1016/j.spl.2016.01.021 STAPRO 7532
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Statistics and Probability Letters
Received date: 6 May 2015 Revised date: 27 January 2016 Accepted date: 27 January 2016 Please cite this article as: Fang, L., Zhu, X., Balakrishnan, N., Stochastic comparisons of parallel and series systems with heterogeneous Birnbaum-Saunders components. Statistics and Probability Letters (2016), http://dx.doi.org/10.1016/j.spl.2016.01.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Stochastic comparisons of parallel and series systems with heterogeneous Birnbaum-Saunders components Longxiang Fanga,b , Xiaojun Zhub,∗, N. Balakrishnanb Department of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, PR China b Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1 a
Abstract In this paper, we discuss stochastic comparisons of lifetimes of parallel and series systems with independent heterogeneous Birnbaum-Saunders components with respect to the usual stochastic order based on vector majorization of parameters. Specifically, let X1 , . . . , Xn be independent random variables with Xi ∼ BS(αi , βi ), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be another set of independent random variables with Xi∗ ∼ BS(αi∗ , βi∗ ), i = 1, . . . , n. Then, we first show that when α1 = . . . = αn = α1∗ = . . . = αn∗ , (β1 , . . . , βn ) ≽m (β1∗ , . . . , βn∗ ) ∗ ∗ ≥st X1:n . implies Xn:n ≥st Xn:n and ( β11 , . . . , β1n ) ≽m ( β1∗ , . . . , β1∗ ) implies X1:n n 1 We subsequently generalize these results to a wider range of the scale parameters. Next, we show that when β1 = . . . = βn = β1∗ = . . . = βn∗ , ∗ ∗ and X1:n ( α11 , . . . , α1n ) ≽m ( α1∗ , . . . , α1∗ ) implies Xn:n ≥st Xn:n ≥st X1:n . Finaln 1 ly, we establish similar results for the log Birnbaum-Saunders distribution. Keywords: Birnbaum-Saunders distribution; Log Birnbaum-Saunders distribution; Usual stochastic order; Parallel systems; Series systems; Majorization 2000 MSC: 60E15; 62G30; 62D05;62N05
Corresponding author Email addresses: [email protected] (Longxiang Fang), [email protected] (Xiaojun Zhu), [email protected] (N. Balakrishnan) ∗
Preprint submitted to Statistics & Probability Letters
January 27, 2016
1. Introduction Let X1:n ≤ . . . ≤ Xn:n denote the order statistics from random variables X1 , . . . , Xn . These quantities play an important role in many areas including statistical inference, operations research, reliability theory, life testing, and quality control. Interested readers may refer to the volumes by Balakrishnan and Rao (1998a,b) for relevant details. Evidently, in reliability theory, Xk:n is the lifetime of a (n − k + 1)-out-of-n system, which is a popular structure of redundancy in fault-tolerant systems. In particular, Xn:n and X1:n correspond to the lifetimes of parallel and series systems. Stochastic comparisons of parallel and series systems with heterogeneous components have been studied by many authors including Dykstra et al. (1997), Khaledi and Kochar (2000, 2006), Balakrishnan and Zhao (2013), Li and Li (2015), Balakrishnan et al. (2015), and Torrado and Kochar (2015). Birnbaum and Saunders (1969a,b) introduced a two-parameter failure time distribution for fatigue failure caused under cyclic loading. This distribution has been used quite effectively for modeling positively skewed data, especially for data on crack growth and life-times. A random variable X is said to have the Birnbaum-Saunders distribution (“BS” in short) if its cumulative distribution function (cdf) is given by √ )] [ (√ 1 x β F (x; α, β) = Φ − , x > 0, α β x where Φ(·) denotes the standard normal cdf, and α > 0 and β > 0 are the shape and scale parameters, respectively. In this case, we denote it by X ∼ BS(α, β). Many authors have studied different aspects of this model and for some recent work, one may refer to Chang and Tang (1993), Dupuis and Mills (1994), D´ıaz-Garc´ıa and Leiva-S´ anchez (2005), Ng et al. (2003, 2006), Kundu et al. (2008), Leiva et al. (2008), Sanhueza et al. (2008), Zhu and Balakrishnan (2015), and the references cited therein. For a brief survey on BS distribution and its properties, see Johnson et al. (1995). Furthermore, Rieck and Nedelman (1991) introduced log Birnbaum-Saunders distribution and showed that 2
it can be obtained as a special case of the sinh-normal distribution. This distribution has many interesting properties in addition to being useful as a log-linear model for lifetime data. A random variable Y is said to have a log Birnbaum-Saunders distribution (“LBS” in short) if its cdf is given by )] [ ( 2 y − ln β , y ∈ R, F (y; α, β) = Φ sinh α 2 where Φ(·) is the standard normal cdf as before, and sinh(y) is the hyperbolic y −y sine function defined as sinh(y) = e −e . We denote it by Y ∼ LBS(α, β). 2 It is clear that X ∼ BS(α, β) ⇐⇒ ln X ∼ LBS(α, β).
(1.1)
In this paper, we study the lifetimes of parallel and series systems with independent BS and LBS components with respect to the usual stochastic order based on vector majorization of parameters. Specifically, let X1 , . . . , Xn be independent random variables with Xi ∼ BS(αi , βi ) (or LBS(αi , βi )), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be another set of independent random variables with Xi∗ ∼ BS(αi∗ , βi∗ ) (or LBS(αi∗ , βi∗ )), i = 1, . . . , n. Then, we first show that when α1 = . . . = αn = α1∗ = . . . = αn∗ , ∗ (β1 , . . . , βn ) ≽m (β1∗ , . . . , βn∗ ) =⇒ Xn:n ≥st Xn:n
and
(
1 1 ,..., β1 βn
)
≽m
(
1 1 ,..., ∗ ∗ β1 βn
)
∗ =⇒ X1:n ≥st X1:n .
