Stochastic comparisons of parallel systems with exponentiated Weibull components

Statistics and Probability Letters xx (xxxx) xxx–xxx

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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

Stochastic comparisons of parallel systems with exponentiated Weibull components Q1

Longxiang Fang a,b,∗ , Xinsheng Zhang b a

Department of Statistics, Anhui Normal University, Wuhu 241000, PR China

b

Department of Statistics, Fudan University, Shanghai 200433, PR China

article

info

Article history: Received 25 March 2014 Received in revised form 21 October 2014 Accepted 27 October 2014 Available online xxxx MSC: 60E15 62N05 62G30 62D05

abstract The exponentiated Weibull (EW) distribution is the first generalization of the twoparameter Weibull distribution to accommodate nonmonotone hazard rates, including the bathtub shaped hazard rate. In this paper, we discuss stochastic comparisons of parallel systems with exponentiated Weibull components in terms of the usual stochastic order, dispersive order and the likelihood ratio order. We give some sufficient conditions for stochastic comparisons between lifetimes of parallel systems. © 2014 Elsevier B.V. All rights reserved.

Keywords: Exponentiated Weibull distribution Parallel systems Usual stochastic order Dispersive order Likelihood ratio order Majorization

1. Introduction

1

Q2

In probability and statistics theory, the two-parameter Weibull distribution has been most commonly used in reliability and life testing, extreme value theory and many other areas due to its distribution with several desirable properties and nice physical interpretations. Recently, Prabhakar Murthy et al. (2004) have studied two-parameter Weibull models in detail. But often, the hazard rate of lifetime data has increasing (decreasing) shape, the bathtub shape or upside-down bathtub shape. Since two-parameter Weibull distribution has monotone hazard rate, it is not a proper statistical model to accommodate nonmonotone hazard rates. To this end, this has lead to the need to seek generalizations of the two-parameter Weibull distribution. The exponentiated Weibull (EW) family of distributions was constructed by Mudholkar and Srivastava (1993) by using exponentiation on the two-parameter Weibull family of distributions. That is, the EW distribution with three parameters is a generalization of the commonly known two-parameter Weibull distribution. And the EW distribution is quite adequate for modeling non-monotone failure rates, including the bathtub shaped hazard rate, which are quite common in reliability and biological studies. It has been shown in the literature that the EW distribution has significantly better fit than traditional models based on the exponential, gamma, Weibull and log-normal distributions in many cases.

∗

Corresponding author at: Department of Statistics, Anhui Normal University, Wuhu 241000, PR China. E-mail address: [email protected] (L. Fang).

http://dx.doi.org/10.1016/j.spl.2014.10.017 0167-7152/© 2014 Elsevier B.V. All rights reserved.

2 3 4 5 6 7 8 9 10 11 12 13

2

1 2

L. Fang, X. Zhang / Statistics and Probability Letters xx (xxxx) xxx–xxx

A random variable X is said to have the exponentiated Weibull (EW) distribution if its cumulative distribution function is given by



3

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

40

β

F (x; α, β, λ) = 1 − e−(λx)

α

,

x > 0, α > 0, β > 0, λ > 0,

(1.1)

