Characterization of MRL order of fail-safe systems with heterogeneous exponential components

Journal of Statistical Planning and Inference 139 (2009) 3027 -- 3037

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Characterization of MRL order of fail-safe systems with heterogeneous exponential components Peng Zhaoa,∗ , N. Balakrishnanb a b

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1

A R T I C L E

I N F O

Article history: Received 27 October 2008 Received in revised form 17 February 2009 Accepted 17 February 2009 Available online 26 February 2009 MSC: primary 60E15 secondary 60K10 Keywords: Stochastic order Hazard rate order Reversed hazard rate order

A B S T R A C T

Let X1 , . . . , Xn be independent exponential random variables with Xi having hazard rate i , i = 1, . . . , n, and Y1 , . . . , Yn be another independent random sample from an exponential distribution with common hazard rate . The purpose of this paper is to examine the mean residual life order between the second order statistics X2:n and Y2:n from these two sets of variables. It is proved that X2:n is larger than Y2:n in terms of the mean residual life order if and only if



n(n − 1)

(2n − 1)  n 1 i=1

i

−

n−1

,



 where  = ni=1 i and i =  − i . It is also shown that X2:n is smaller than Y2:n in terms of the mean residual life order if and only if



min1  i  n i . n−1

These results extend the corresponding ones based on hazard rate order and likelihood ratio order established by Paltanea [2008. On the comparison in hazard rate ordering of fail-safe systems. Journal of Statistical Planning and Inference 138, 1993–1997] and Zhao et al. [2009. Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables. Journal of Multivariate Analysis 100, 952–962], respectively. © 2009 Elsevier B.V. All rights reserved.

1. Introduction It is well known that order statistics play a very important role in statistical inference, life testing, reliability theory, and many other areas. Let X1:n  X2:n  · · ·  Xn:n denote the order statistics from random variables X1 , X2 , . . . , Xn . Then, the k-th order statistic Xk:n is the lifetime of a (n − k + 1)-out-of-n system, which is a very popular structure of redundancy in fault-tolerant systems that has been used extensively in industrial and military systems. In particular, Xn:n and X1:n correspond to the lifetimes of parallel and series systems, respectively. Order statistics have been studied quite extensively in the case when the observations are independent and identically distributed (i.i.d.). However, in some practical situations, observations are non-i.i.d. Due to the complicated form of distributions of order statistics in the non-i.i.d. case, only limited results are found in the literature. Interested readers may refer to David and Nagaraja (2003) and Balakrishnan and Rao (1998a, b) for comprehensive discussions

∗ Corresponding author. E-mail addresses: [email protected] (P. Zhao), [email protected] (N. Balakrishnan). 0378-3758/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2009.02.006

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on this topic, and the recent review article of Balakrishnan (2007) for an elaborate review of developments on the independent and non-identically distributed (i.ni.d.) case. Because of its nice mathematical form and the unique memoryless property, the exponential distribution has been widely used in statistics, reliability, operations research, life testing, and some other applied fields. Readers may refer to Barlow and Proschan (1975) and Balakrishnan and Basu (1995) for an encyclopedic treatment to developments on the exponential distribution. Here, we will focus on the second order statistics, viz., the lifetimes of the (n − 1)-out-of-n systems, which are commonly referred to as fail-safe systems; see Barlow and Proschan (1965). Pledger and Proschan (1971) were among the first to carry out a stochastic comparison of order statistics from non-i.i.d. exponential random variables with corresponding ones from i.i.d. exponential random variables. Since then, many researchers have studied this problem with many different goals and viewpoints; see, for example, Proschan and Sethuraman (1976), Kochar and Rojo (1996), Dykstra et al. (1997), Bon and Paltanea (1999, 2006), Khaledi and Kochar (2000, 2002), Kochar and Xu (2007), Paltanea (2008), and Zhao et al. (2009). In the following, we first recall some notions on stochastic orders that are most pertinent to the main results developed in the subsequent sections. Throughout, the term increasing is used for monotone non-decreasing and similarly the term decreasing is used for monotone non-increasing. Definition 1.1. For two random variables X and Y with their densities f , g and distribution functions F, G, respectively, let F¯ = 1 − F and G¯ = 1 − G denote their survival functions. As the ratios in the statements below are well defined: (i) (ii) (iii) (iv)