We subsequently generalize these results to a wider range of the scale parameters as follows: ∗ (β1 , . . . , βn ) ≽w (β1∗ , . . . , βn∗ ) =⇒ Xn:n ≥st Xn:n
and
(
1 1 ,..., β1 βn
)
≽w
(
1 1 ,..., ∗ ∗ β1 βn 3
)
∗ ≥st X1:n . =⇒ X1:n
Finally, we establish that when β1 = . . . = βn = β1∗ = . . . = βn∗ , ( ) ( ) 1 1 1 1 ∗ ∗ ,..., ≽m , . . . , ∗ =⇒ Xn:n ≥st Xn:n and X1:n ≥st X1:n . ∗ α1 αn α1 αn Since BS distribution has become one of the popular lifetime models in reliability literature and that we discuss stochastic orderings for largest and smallest order statistics, the results established have directly relate to some key distributional properties and features of parallel and series systems, two most common coherent systems, with BS components. Furthermore, these results may also be useful in establishing some statistical properties of estimators of the scale and shape parameters of the BS distribution. We are currently looking into this problem and hope to report the findings in a future paper. Throughout this paper, “increasing” and “decreasing” are used to mean “nondecreasing” and “nonincreasing”, respectively. 2. Preliminaries In this section, we provide a brief review of definitions of usual stochastic order, majorization and weak majorization. For more details, one may refer to Shaked and Shanthikumar (2007) and Marshall et al. (2011). Definition 2.1 We say that Y is smaller than X in the usual stochastic ¯ order, denoted by X ≥st Y , if F¯ (x) ≥ G(x) for all x. Definition 2.2 Let λ = (λ1 , . . . , λn ) and λ∗ = (λ∗1 , . . . , λ∗n ) be two real vectors, and λ[1] ≥ . . . ≥ λ[n] and λ∗[1] ≥ . . . ≥ λ∗[n] denote their ordered components. Then: (1) λ∗ is said to be majorized by λ, denoted by λ ≽m λ∗ , if k ∑ i=1
for k = 1, 2, . . . , n − 1, and
n ∑
i=1
λ[i] ≥
λi =
n ∑
i=1
4
k ∑ i=1
λ∗i ;
λ∗[i]
(2) λ∗ is said to be weak lower majorized by λ, denoted by λ ≽w λ∗ , if k ∑ i=1
for k = 1, 2, . . . , n − 1, and
n ∑
i=1
k ∑
λ[i] ≥
λi ≥
n ∑
i=1
λ∗[i]
i=1
λ∗i .
The following lemma gives a relationship between these two forms of majorization. Lemma 2.3 (Peˇ caric et al. (1992)) Let λ = (λ1 , . . . , λn ) and λ∗ = (λ∗1 , . . . , λ∗n ) be two real vectors. Then, λ ≽w λ∗ if and only if there exists an n-dimensional vector µ such that λ ≽m µ and µ ≥ λ∗ (i.e., µi ≥ λ∗i , i = 1, . . . , n). Before we present our main results, we need the following well-known concept and lemmas. Definition 2.4 Let λ = (λ1 , . . . , λn ) and µ = (µ1 , . . . , µn ) be two real vectors. A real-valued function ϕ(λ):Rn → R is said to be a Schur-concave (Schurconvex) function if for all λ ≽m µ, we have ϕ(λ) ≤ (≥)ϕ(µ). Lemma 2.5 (Marshall et al. (2011)) A permutation-symmetric differentiable function ϕ(X) is Schur-concave (Schur-convex) if and only if ( ) ∂ϕ(X) ∂ϕ(X) (Xi − Xj ) − ≤ 0 (≥ 0) ∂Xi ∂Xj for all i ̸= j. Lemma 2.6 (Fang and Zhang (2010)) (1) Let g(u) =
∫u
−∞
(2) Let h(u) =
u2 2 t2 e− 2
e−
u2 2 ∫ +∞ − t2 e 2 u
dt
for all u ∈ R. Then, g(u) is a decreasing function;
dt
for all u ∈ R. Then, h(u) is an increasing function.
e−
Finally, we consider the following scale model. Suppose F is an absolutely continuous distribution function with density function f . Then, the independent random variables Xλ1 , . . . , Xλn are said to follow the scale model if Xλi ∼ F (λi x), where λi > 0, for i = 1, . . . , n. In this case, F is referred to as the baseline distribution and the λ′i s are the scale parameters. Stochastic comparisons among order statistics when the variables are from the scale 5
model have been discussed by many authors; see, for example, Pledger and Proschan (1971), Khaledi et al. (2011), and Kochar and Torrado (2015). We first present the following lemma which is necessary to prove the main result in the next section. Lemma 2.7 Let X1 , . . . , Xn be independent random variables with Xi ∼ F (λi x), i = 1, . . . , n, where F is an absolutely continuous distribution function with density function f and reverse hazard rate re, and X1∗ , . . . , Xn∗ be independent random variables with Xi∗ ∼ F (λ∗i x), i = 1, . . . , n. If re(x) is decreasing in x, then ∗ . (λ1 , . . . , λn ) ≽m (λ∗1 , . . . , λ∗n ) =⇒ Xn:n ≥st Xn:n
The proof of Lemma 2.7 is similar to that of Theorem 3.2 in Khaledi et al. (2011), and is therefore not presented here for the sake of conciseness. 3. Main results First, we present stochastic comparison results for the lifetimes of parallel and series systems with independent heterogeneous BS components with respect to the usual stochastic order based on vector majorization of scale parameters. Theorem 3.1 Let X1 , . . . , Xn be independent random variables with Xi ∼ BS(α, βi ), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be independent random variables with Xi∗ ∼ BS(α, βi∗ ), i = 1, . . . , n. Then, we have: ∗ (1) (β1 , . . . , βn ) ≽m (β1∗ , . . . , βn∗ ) implies Xn:n ≥st Xn:n ; 1 1 1 1 ∗ (2) ( β1 , . . . , βn ) ≽m ( β ∗ , . . . , β ∗ ) implies X1:n ≥st X1:n . n 1 Proof (1) Note that when the shape parameter of all BS variables is the same and fixed, then the random variables belong to the scale model. To ∗ , it is sufficient to prove, for x > 0, that prove that Xn:n ≥st Xn:n ( ) − 12 (√ βx −√ βx )2 f (x; α, β) sgn β 1 β 3 e 2α √ re(x) = = ( )2 + ( )2 ∫ α1 (√ βx − βx ) − u2 F (x; α, β) x x e 2 du −∞ 6