Here, α and β are shape parameters and λ is a scale parameter, respectively. EW (β, α, λ) would be used to denote a EW distribution. From (1.1), we know that the EW distribution includes many distributions as special cases. If α = 1, it reduces to the standard two-parameter Weibull distribution. If β = 1, it reduces to the generalized exponential(EE) distribution due to Gupta and Kundu (1999). If α = 1 and β = 1, it becomes the one-parameter exponential distribution. The particular case for β = 2 is the Burr type X distribution studied by Sartawi and Abu-Salih (1991), Aludaat et al. (2008) and among others. The particular case for β = 2 and α = 1 is the Rayleigh distribution. The detailed review of some basic properties of the EW distribution is contained in Nadarajah and Kotz (2006) and Nadarajah et al. (2013). We give two applications of the EW distribution as the following: (i) Construct a parallel system consisting of n independent and identically distributed components with common twoparameter Weibull distribution EW (β, 1, λ). Then, the lifetime of the system has a EW distribution EW (β, n, λ). In other words, the family of EW distributions is closed under maxima, that is, if X1 , . . . , Xn are independent random variables n with EW (β, αi , λ), i = 1, . . . , n, then max(X1 , . . . , Xn ) also has a EW distribution EW (β, i=1 αi , λ). Therefore, the EW distribution is closed under parallel structure with respect to the shape parameter α while the Weibull distribution which is closed under series structure. (ii) Let X1 , . . . , Xn be independent Weibull random variables such that X1 , . . . , Xp with EW (β1 , 1, λ1 ) and Xp+1 , . . . , Xn with EW (β2 , 1, λ2 ). This model is called as multiple-outlier Weibull model. Note that max(X1 , . . . , Xn ) = max(Y1 , Y2 ), where Y1 and Y2 are independent random variables such that Y1 ∼ EW (β1 , p, λ1 ) and Y2 ∼ EW (β2 , n − p, λ2 ). Thus, a parallel system with the multiple-outlier Weibull components can be considered as a parallel system with two heterogeneous EW components. Let X1 , . . . , Xn be n random variables and let Xi:n denote their ith order statistic. In reliability theory, k-out-of-n systems consisting of n components work as long as at least k components are working. The survival function of a k-out-of-n system is the same as that of the (n − k + 1)th order statistic Xn−k+1:n of a set of n random variables. In particular, the parallel systems are 1-out-of-n systems. Thus, the study of parallel systems is equivalent to the study of the largest order statistics. For k-out-of-n systems or (n − k + 1)-out-of-n systems, order statistics have been studied in the literature, for example, see Gurlerand Bairamov (2009), Gurler and Capar (2011) and Gurler (2012). Generally, the results of stochastic comparisons of Q3 order statistics can be seen in Balakrishnan et al. (2014), Khaledi and Kochar (2000, 2002), Khaledi et al. (2011), Kochar and Xu (2009a,b), Kochar (2012), Misra et al. (2011) and so on. In this paper, we will investigate stochastic comparisons of parallel systems with exponentiated Weibull components. Some recent results on the EW distribution can be found in the literature of Gurler and Capar (2011), Mudholkar and Srivastava (1993), Mudholkar et al. (1995) and Shen et al. (2009). To continue our discussion, we need definitions of some stochastic orders and the concept of majorization which is given in Section 2. In this paper, we compare the lifetime of two parallel systems with independent EW components. Firstly, let X1 , . . . , Xn be independent random variables with Xi ∼ EW (βi , α, λ), i = 1, . . . , n. Let X1∗ , . . . , Xn∗ be independent random variables with Xi∗ ∼ EW (βi∗ , α, λ), i = 1, . . . , n. We will show that Xn:n ≥st Xn∗:n under the condition of (β1 , . . . , βn ) ≽m (β1∗ , . . . , βn∗ ). Secondly, let X1 , . . . , Xn and X1∗ , . . . , Xn∗ be two independent EW random variables with common shape parameter β and varying other two parameters. We consider stochastic comparisons in terms of the usual stochastic order, dispersive order and the likelihood ratio order. 2. Preliminaries

45

We first introduce some notations. Suppose that the random variables X and Y have distribution functions F (x) and G(x), density functions f (x) and g (x), the survival functions F (x) = 1 − F (x) and G(x) = 1 − G(x), respectively. And we define the quantile functions as F −1 (p) = inf{x : F (x) ≥ p} and G−1 (p) = inf{x : G(x) ≥ p}, for all real values p ∈ (0, 1). In this paper, all the random variables are assumed to be nonnegative and having support (0, +∞). Let R = (−∞, +∞), R+ = (0, +∞), Rn = {(x1 , . . . , xn ) : xi ∈ R for all i}.