X is said to be smaller than Y X is said to be smaller than Y X is said to be smaller than Y X is said to be smaller than Y ∞ t

¯ F(x)dx  ¯F(t)

∞ t

in the likelihood ratio order (denoted by X  lr Y) if g(x)/f (x) is increasing in x. ¯ ¯ F(x) is increasing in x. in the hazard rate order (denoted by X  hr Y) if G(x)/ ¯ ¯  F(x). in the stochastic order (denoted by X  st Y) if G(x) in the mean residual life order (denoted by X  mrl Y) if, for all t  0,

¯ G(x)dx . ¯G(t)

The hazard rate order in (ii) implies both the usual stochastic order in (iii) and the mean residual life order in (iv), but neither the usual stochastic order nor the mean residual life order implies the other. For a comprehensive discussion on stochastic orders, ¨ interested readers may refer to Muller and Stoyan (2002) and Shaked and Shanthikumar (2007). Let X1 , . . . , Xn be independent exponential random variables with Xi having hazard rate i , i = 1, . . . , n, and Y1 , . . . , Yn be a random sample from an exponential distribution with common hazard rate . Bon and Paltanea (2006) then showed that ⎛ Xk:n  st Yk:n

1 ⇐⇒   ⎝ n (k)

⎞1/k

 1  i1 <···
i1 · · · ik ⎠

for 1  k  n. Recently, Paltanea (2008) improved this result partially for the special case k = 2 as follows:   1  i
(1)

2

and X2:n  hr Y2:n ⇐⇒   where i =

n

j=1 j

X2:n  lr Y2:n where (k) =

min1  i  n i , n−1

(2)

− i . Zhao et al. (2009) obtained the corresponding characterization on the likelihood ratio order as follows:

  (3) − (1)(2) 1 , ⇐⇒   lr = 2(1) + 2n − 1 2 (1) − (2)

n

k i=1 i , k = 1, 2, 3,

X2:n  lr Y2:n ⇐⇒  

(3)

and

n

i=1 i

− max1  i  n i . n−1

(4)

In a similar way, in this paper, the corresponding analogues based on the mean residual life order are established. More precisely, it is proved that X2:n  mrl Y2:n ⇐⇒   mrl =

 n(n − 1)

(2n − 1) n 1 n=1

i

−

n−1



,

(5)

P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 139 (2009) 3027 -- 3037

where  =

n

i=1 i

3029

and i =  − i , and

X2:n  mrl Y2:n ⇐⇒  

min1  i  n i . n−1

(6)

Thus, the two results in (5) and (6) form nice extensions of those in (1) and (2) due to Paltanea (2008) based on hr-order, and in (3) and (4) due to Zhao et al. (2009) based on lr-order. 2. Preliminaries In this section, we first recall some basic definitions and notions and then present some useful lemmas. 2.1. Majorization and related orders Definition 2.1. Let {x(1)  · · ·  x(n) } be an increasing arrangement of the components of x = (x1 , . . . , xn ). Then: m

(i) A vector x = (x1 , . . . , xn ) ∈ Rn is said to majorize another vector y = (y1 , . . . , yn ) ∈ Rn (written as x  y) if j 

j 

x(i) 

i=1

and

for j = 1, . . . , n − 1

y(i)

i=1

n

i=1 x(i)

=

n

i=1 y(i) .

w

n

(ii) A vector x ∈ R is said to weakly majorize another vector y ∈ Rn (written as x  y) if j 

x(i) 

i=1

j 

for j = 1, . . . , n.

y(i)

i=1 p

(iii) A vector x ∈ Rn+ is said to be p-larger than another vector y ∈ Rn+ (written as x  y) if j 

x(i) 

i=1

j 

y(i)

for j = 1, . . . , n.