sgn
is a decreasing √ function in x, where ‘ = ’ means equality of signs. It is obvious √ that βx − βx is an increasing function in x. So, from Part (1) of Lemma 2.6, we have the composite function √x √β 2 − 12 ( β − x) 2α e √ ∫ α1 (√ βx − βx ) − u2 e 2 du −∞ 1
3
to be a decreasing function in x. Moreover, ( βx ) 2 + ( βx ) 2 is a decreasing function in x. From these observations, we readily have re(x) to be a decreasing function in x. Thus, using Lemma 2.7, the proof gets completed. (2) Let Yi = X1i and Yi∗ = X1∗ , i = 1, . . . , n. From Saunders (1974), i it is known that BS random variables are closed under reciprocation. So, Y1 , . . . , Yn are independent random variables with Yi ∼ BS(α, β1i ), i = 1, . . . , n, and Y1∗ , . . . , Yn∗ are independent random variables with Yi∗ ∼ BS(α, 1 ), i = 1, . . . , n. So, from Part (1), we have ( β11 , . . . , β1n ) ≽m ( β1∗ , . . . , β1∗ ) βi∗ n 1 ∗ ∗ implying that Yn:n ≥st Yn:n , that is, X1:n ≥st X1:n , and this completes the proof of the theorem. We now generalize Theorem 3.1 to a wider range of the scale parameters as follows. Theorem 3.2 Let X1 , . . . , Xn be independent random variables with Xi ∼ BS(α, βi ), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be independent random variables with Xi∗ ∼ BS(α, βi∗ ), i = 1, . . . , n. If (β1 , . . . , βn ) ≥ (β1∗ , . . . , βn∗ ), that is, ∗ ∗ βi ≥ βi∗ , i = 1, . . . , n, then Xn:n ≥st Xn:n and X1:n ≥st X1:n . Proof The survival functions of Xn:n and X1:n are, for x > 0, √ )] [ (√ n ∏ 1 x βi F Xn:n (x) = 1 − Φ − α βi x i=1 and
√ )]} [ (√ n { ∏ 1 x βi − F X1:n (x) = 1−Φ , α βi x i=1
respectively. By√using the definition of the usual stochastic order and the √ fact that βx − βx is a decreasing function with respect to β, the required results follow readily. 7
Now, we discuss stochastic comparisons of the lifetimes of parallel and series systems with independent BS components with respect to the usual stochastic order based on vector weak lower majorization of scale parameters. Theorem 3.3 Let X1 , . . . , Xn be independent random variables with Xi ∼ BS(α, βi ), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be independent random variables with Xi∗ ∼ BS(α, βi∗ ), i = 1, . . . , n. Then, we have: ∗ (1) (β1 , . . . , βn ) ≽w (β1∗ , . . . , βn∗ ) implies Xn:n ≥st Xn:n ; 1 1 1 1 ∗ (2) ( β1 , . . . , βn ) ≽w ( β ∗ , . . . , β ∗ ) implies X1:n ≥st X1:n . n 1 Proof (1) If (β1 , . . . , βn ) ≽w (β1∗ , . . . , βn∗ ), then by Lemma 2.3, there exists a vector (µ1 , . . . , µn ) such that (β1 , . . . , βn ) ≽m (µ1 , . . . , µn ) and (µ1 , . . . , µn ) ≥ (β1∗ , . . . , βn∗ ). Now, let Y1 , . . . , Yn be independent random variables with Yi ∼ BS(α, µi ), i = 1, . . . , n. Then, from Part (1) of Theorem 3.1, it is known that Xn:n ≥st Yn:n . Since (µ1 , . . . , µn ) ≥ (β1∗ , . . . , βn∗ ), we have ∗ µi ≥ βi∗ , i = 1, . . . , n, and so Yn:n ≥st Xn:n according to Theorem 3.2. Thus, ∗ we have Xn:n ≥st Xn:n . (2) If ( β11 , . . . , β1n ) ≽w ( β1∗ , . . . , β1∗ ), then by Lemma 2.3, there exists a vecn 1 tor ( θ11 , . . . , θ1n ) such that ( β11 , . . . , β1n ) ≽m ( θ11 , . . . , θ1n ) and βi∗ ≥ θi , i = 1, . . . , n. Now, let Z1 , . . . , Zn be independent random variables with Zi ∼ BS(α, θi ), i = 1, . . . , n. Then, from Part (2) of Theorem 3.1 and Theorem ∗ ∗ 3.2, we have Z1:n ≥st X1:n and X1:n ≥st Z1:n , and thus X1:n ≥st X1:n holds. Finally, we consider stochastic comparisons of the lifetimes of parallel and series systems with independent BS components with respect to the usual stochastic order based on vector majorization of shape parameters. Theorem 3.4 Let X1 , . . . , Xn be independent random variables with Xi ∼ BS(αi , β), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be independent random variables with Xi∗ ∼ BS(αi∗ , β), i = 1, . . . , n. Then, ( α11 , . . . , α1n ) ≽m ( α1∗ , . . . , α1∗ ) n 1 ∗ ∗ ≥st X1:n . and X1:n implies Xn:n ≥st Xn:n Proof Set λ1 = α11 , . . . , λn = α1n and λ∗1 = α1∗ , . . . , λ∗n = α1∗ . Then, 1
8
n
(λ1 , . . . , λn ) ≽m (λ∗1 , . . . , λ∗n ) and the survival function of Xn:n is, for x > 0, √ )] [ (√ n ∏ x β F Xn:n (x) = 1 − Φ λi − β x i=1 ∫ n a(x;λi ,β) ∏ u2 1 √ e− 2 du = 1− 2π i=1 −∞ (√ √ ) x − βx = λi l(x; β), say. by letting a(x; λi , β) = λi β Upon differentiating with respect to λi , i = 1, . . . , n, we get, for x > 0, )2 ( − 12