46

Definition 2.1. Let X and Y be two nonnegative random variables having support (0, +∞),

41 42 43 44

47 48 49 50 51 52 53 54

(i) X is said to be less dispersive than Y if F −1 (β) − F −1 (α) ≤ G−1 (β) − G−1 (α), for 0 < α ≤ β < 1, in symbols, Y ≥disp X . g (x) (ii) X is said to be smaller than Y in the likelihood ratio order if f (x) is increasing in x ∈ R+ , in symbols, Y ≥lr X . (iii) X is said to be smaller than Y in the usual stochastic order if F (x) ≥ G(x) for all x; in symbols, Y ≥st X . For a comprehensive discussion on various stochastic orders, one may refer to Shaked and Shanthikumar (1994). Majorization is a very interesting topic in statistics, which is a pre-ordering on vectors by sorting all components in increasing order.

L. Fang, X. Zhang / Statistics and Probability Letters xx (xxxx) xxx–xxx

3

Definition 2.2. Let λ = (λ1 , λ2 , . . . , λn ), λ∗ = (λ∗1 , λ∗2 , . . . , λ∗n ) denote two n-dimensional real vectors. Let λ(1) ≤ λ(2) ≤ · · · ≤ λ(n) , λ∗(1) ≤ λ∗(2) ≤ · · · ≤ λ∗(n) be their ordered components. λ∗ ∈ Rn is said to be majorized by λ ∈ Rn , in symbols

λ ≽m λ∗ , if

j

j

i=1 λ(i) ≤

∗ i=1 λ(i) ; for j = 1, 2, . . . , n − 1, and

i =1 λ i =

n

∗ i=1 λi .

n

4 5

Lemma 2.1 (Marshall and Olkin, 1979, page 57). A permutation-symmetric differentiable function φ(X ) is Schur-concave (Schurconvex) if and only if

(Xi − Xj )

∂φ(X ) ∂φ(X ) − ∂ Xi ∂ Xj



2 3

For more details on majorization and their applications, the reader may refer to Marshall and Olkin (1979). The following six useful results will be needed to prove our main results.



1

≤ 0(≥ 0),

6 7

8

for all i ̸= j.

9

Lemma 2.2 (Khaledi and Kochar, 2000). For t > 0, the functions

1−e−t t

t 2 e−t

and (1−e−t )2 are both decreasing.

10

Lemma 2.3. Let function h : (0, ∞) × (0, ∞) → (0, ∞) be defined as t −yt

ye

h(y, t ) =

ln y

1 − e−y

t

11

.

12

Then, for each y > 0, h(y, t ) is decreasing with respect to t.

13

Proof. For each fixed y > 0, we have

14

2 t −2yt

∂ h(y, t ) (ln y) y e t t (ey − 1 − yt ey ). = ∂t (1 − e−yt )2

15

t

t

It is easy to verify that the function ex − 1 − xex < 0 for any x > 0. So, we get ey − 1 − yt ey < 0. That is, t > 0, which implies that h(y, t ) is decreasing with respect to t.

∂ h(y,t ) ∂t

< 0 for any

Lemma 2.4. For x > 0, β > 0, the function

w(t ) =

e

txβ

16 17

18

β txβ

− 1 − tx e (etxβ − 1)2

19

is increasing in t > 0.

20

Proof. Taking the derivative of w(t ) with respect to t, we have

21

w ′ (t ) =

β txβ

x e β etx

(

− 1)

β

3

β

[txβ etx + txβ − 2etx + 2]. β

22

β

Let us define the function v(t ) = txβ etx + txβ − 2etx + 2, t > 0. Then, 2β txβ

v ′ (t ) = tx e

β txβ

+ xβ − x e ,

24

and we have

25

2β txβ

v (t ) = x e ′′

3β txβ

+ tx e

2β txβ

−x e

3β txβ

= tx e

> 0.