i=1 m

p

w

Functions that preserve the majorization order are said to be Schur-convex. Clearly, x  y implies x  y, and x  y is equivalent w

to log(x)  log(y), where log(x) is the vector of logarithms of the coordinates of x. Also, Khaledi and Kochar (2002) showed that p

m

p

x  y implies x  y for x, y ∈ Rn+ . The converse is, however, not true. For example, (1, 5.5) (2, 3), but clearly the majorization order does not hold. For more details on majorization and p-larger orders and their applications, see Marshall and Olkin (1979), Bon and Paltanea (1999), and Khaledi and Kochar (2002). rm

Definition 2.2. The vector x in Rn+ is said to reciprocal majorize another vector y (written as x  y) if j j   1 1  x(i) y(i) i=1

i=1

for j = 1, . . . , n. In the special case when n = 2, the following implication holds: p

m

rm

(a1 , a2 ) (b1 , b2 ) ⇒ (a1 , a2 ) (b1 , b2 ) ⇒ (a1 , a2 )  (b1 , b2 ) for any two non-negative vectors (a1 , a2 ) and (b1 , b2 ). We give a short explanation for the second implication. Without loss of p

generality, let us assume that a1  a2 and b1  b2 . From the definition of the  order, we have a1  b1 and a1 a2  b1 b2 , which implies that there exists some a2 such that a2  a2 and a1 a2 = b1 b2 . Since (1/a1 ) + 1/a2  (1/a1 ) + 1/a2 , it is enough to show that (1/a1 ) + 1/a2  (1/b1 ) + 1/b2 , which is actually equivalent to showing (1/a1 ) + a1 /(b1 b2 )  (1/b1 ) + 1/b2 . This inequality is readily rm

p

seen to be true by observing that (a1 − b1 )(a1 − b2 )  0. On the other hand, the  order does not imply the  order. For example, rm

rm

p

from the definition of the  order, it follows that (1, 4)  ( 43 , 2), but clearly the  order does not hold between these two vectors.

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2.2. Some useful lemmas The following lemma, due to Marshall and Olkin (1979), will be used to establish Lemmas 2.4 and 2.5 below. Lemma 2.3 (Marshall and Olkin, 1979, p. 57). Let I ⊂ R be an open interval and let : In −→ R be continuously differentiable. Then, necessary and sufficient conditions for  to be Schur-convex on In , are that for all i  j,   (zi − zj )

* * (z) − (z)  0 *zi *zj

for all z ∈ In , where */ *zi (z) denotes the partial derivative with respect to its i-th argument. ∗

∗

Let X1 , X2 , Y1 , Y2 be independent exponential random variables with respective hazard rates 1 , 2 , 1 , 2 . Bon and Paltanea (1999) then proved the equivalence p

∗

∗

X1 + X2  hr (  st )Y1 + Y2 ⇐⇒ (1 , 2 ) (1 , 2 ).

(7)

Zhao et al. (2009) further established the equivalence w

∗

∗

X1 + X2  lr Y1 + Y2 ⇐⇒ (1 , 2 ) (1 , 2 ).

(8)

Zhao and Balakrishnan (2009) established and proved the following equivalence, similar to those in (7) and (8), for the mean residual life order. But, the simpler proof presented here was supplied by the referee. Lemma 2.4. Let X1 , X2 be independent exponential random variables with respective hazard rates 1 , 2 , and Y1 , Y2 be another set of ∗ ∗ independent exponential random variables with respective hazard rates 1 , 2 . Then, rm

∗

∗

X1 + X2  mrl Y1 + Y2 ⇐⇒ (1 , 2 )  (1 , 2 ). rm

∗

∗

∗

∗

Proof. Let us assume (1 , 2 )  (1 , 2 ). Without loss of generality, we can suppose 1  2 and 1  2 . Denote

2 =

1

∗1

+

1 1

∗2

−

1

.