a(x;λi ,β)
∂F Xn:n (x) e = − ∫ a(x;λ ,β) u2 l(x; β)FXn:n (x). i ∂λi e− 2 du −∞
So, we obtain
∂F Xn:n (x) ∂F Xn:n (x) − ∂λi ∂λj ( )2 ( )2 1 1 − 2 a(x;λi ,β) [ − 2 a(x;λj ,β) ] e e = l(x; β)FXn:n (x) ∫ a(x;λ ,β) u2 − ∫ a(x;λ ,β) u2 . j i − 2 − 2 e du e du −∞ −∞
From Part (1) of Lemma 2.6, we have ( ) ∂F Xn:n (x) ∂F Xn:n (x) (λi − λj ) − ≥ 0. ∂λi ∂λj
Thus, by using Lemma 2.5, F Xn:n (x) is a Schur-convex function with respect ∗ to (λ1 , . . . , λn ), and so we obtain Xn:n ≥st Xn:n . Similarly, the survival function of X1:n is, for x > 0, ( )] n [ ∏ F X1:n (x) = 1 − Φ a(x; λi , β) =
i=1 n ∫ +∞ ∏ i=1
a(x;λi ,β)
9
u2 1 √ e− 2 du. 2π
Upon differentiating with respect to λi , i = 1, . . . , n, we get, for x > 0, ( )2 − 12
a(x;λi ,β)
∂F X1:n (x) e = − ∫ +∞ ∂λi
2
u √1 e− 2 a(x;λi ,β) 2π
du
l(x; β)F X1:n (x).
So, we obtain
∂F X1:n (x) ∂F X1:n (x) − ∂λi ∂λj ( )2 ( )2 1 1 − 2 a(x;λj ,β) − 2 a(x;λi ,β) [ ] e e = l(x; β)F X1:n (x) ∫ +∞ . − ∫ +∞ u2 u2 √1 e− 2 du √1 e− 2 du a(x;λj ,β) 2π a(x;λi ,β) 2π
From Part (2) of Lemma 2.6, we have ( ) ∂F X1:n (x) ∂F X1:n (x) (λi − λj ) − ≤ 0. ∂λi ∂λj
Thus, by using Lemma 2.5, F X1:n (x) is a Schur-concave function with respect ∗ to (λ1 , . . . , λn ), and so we obtain X1:n ≥st X1:n . Hence, the theorem. Upon using the simple relationship between BS and LBS distributions in (1.1) and the definition of the usual stochastic order, the following results can be readily deduced. Corollary 3.5 Let X1 , . . . , Xn be independent random variables with Xi ∼ LBS(α, βi ), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be independent random variables with Xi∗ ∼ LBS(α, βi∗ ), i = 1, . . . , n. Then, we have: ∗ (1) (β1 , . . . , βn ) ≽w (β1∗ , . . . , βn∗ ) implies Xn:n ≥st Xn:n ; 1 1 1 1 ∗ (2) ( β1 , . . . , βn ) ≽w ( β ∗ , . . . , β ∗ ) implies X1:n ≥st X1:n . n 1 Corollary 3.6 Let X1 , . . . , Xn be independent random variables with Xi ∼ LBS(αi , β), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be(independent ) random ( variables ) with Xi∗ ∼ LBS(αi∗ , β), i = 1, . . . , n. Then, ∗ ∗ ≥st X1:n . and X1:n implies Xn:n ≥st Xn:n
10
1 , . . . , α1n α1
≽m
1 , . . . , α1∗ α∗1 n
Acknowledgments This research was supported by the Provincial Natural Science Research Project of Anhui Colleges (No. KJ2013A137), the National Natural Science Foundation of Anhui Province (No. 1408085MA07), and the PhD research startup foundation of Anhui Normal University (No. 2014bsqdjj34) which facilitated the research visit of the first author to McMaster University, Canada. The research work of the last author was supported by an Individual Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. Our sincere thanks also go to reviewers for their useful comments and suggestions on an earlier version of this manuscript which led to this improved one. References Balakrishnan, N., Haidari, A., Masoumifard, K., 2015. Stochastic comparisons of series and parallel systems with generalized exponential components. IEEE Transactions on Reliability 64, 333-348. Balakrishnan, N., Rao, C.R. (Eds.), 1998a. Handbook of Statistics, Vol. 16Order Statistics: Theory and Methods. North-Holland, Amsterdam. Balakrishnan, N., Rao, C.R. (Eds.), 1998b. Handbook of Statistics, Vol. 17Order Statistics: Applications. North-Holland, Amsterdam. Balakrishnan, N., Zhao, P., 2013. Ordering properties of order statistics from heterogeneous populations: a review with an emphasis on some recent developments. Probability in the Engineering and Informational Sciences 27, 403-469 (with discussions). Birnbaum, Z.W., Saunders, S.C., 1969a. A new family of life distributions. Journal of Applied Probability 6, 319-327.
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Pledger, P., Proschan, F., 1971. Comparisons of order statistics and of spacings from heterogeneous distributions. In: Rustagi, J.S. (Ed.), Optimizing Methods in Statistics. Academic Press, NewYork, pp. 89-113. Sanhueza, A., Leiva, V., Balakrishnan, N., 2008. The generalized BirnbaumSaunders distribution and its theory, methodology, and application. Communications in Statistics - Theory and Methods 37, 645-670. Saunders, S.C., 1974. A family of random variables closed under reciprocation. Journal of the American Statistical Association 69, 533-539. Shaked, M., Shanthikumar, J.G., 2007. Stochastic Orders. Springer, New York. Torrado, N., Kochar, S.C., 2015. Stochastic order relations among parallel systems from Weibull distributions. Journal of Applied Probability, 52, 102-116. Zhu, X., Balakrishnan, N., 2015. Birnbaum-Saunders distribution based on Laplace kernel and some properties and inferential issues. Statistics & Probability Letters 101, 1-10.