26

It is easy to see that, for t > 0,

27

v (t ) > 0 H⇒ v (t ) > v (0) = 0 H⇒ v(t ) > v(0) = 0 H⇒ w (t ) > 0, ′′

′

′

′

which implies that w(x) is increasing in t > 0.

 n  



n

pi

i =1

i=1

pi ai bi

 ≥

n

 i =1

p i ai

 n 

28 29

Lemma 2.5 (Mitrinović et al., 1993, P. 240). Let a = (a1 , . . . , an ), b = (b1 , . . . , bn ) are two non-increasing sequences, and p = (p1 , . . . , pn ) is a non-negative sequence, then



23

30 31

 p i bi

,

32

i=1

with equality holding if and only if at least one of the sequences a or b is constant.

33

Lemma 2.6. For x > 0, β > 0, the function k(t ) =

34

Proof. From Lemma 2.4, the proof is obvious.

t β etx −1

is convex in t > 0.

35

4

1

2 3 4

5 6 7

L. Fang, X. Zhang / Statistics and Probability Letters xx (xxxx) xxx–xxx

3. Main results In this section, we provide some comparison results on the lifetimes of parallel systems arising from independent heterogeneous EW random variables. The following result considers the comparison on the lifetimes of parallel systems in terms of the usual stochastic order with respect to the shape parameter β . Theorem 3.1. Let X1 , . . . , Xn be independent random variables with Xi ∼ EW (βi , α, λ), i = 1, . . . , n. Let X1∗ , . . . , Xn∗ be independent random variables with Xi∗ ∼ EW (βi∗ , α, λ), i = 1, . . . , n. Then (β1 , . . . , βn ) ≽m (β1∗ , . . . , βn∗ ) implies that Xn:n ≥st Xn∗:n . Proof. We prove that Xn:n ≥st Xn∗:n , it is sufficient to prove n  

FXn:n (x; α, β, λ) =

8

1−e

−(λx)βi

α

i=1 9 10

is Schur-concave function with respect to (β1 , . . . , βn ) for x > 0. Note that, β

11

12 13

∂ FXn:n (x; α, β, λ) α(λx)βi ln(λx)e−(λx) i = FXn:n (x; α, β, λ) · . β ∂βi 1 − e−(λx) i Since the function FXn:n (x; α, β, λ) is permutation symmetric, each partial derivative has the same structure. From Lemma 2.3, we have

  ∂ FXn:n (x; α, β, λ) ∂ FXn:n (x; α, β, λ) − (βi − βj ) ∂βi ∂βj  βj  β βi (λx) ln(λx)e−(λx) i (λx)βj ln(λx)e−(λx) = α FXn:n (x; α, β, λ)(βi − βj ) − βj β 1 − e−(λx) i 1 − e−(λx) = α FXn:n (x; α, β, λ)(βi − βj )[h(λx, βi ) − h(λx, βj )] ≤ 0.

14

15

16 17 18 19

20 21

22

23 24 25

26

Thus, using Lemma 2.1, FXn:n (x; α, β, λ) is Schur-concave in β. So, we have Xn:n ≥st Xn∗:n holds. The following two results discuss the variability ordering of lifetimes of parallel systems with independent heterogeneous EW components in terms of the dispersive order. Theorem 3.2. Let X1 , . . . , Xn be independent random variables with Xi ∼ EW (β, αi , λi ), i = 1, . . . , n. Let X1∗ , . . . , Xn∗ be in-

ˆ i = 1, . . . , n. Let γ = dependent random variables with Xi∗ ∼ EW (β, αi∗ , λ), n

i=1

αi

(λi ) γ , or (2) β > 1, λˆ ≥

n

i=1

α β i (λ ) γ i

β−1

λmin

n

i=1

28

fXn:n (x; α, β, λ) ≤ fXn∗:n (FX−∗1 FXn:n (x; α, β, λ)).