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S0167-7152(16)00014-6 http://dx.doi.org/10.1016/j.spl.2016.01.021 STAPRO 7532
To appear in:
Statistics and Probability Letters
Received date: 6 May 2015 Revised date: 27 January 2016 Accepted date: 27 January 2016 Please cite this article as: Fang, L., Zhu, X., Balakrishnan, N., Stochastic comparisons of parallel and series systems with heterogeneous Birnbaum-Saunders components. Statistics and Probability Letters (2016), http://dx.doi.org/10.1016/j.spl.2016.01.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Stochastic comparisons of parallel and series systems with heterogeneous Birnbaum-Saunders components Longxiang Fanga,b , Xiaojun Zhub,∗, N. Balakrishnanb Department of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, PR China b Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1 a
Abstract In this paper, we discuss stochastic comparisons of lifetimes of parallel and series systems with independent heterogeneous Birnbaum-Saunders components with respect to the usual stochastic order based on vector majorization of parameters. Specifically, let X1 , . . . , Xn be independent random variables with Xi ∼ BS(αi , βi ), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be another set of independent random variables with Xi∗ ∼ BS(αi∗ , βi∗ ), i = 1, . . . , n. Then, we first show that when α1 = . . . = αn = α1∗ = . . . = αn∗ , (β1 , . . . , βn ) ≽m (β1∗ , . . . , βn∗ ) ∗ ∗ ≥st X1:n . implies Xn:n ≥st Xn:n and ( β11 , . . . , β1n ) ≽m ( β1∗ , . . . , β1∗ ) implies X1:n n 1 We subsequently generalize these results to a wider range of the scale parameters. Next, we show that when β1 = . . . = βn = β1∗ = . . . = βn∗ , ∗ ∗ and X1:n ( α11 , . . . , α1n ) ≽m ( α1∗ , . . . , α1∗ ) implies Xn:n ≥st Xn:n ≥st X1:n . Finaln 1 ly, we establish similar results for the log Birnbaum-Saunders distribution. Keywords: Birnbaum-Saunders distribution; Log Birnbaum-Saunders distribution; Usual stochastic order; Parallel systems; Series systems; Majorization 2000 MSC: 60E15; 62G30; 62D05;62N05
Corresponding author Email addresses: [email protected] (Longxiang Fang), [email protected] (Xiaojun Zhu), [email protected] (N. Balakrishnan) ∗
Preprint submitted to Statistics & Probability Letters
January 27, 2016
1. Introduction Let X1:n ≤ . . . ≤ Xn:n denote the order statistics from random variables X1 , . . . , Xn . These quantities play an important role in many areas including statistical inference, operations research, reliability theory, life testing, and quality control. Interested readers may refer to the volumes by Balakrishnan and Rao (1998a,b) for relevant details. Evidently, in reliability theory, Xk:n is the lifetime of a (n − k + 1)-out-of-n system, which is a popular structure of redundancy in fault-tolerant systems. In particular, Xn:n and X1:n correspond to the lifetimes of parallel and series systems. Stochastic comparisons of parallel and series systems with heterogeneous components have been studied by many authors including Dykstra et al. (1997), Khaledi and Kochar (2000, 2006), Balakrishnan and Zhao (2013), Li and Li (2015), Balakrishnan et al. (2015), and Torrado and Kochar (2015). Birnbaum and Saunders (1969a,b) introduced a two-parameter failure time distribution for fatigue failure caused under cyclic loading. This distribution has been used quite effectively for modeling positively skewed data, especially for data on crack growth and life-times. A random variable X is said to have the Birnbaum-Saunders distribution (“BS” in short) if its cumulative distribution function (cdf) is given by √ )] [ (√ 1 x β F (x; α, β) = Φ − , x > 0, α β x where Φ(·) denotes the standard normal cdf, and α > 0 and β > 0 are the shape and scale parameters, respectively. In this case, we denote it by X ∼ BS(α, β). Many authors have studied different aspects of this model and for some recent work, one may refer to Chang and Tang (1993), Dupuis and Mills (1994), D´ıaz-Garc´ıa and Leiva-S´ anchez (2005), Ng et al. (2003, 2006), Kundu et al. (2008), Leiva et al. (2008), Sanhueza et al. (2008), Zhu and Balakrishnan (2015), and the references cited therein. For a brief survey on BS distribution and its properties, see Johnson et al. (1995). Furthermore, Rieck and Nedelman (1991) introduced log Birnbaum-Saunders distribution and showed that 2
it can be obtained as a special case of the sinh-normal distribution. This distribution has many interesting properties in addition to being useful as a log-linear model for lifetime data. A random variable Y is said to have a log Birnbaum-Saunders distribution (“LBS” in short) if its cdf is given by )] [ ( 2 y − ln β , y ∈ R, F (y; α, β) = Φ sinh α 2 where Φ(·) is the standard normal cdf as before, and sinh(y) is the hyperbolic y −y sine function defined as sinh(y) = e −e . We denote it by Y ∼ LBS(α, β). 2 It is clear that X ∼ BS(α, β) ⇐⇒ ln X ∼ LBS(α, β).
(1.1)
In this paper, we study the lifetimes of parallel and series systems with independent BS and LBS components with respect to the usual stochastic order based on vector majorization of parameters. Specifically, let X1 , . . . , Xn be independent random variables with Xi ∼ BS(αi , βi ) (or LBS(αi , βi )), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be another set of independent random variables with Xi∗ ∼ BS(αi∗ , βi∗ ) (or LBS(αi∗ , βi∗ )), i = 1, . . . , n. Then, we first show that when α1 = . . . = αn = α1∗ = . . . = αn∗ , ∗ (β1 , . . . , βn ) ≽m (β1∗ , . . . , βn∗ ) =⇒ Xn:n ≥st Xn:n
and
(
1 1 ,..., β1 βn
)
≽m
(
1 1 ,..., ∗ ∗ β1 βn
)
∗ =⇒ X1:n ≥st X1:n .
We subsequently generalize these results to a wider range of the scale parameters as follows: ∗ (β1 , . . . , βn ) ≽w (β1∗ , . . . , βn∗ ) =⇒ Xn:n ≥st Xn:n
and
(
1 1 ,..., β1 βn
)
≽w
(
1 1 ,..., ∗ ∗ β1 βn 3
)
∗ ≥st X1:n . =⇒ X1:n
Finally, we establish that when β1 = . . . = βn = β1∗ = . . . = βn∗ , ( ) ( ) 1 1 1 1 ∗ ∗ ,..., ≽m , . . . , ∗ =⇒ Xn:n ≥st Xn:n and X1:n ≥st X1:n . ∗ α1 αn α1 αn Since BS distribution has become one of the popular lifetime models in reliability literature and that we discuss stochastic orderings for largest and smallest order statistics, the results established have directly relate to some key distributional properties and features of parallel and series systems, two most common coherent systems, with BS components. Furthermore, these results may also be useful in establishing some statistical properties of estimators of the scale and shape parameters of the BS distribution. We are currently looking into this problem and hope to report the findings in a future paper. Throughout this paper, “increasing” and “decreasing” are used to mean “nondecreasing” and “nonincreasing”, respectively. 2. Preliminaries In this section, we provide a brief review of definitions of usual stochastic order, majorization and weak majorization. For more details, one may refer to Shaked and Shanthikumar (2007) and Marshall et al. (2011). Definition 2.1 We say that Y is smaller than X in the usual stochastic ¯ order, denoted by X ≥st Y , if F¯ (x) ≥ G(x) for all x. Definition 2.2 Let λ = (λ1 , . . . , λn ) and λ∗ = (λ∗1 , . . . , λ∗n ) be two real vectors, and λ[1] ≥ . . . ≥ λ[n] and λ∗[1] ≥ . . . ≥ λ∗[n] denote their ordered components. Then: (1) λ∗ is said to be majorized by λ, denoted by λ ≽m λ∗ , if k ∑ i=1
for k = 1, 2, . . . , n − 1, and
n ∑
i=1
λ[i] ≥
λi =
n ∑
i=1
4
k ∑ i=1
λ∗i ;
λ∗[i]
(2) λ∗ is said to be weak lower majorized by λ, denoted by λ ≽w λ∗ , if k ∑ i=1
for k = 1, 2, . . . , n − 1, and
n ∑
i=1
k ∑
λ[i] ≥
λi ≥
n ∑
i=1
λ∗[i]
i=1
λ∗i .
The following lemma gives a relationship between these two forms of majorization. Lemma 2.3 (Peˇ caric et al. (1992)) Let λ = (λ1 , . . . , λn ) and λ∗ = (λ∗1 , . . . , λ∗n ) be two real vectors. Then, λ ≽w λ∗ if and only if there exists an n-dimensional vector µ such that λ ≽m µ and µ ≥ λ∗ (i.e., µi ≥ λ∗i , i = 1, . . . , n). Before we present our main results, we need the following well-known concept and lemmas. Definition 2.4 Let λ = (λ1 , . . . , λn ) and µ = (µ1 , . . . , µn ) be two real vectors. A real-valued function ϕ(λ):Rn → R is said to be a Schur-concave (Schurconvex) function if for all λ ≽m µ, we have ϕ(λ) ≤ (≥)ϕ(µ). Lemma 2.5 (Marshall et al. (2011)) A permutation-symmetric differentiable function ϕ(X) is Schur-concave (Schur-convex) if and only if ( ) ∂ϕ(X) ∂ϕ(X) (Xi − Xj ) − ≤ 0 (≥ 0) ∂Xi ∂Xj for all i ̸= j. Lemma 2.6 (Fang and Zhang (2010)) (1) Let g(u) =
∫u
−∞
(2) Let h(u) =
u2 2 t2 e− 2
e−
u2 2 ∫ +∞ − t2 e 2 u
dt
for all u ∈ R. Then, g(u) is a decreasing function;
dt
for all u ∈ R. Then, h(u) is an increasing function.