β β n  βαi λ xβ−1 e−(λi x)

i

β

1 − e−(λi x)

   αγi  β−β 1 n   −(λi x)β  − γ β λ − ln 1 − 1−e i=1

  αγi  − γ1 n  n  −(λi x)β −(λi x)β αi · 1− 1−e (1 − e ) ≤ 0. i =1

We denote the left-hand side of (3.3) by η(x). Note that

 − ln 1 −

n  

1−e

−(λi x)β

 αγi 

β

≥ λmin xβ .

i=1

33

(3.2)

Thus, we can see after some simplifications that (3.2) is equivalent to

i=1

32

αi∗ . If (1) 0 < β ≤ 1, λˆ ≥

densities fXn:n (x; α, β, λ) and fXn∗:n (x; α∗ , β,  λ). We will prove Xn:n ≥disp Xn∗:n by proving, for all x > 0,

29

31

i=1

Proof. Let FXn:n (x; α, β, λ) and FXn∗:n (x; α∗ , β,  λ) denote the distribution functions of Xn:n and Xn∗:n with corresponding

i =1

30

n

holds, we have Xn:n ≥disp Xn∗:n .

n:n

27

αi =

Therefore,

η(x) ≤ β x

β−1

 n i=1

  αγi  β β n  1 αi λi e−(λi x) β− 1  − γ λ · λ − 1 . min β −(λi x)β 1 − e−(λi x) i=1 1 − e

(3.3)

L. Fang, X. Zhang / Statistics and Probability Letters xx (xxxx) xxx–xxx

Denote

β

β

αi λi e−(λi x)

n

i=1

β−1 − γ λ · λmin

β 1−e−(λi x)





n

i =1

1 β 1−e−(λi x)

 αγi

 −1

5

, σ (x). In order to verify (3.3) to hold, we must prove

σ (x) ≤ 0 for ∀x > 0. Note that limx→+∞ σ (x) = 0. So, it is sufficient to show σ (x) is non-decreasing in x by proving σ (x) ≤ 0 for ∀x > 0. Differentiating σ (x) with respect to x, we have σ ′ (x) = −

β 2β n  βαi λ xβ−1 e−(λi x)

β−1

+ λ · λmin

i

(1 − e−(λi x)β )2

i=1

ˆ≥ (1) If 0 < β ≤ 1, λ ˆ≥ (2) If β > 1, λ

n

n

i=1

i=1 α β i (λ ) γ i

β−1

λmin

n   i=1

 αγi  β  β n βαi λi xβ−1 e−(λi x) . · β β 1 − e−(λi x) 1 − e−(λi x) i=1 1

(3.4)

αi α  β i β−1 (λi ) γ , we have  λ · λmin ≥ ni=1 (λi ) γ . α  β−1 β i , we have  λ · λmin ≥ ni=1 (λi ) γ .

σ ′ (x) ≥ β x−β−1

i=1

−(λi x)β 

αi (λi x)β e β 1 − e−(λi x)

γ n  i=1

 β

αi (1−e−(λi x) ) (λi x)β

γ

n  αi (λi x) e β 1 − e−(λi x) i=1

−(λi x)β

(λi x)β

i=1

4

5

β  n  αi (λi x)2β e−(λi x) − . −(λi x)β )2 i=1 (1 − e

8 9

10

11

n  αi (1 − e

≥

3

7

From Lemmas 2.2 and 2.5, it is easy to see β −(λi x)β

2

6

Since the weighted geometric mean is always greater than or equal to the weighted harmonic mean, from (3.4) and above (1)-(2), we have

 n

1

2β −(λi x)β

n αi (λi x) e )  · . −(λi x)β )2 i=1 (1 − e

(3.5)

By (3.5), we easily get σ ′ (x) ≥ 0. This proves that σ (x) is non-decreasing in x.