e−
Finally, we consider the following scale model. Suppose F is an absolutely continuous distribution function with density function f . Then, the independent random variables Xλ1 , . . . , Xλn are said to follow the scale model if Xλi ∼ F (λi x), where λi > 0, for i = 1, . . . , n. In this case, F is referred to as the baseline distribution and the λ′i s are the scale parameters. Stochastic comparisons among order statistics when the variables are from the scale 5
model have been discussed by many authors; see, for example, Pledger and Proschan (1971), Khaledi et al. (2011), and Kochar and Torrado (2015). We first present the following lemma which is necessary to prove the main result in the next section. Lemma 2.7 Let X1 , . . . , Xn be independent random variables with Xi ∼ F (λi x), i = 1, . . . , n, where F is an absolutely continuous distribution function with density function f and reverse hazard rate re, and X1∗ , . . . , Xn∗ be independent random variables with Xi∗ ∼ F (λ∗i x), i = 1, . . . , n. If re(x) is decreasing in x, then ∗ . (λ1 , . . . , λn ) ≽m (λ∗1 , . . . , λ∗n ) =⇒ Xn:n ≥st Xn:n
The proof of Lemma 2.7 is similar to that of Theorem 3.2 in Khaledi et al. (2011), and is therefore not presented here for the sake of conciseness. 3. Main results First, we present stochastic comparison results for the lifetimes of parallel and series systems with independent heterogeneous BS components with respect to the usual stochastic order based on vector majorization of scale parameters. Theorem 3.1 Let X1 , . . . , Xn be independent random variables with Xi ∼ BS(α, βi ), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be independent random variables with Xi∗ ∼ BS(α, βi∗ ), i = 1, . . . , n. Then, we have: ∗ (1) (β1 , . . . , βn ) ≽m (β1∗ , . . . , βn∗ ) implies Xn:n ≥st Xn:n ; 1 1 1 1 ∗ (2) ( β1 , . . . , βn ) ≽m ( β ∗ , . . . , β ∗ ) implies X1:n ≥st X1:n . n 1 Proof (1) Note that when the shape parameter of all BS variables is the same and fixed, then the random variables belong to the scale model. To ∗ , it is sufficient to prove, for x > 0, that prove that Xn:n ≥st Xn:n ( ) − 12 (√ βx −√ βx )2 f (x; α, β) sgn β 1 β 3 e 2α √ re(x) = = ( )2 + ( )2 ∫ α1 (√ βx − βx ) − u2 F (x; α, β) x x e 2 du −∞ 6
sgn
is a decreasing √ function in x, where ‘ = ’ means equality of signs. It is obvious √ that βx − βx is an increasing function in x. So, from Part (1) of Lemma 2.6, we have the composite function √x √β 2 − 12 ( β − x) 2α e √ ∫ α1 (√ βx − βx ) − u2 e 2 du −∞ 1
3
to be a decreasing function in x. Moreover, ( βx ) 2 + ( βx ) 2 is a decreasing function in x. From these observations, we readily have re(x) to be a decreasing function in x. Thus, using Lemma 2.7, the proof gets completed. (2) Let Yi = X1i and Yi∗ = X1∗ , i = 1, . . . , n. From Saunders (1974), i it is known that BS random variables are closed under reciprocation. So, Y1 , . . . , Yn are independent random variables with Yi ∼ BS(α, β1i ), i = 1, . . . , n, and Y1∗ , . . . , Yn∗ are independent random variables with Yi∗ ∼ BS(α, 1 ), i = 1, . . . , n. So, from Part (1), we have ( β11 , . . . , β1n ) ≽m ( β1∗ , . . . , β1∗ ) βi∗ n 1 ∗ ∗ implying that Yn:n ≥st Yn:n , that is, X1:n ≥st X1:n , and this completes the proof of the theorem. We now generalize Theorem 3.1 to a wider range of the scale parameters as follows. Theorem 3.2 Let X1 , . . . , Xn be independent random variables with Xi ∼ BS(α, βi ), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be independent random variables with Xi∗ ∼ BS(α, βi∗ ), i = 1, . . . , n. If (β1 , . . . , βn ) ≥ (β1∗ , . . . , βn∗ ), that is, ∗ ∗ βi ≥ βi∗ , i = 1, . . . , n, then Xn:n ≥st Xn:n and X1:n ≥st X1:n . Proof The survival functions of Xn:n and X1:n are, for x > 0, √ )] [ (√ n ∏ 1 x βi F Xn:n (x) = 1 − Φ − α βi x i=1 and
√ )]} [ (√ n { ∏ 1 x βi − F X1:n (x) = 1−Φ , α βi x i=1
respectively. By√using the definition of the usual stochastic order and the √ fact that βx − βx is a decreasing function with respect to β, the required results follow readily. 7
Now, we discuss stochastic comparisons of the lifetimes of parallel and series systems with independent BS components with respect to the usual stochastic order based on vector weak lower majorization of scale parameters. Theorem 3.3 Let X1 , . . . , Xn be independent random variables with Xi ∼ BS(α, βi ), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be independent random variables with Xi∗ ∼ BS(α, βi∗ ), i = 1, . . . , n. Then, we have: ∗ (1) (β1 , . . . , βn ) ≽w (β1∗ , . . . , βn∗ ) implies Xn:n ≥st Xn:n ; 1 1 1 1 ∗ (2) ( β1 , . . . , βn ) ≽w ( β ∗ , . . . , β ∗ ) implies X1:n ≥st X1:n . n 1 Proof (1) If (β1 , . . . , βn ) ≽w (β1∗ , . . . , βn∗ ), then by Lemma 2.3, there exists a vector (µ1 , . . . , µn ) such that (β1 , . . . , βn ) ≽m (µ1 , . . . , µn ) and (µ1 , . . . , µn ) ≥ (β1∗ , . . . , βn∗ ). Now, let Y1 , . . . , Yn be independent random variables with Yi ∼ BS(α, µi ), i = 1, . . . , n. Then, from Part (1) of Theorem 3.1, it is known that Xn:n ≥st Yn:n . Since (µ1 , . . . , µn ) ≥ (β1∗ , . . . , βn∗ ), we have ∗ µi ≥ βi∗ , i = 1, . . . , n, and so Yn:n ≥st Xn:n