13

Theorem 3.3. Let X1 , . . . , Xn be independent random variables with Xi ∼ EW (β, αi , λi ), i = 1, . . . , n. Let X1∗ , . . . , Xn∗ be

˜ i = 1, . . . , n. Let γ = independent random variables with Xi∗ ∼ EW (β, αi∗ , λ), n

˜≥ Xn:n ≥disp Xn∗:n H⇒ Xn:n ≥st Xn∗:n H⇒ λ



12

n

i =1

αi =

n

i=1

αi∗ . Then

14 15

αi

(λi ) γ .

16

i=1

Proof. We all know that the dispersive order implies the usual stochastic order for non-negative random variables. So,

17

(λi ) . For FXn:n (x; α, β, λ) and

18

Xn:n ≥disp Xn∗:n ∗ ∗

Xn:n ≥st Xn∗:n

H⇒ is obvious. Now, we prove that FXn:n (x; α , β,  λ), by using Taylor expansion, we have FXn:n (x; α, β, λ) = (xβ )γ

n  

β

λi +

o(xβ )

i =1

αi

xβ

Xn:n ≥st Xn∗:n

˜≥ implies λ

n

i=1

αi γ

19

,

(3.6)

and

20

21

β

o( x )  β FXn∗:n (x; α∗ , β,  λ) = (1 − e−(λx) )γ = (xβ )γ λ˜ β + β x



γ

.

(3.7)

If Xn:n ≥st Xn∗:n holds, from (3.6) and (3.7), we get the following result for all x > 0, n



β

λi +

i =1

o(xβ )

αi

xβ



≤ λ˜ β +

o(xβ ) xβ

γ

22

23

.

24

Thus,

25

lim

x→0+

n   i =1

β

λi +

o(xβ )

αi

xβ

˜ β )γ ≥ So, we can obtain (λ



≤ lim λ˜ β + x→0+

n

i =1

o(xβ )

γ

xβ

β (λi )αi , that is λ˜ ≥

n

i=1

.

26

αi

(λi ) γ .

27

⇐⇒ λ˜ ≥

αi γ

From Theorems 3.2 and 3.3, we can obtain that when 0 < β ≤ 1. i=1 (λi ) Lastly, we get some new results on the lifetimes of parallel systems in terms of the likelihood ratio order. Xn:n ≥disp Xn∗:n

n

28 29

6

1 2 3 4

5

6

L. Fang, X. Zhang / Statistics and Probability Letters xx (xxxx) xxx–xxx

Theorem 3.4. Let X1 , . . . , Xn be independent random variables with Xi ∼ EW (β, αi , λi ), i = 1, . . . , n. Let X1∗ , . . . , Xn∗ be

¯ i = 1, . . . , n. Let γ = independent random variables with Xi∗ ∼ EW (β, αi∗ , λ), implies that Xn:n ≥lr Xn∗:n .

8 9

10

12

13

14

15

16



n 

fXn:n (x; α, β, λ)

¯ fXn∗:n (x; α∗ , β, λ)

=

αi =

n

i =1

 αi∗ . Then λ¯ = ( ni=1

αi β β1 λ ) γ i

1 − e−(λi x)

αi

β

β

n 

αi λi

(λi x)β −1 i =1 e γ λ¯ β ¯ β e(λx) −1

i=1

(1 − e−(λ¯ x)β )γ

= δ(x)φ(x).

Firstly, we prove δ(x) is increasing in x > 0. The derivative of δ(x) is

δ ′ (x) = γ β xβ−1



β

1 − e−(λi x)

αi 

i =1

(1 − e−(λ¯ x)β )γ

n 1 

γ

i =1

 β αi λi λ¯ β − . β β e(λi x) − 1 e(λ¯ x) − 1

From Lemma 2.6, we immediately get δ (x) > 0 for all x > 0. Secondly, we prove φ(x) is also increasing in x > 0. The derivative of φ(x) is ′

β xβ−1

φ ′ ( x) =

 n

γ λ¯ β

¯ β e(λx) −1 11

i =1

Proof. For x > 0, the ratio of the density functions of Xn:n and Xn∗:n is

n  7

n

i =1

β

αi λi β (λ i e x) − 1

¯ β λ¯ β e(λx)



 −

β

e(λ¯ x) − 1

 n i =1

β

2β

αi λi e(λi x) (e(λi x)β − 1)2

 .

From Lemmas 2.2 and 2.5, we have

φ ′ ( x) ≥

β xβ−1

 n

γ λ¯ β ¯ β e(λx) −1

i=1

β

2β

αi λi e−(λi x) (1 − e−(λi x)β )2



β n 1  αi (1 − e−(λi x) )

β

γ

λi

i=1

λ¯ β 1 − e−(λ¯ x)

β

 −1 .

Denote

ν(x) =

β n 1  αi (1 − e−(λi x) )

γ

β

λi

i=1

¯ β

−

1 − e−(λx)

λ¯ β

.

By the weighted arithmetic–geometric mean inequality, for x > 0, we have

ν (x) = β x ′

β−1



n 1 

γ

αi e

−(λi x)β

−(λ¯ x)β

−e



≥ 0,

i=1

17

that is, ν(x) is increasing. Note that ν(0) = 0, we have ν(x) ≥ 0 for x > 0. Thus, we have proved that φ(x) is increasing in

18

¯ =( x > 0. So, when λ

19

Acknowledgments

20 21 22

23

24 25 26 27 28 29 30 31 32 33

n

αi

i=1 γ

β

1

λi ) β holds,

fXn:n (x;α,β,λ) ¯ fX ∗ (x;α∗ ,β,λ) n:n

is increasing in x > 0, that is Xn:n ≥lr Xn∗:n .

This research is supported by the National Natural Science Foundation of China (No. 11071045); the National Statistical Science Research Project of China (No. 2012LY158); the Provincial Natural Science Research Project of Anhui Colleges (No. Q4 KJ2013A137), the Natural Science Foundation of Anhui Province (No. 1408085MA07). References Aludaat, K.M., Alodat, M.T., Alodat, T.T., 2008. Parameter estimation of Burr type X distribution for grouped data. Appl. Math. Sci. 2, 415–423. Balakrishnan, N., Haidari, A., Masoumifard, K., 2014. Stochastic comparisons of series and parallel systems with generalized exponential components. IEEE Trans. Reliab. 99, 1–16. Gupta, R.D., Kundu, D., 1999. Generalized exponential distributions. Aust. N. Z. J. Stat. 41, 173–188. Gurler, S., 2012. On residual lifetimes in sequential (n − k + 1)-out-of-n systems. Statist. Papers 51, 1–9. Gurler, S., Bairamov, I., 2009. Parallel and k-out-of-n: G systems with nonidentical components and their mean residual life functions. Appl. Math. Model. 33, 1116–1125. Gurler, S., Capar, S., 2011. An algorithm for mean residual life computation of (n − k + 1)-out-of-n systems: an application of exponentiated Weibull distribution. Appl. Math. Comput. 217, 7806–7811. Khaledi, B., Farsinejad, S., Kochar, S.C., 2011. Stochastic comparisons of order statistics in the scale model. J. Statist. Plann. Inference 141, 276–286. Khaledi, B., Kochar, S.C., 2000. Some new results on stochastic comparisons of parallel systems. J. Appl. Probab. 37, 1123–1128. Khaledi, B., Kochar, S.C., 2002. In: Misra, J.C. (Ed.), Stochastic Orderings Among Order Statistics and Sample Spacings. Uncertainty and Optimality — Probability, Statistics and Operations Research. World Scientific Publications, Singapore, pp. 167–203. Kochar, S.C., 2012. Stochastic comparisons of order statistics and spacings-a review. ISRN Probab. Stat. 47. Article ID 839473.

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