according to Theorem 3.2. Thus, ∗ we have Xn:n ≥st Xn:n . (2) If ( β11 , . . . , β1n ) ≽w ( β1∗ , . . . , β1∗ ), then by Lemma 2.3, there exists a vecn 1 tor ( θ11 , . . . , θ1n ) such that ( β11 , . . . , β1n ) ≽m ( θ11 , . . . , θ1n ) and βi∗ ≥ θi , i = 1, . . . , n. Now, let Z1 , . . . , Zn be independent random variables with Zi ∼ BS(α, θi ), i = 1, . . . , n. Then, from Part (2) of Theorem 3.1 and Theorem ∗ ∗ 3.2, we have Z1:n ≥st X1:n and X1:n ≥st Z1:n , and thus X1:n ≥st X1:n holds. Finally, we consider stochastic comparisons of the lifetimes of parallel and series systems with independent BS components with respect to the usual stochastic order based on vector majorization of shape parameters. Theorem 3.4 Let X1 , . . . , Xn be independent random variables with Xi ∼ BS(αi , β), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be independent random variables with Xi∗ ∼ BS(αi∗ , β), i = 1, . . . , n. Then, ( α11 , . . . , α1n ) ≽m ( α1∗ , . . . , α1∗ ) n 1 ∗ ∗ ≥st X1:n . and X1:n implies Xn:n ≥st Xn:n Proof Set λ1 = α11 , . . . , λn = α1n and λ∗1 = α1∗ , . . . , λ∗n = α1∗ . Then, 1
8
n
(λ1 , . . . , λn ) ≽m (λ∗1 , . . . , λ∗n ) and the survival function of Xn:n is, for x > 0, √ )] [ (√ n ∏ x β F Xn:n (x) = 1 − Φ λi − β x i=1 ∫ n a(x;λi ,β) ∏ u2 1 √ e− 2 du = 1− 2π i=1 −∞ (√ √ ) x − βx = λi l(x; β), say. by letting a(x; λi , β) = λi β Upon differentiating with respect to λi , i = 1, . . . , n, we get, for x > 0, )2 ( − 12
a(x;λi ,β)
∂F Xn:n (x) e = − ∫ a(x;λ ,β) u2 l(x; β)FXn:n (x). i ∂λi e− 2 du −∞
So, we obtain
∂F Xn:n (x) ∂F Xn:n (x) − ∂λi ∂λj ( )2 ( )2 1 1 − 2 a(x;λi ,β) [ − 2 a(x;λj ,β) ] e e = l(x; β)FXn:n (x) ∫ a(x;λ ,β) u2 − ∫ a(x;λ ,β) u2 . j i − 2 − 2 e du e du −∞ −∞
From Part (1) of Lemma 2.6, we have ( ) ∂F Xn:n (x) ∂F Xn:n (x) (λi − λj ) − ≥ 0. ∂λi ∂λj
Thus, by using Lemma 2.5, F Xn:n (x) is a Schur-convex function with respect ∗ to (λ1 , . . . , λn ), and so we obtain Xn:n ≥st Xn:n . Similarly, the survival function of X1:n is, for x > 0, ( )] n [ ∏ F X1:n (x) = 1 − Φ a(x; λi , β) =
i=1 n ∫ +∞ ∏ i=1
a(x;λi ,β)
9
u2 1 √ e− 2 du. 2π
Upon differentiating with respect to λi , i = 1, . . . , n, we get, for x > 0, ( )2 − 12
a(x;λi ,β)
∂F X1:n (x) e = − ∫ +∞ ∂λi
2
u √1 e− 2 a(x;λi ,β) 2π
du
l(x; β)F X1:n (x).
So, we obtain
∂F X1:n (x) ∂F X1:n (x) − ∂λi ∂λj ( )2 ( )2 1 1 − 2 a(x;λj ,β) − 2 a(x;λi ,β) [ ] e e = l(x; β)F X1:n (x) ∫ +∞ . − ∫ +∞ u2 u2 √1 e− 2 du √1 e− 2 du a(x;λj ,β) 2π a(x;λi ,β) 2π
From Part (2) of Lemma 2.6, we have ( ) ∂F X1:n (x) ∂F X1:n (x) (λi − λj ) − ≤ 0. ∂λi ∂λj
Thus, by using Lemma 2.5, F X1:n (x) is a Schur-concave function with respect ∗ to (λ1 , . . . , λn ), and so we obtain X1:n ≥st X1:n . Hence, the theorem. Upon using the simple relationship between BS and LBS distributions in (1.1) and the definition of the usual stochastic order, the following results can be readily deduced. Corollary 3.5 Let X1 , . . . , Xn be independent random variables with Xi ∼ LBS(α, βi ), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be independent random variables with Xi∗ ∼ LBS(α, βi∗ ), i = 1, . . . , n. Then, we have: ∗ (1) (β1 , . . . , βn ) ≽w (β1∗ , . . . , βn∗ ) implies Xn:n ≥st Xn:n ; 1 1 1 1 ∗ (2) ( β1 , . . . , βn ) ≽w ( β ∗ , . . . , β ∗ ) implies X1:n ≥st X1:n . n 1 Corollary 3.6 Let X1 , . . . , Xn be independent random variables with Xi ∼ LBS(αi , β), i = 1, . . . , n, and X1∗ , . . . , Xn∗ be(independent ) random ( variables ) with Xi∗ ∼ LBS(αi∗ , β), i = 1, . . . , n. Then, ∗ ∗ ≥st X1:n . and X1:n implies Xn:n ≥st Xn:n
10
1 , . . . , α1n α1
≽m
1 , . . . , α1∗ α∗1 n
Acknowledgments This research was supported by the Provincial Natural Science Research Project of Anhui Colleges (No. KJ2013A137), the National Natural Science Foundation of Anhui Province (No. 1408085MA07), and the PhD research startup foundation of Anhui Normal University (No. 2014bsqdjj34) which facilitated the research visit of the first author to McMaster University, Canada. The research work of the last author was supported by an Individual Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. Our sincere thanks also go to reviewers for their useful comments and suggestions on an earlier version of this manuscript which led to this improved one. References Balakrishnan, N., Haidari, A., Masoumifard, K., 2015. Stochastic comparisons of series and parallel systems with generalized exponential components. IEEE Transactions on Reliability 64, 333-348. Balakrishnan, N., Rao, C.R. (Eds.), 1998a. Handbook of Statistics, Vol. 16Order Statistics: Theory and Methods. North-Holland, Amsterdam. Balakrishnan, N., Rao, C.R. (Eds.), 1998b. Handbook of Statistics, Vol. 17Order Statistics: Applications. North-Holland, Amsterdam. Balakrishnan, N., Zhao, P., 2013. Ordering properties of order statistics from heterogeneous populations: a review with an emphasis on some recent developments. Probability in the Engineering and Informational Sciences 27, 403-469 (with discussions). Birnbaum, Z.W., Saunders, S.C., 1969a. A new family of life distributions. Journal of Applied Probability 6, 319-327.
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