1 m

∗

∗

Clearly, 2  2 > 0 and (1/ 1 , 1/ 2 ) (1/ 1 , 1/ 2 ). Let Z2 be an exponential random variable with the hazard rate 2 , independent of X1 . Since exponential random variable is IFR and X2  mrl Z2 , we have X1 + X2  mrl X1 + Z2 by using Lemma 2.A.8 of Shaked and m

∗

∗

Shanthikumar (2007). It remains to be shown that X1 + Z2  mrl Y1 + Y2 , under the condition (1/ 1 , 1/ 2 ) (1/ 1 , 1/ 2 ). Thus, based on the expression of the mean residual life of convolution of two independent exponential random variables, we observe that it suffices to prove that the symmetric continuous differentiable function : R2+ → R given by ⎧ 2 1/x y e − x2 e1/y ⎪ ⎪ , ⎨ ye1/x − xe1/y (x, y) = ⎪ ⎪ ⎩ x(2x + 1) , x+1

x  y, x=y

is Schur-convex. For x  y, we have  y2 x2 2 2 + − +y −x −y+x x y .  2 ye1/x − xe1/y



* * (x, y) − (x, y) = *x *y

e(1/x)+1/y

x2 e(1/y)−1/x

− y2 e(1/x)−1/y

Suppose x > y, we have 1/y − 1/x > 0. In this case, from the elementary inequalities et > 1 + t + t2 /2 for t > 0

and et < 1 + t + t2 /2 for t < 0,

we obtain  x2 e(1/y)−1/x − y2 e(1/x)−1/y > x2 +

x2 y

− x − y2 −

y2 x

(x2 − y2 ) +y+

2

1 1 − y x

2 .

P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 139 (2009) 3027 -- 3037

3031

Therefore,

* * (x, y) − (x, y)  0. *x *y Upon applying Lemma 2.3, it now follows that  is a Schur-convex function. Conversely, suppose X1 + X2  mrl Y1 + Y2 . Then,

E(X1 + X2 )  E(Y1 + Y2 ) ⇒

1

+

1

1



2

1

∗1

+

1

∗2

, rm

∗

∗

∗

and a limiting argument (t → ∞), under the condition “  mrl ”, assures 1/   1/ 1 . Therefore, (1 , 2 )  (1 , 2 ). Hence, the lemma.  Lemma 2.5. Let k = (1 , . . . , n ) be a positive vector, and am = (1/n)

am  mrl =

 n(n − 1)

where  =

n

i=1 i

(2n − 1) n 1

i

i=1

−

n−1

n

i=1 i

denote the arithmetic mean of i 's. Then,

 > 0,

(9)



and i =  − i .

Proof. From the n-dimensional function (k) =

n

i=1 1/ i ,

we have, for p < q,

 1  1 * * (k) − (k) = − 2 2 *p *q i  q i i  p i =

p − q p q



1

p

+



1

q

.

This implies that (p − q )(*/(*p )(k)) − */ *q (k))  0, and consequently the function (k) is Schur-convex in k, due to Lemma m

2.3. Since (1 , . . . , n ) (am , . . . , am ) and that a Schur-convex function preserves the majorization order, it immediately follows that (k)  n/((n − 1)am ). Thus, we have

mrl = n(n − 1)

2n − 1  n 1 i=1

i

−

n−1





2n − 1 × n(n − 1)

1 = am . n n−1 − (n − 1)am nam



Before stating the next lemma, let us first recall two well-known classes of life distributions, viz., IFR and IMRL, which will be used in the sequel. For a non-negative random variable X with distribution function F and reliability function F¯ = 1 − F, let Xt = X − t|(X > t) denote the residual lifetime of X at time t  0. Then, X is said to have increasing failure rate (IFR) if its failure rate ¯ f (x)/ F(x) is increasing in x  0; X is said to have increasing mean residual life (IMRL) if E(Xs )  E(Xt ) for all t  s  0. For more details on these classes of life distributions, one may refer to Barlow and Proschan (1975) and Marshall and Olkin (2007). The next lemma is essential for proving the main results presented in Section 3. Lemma 2.6. Let X be an exponential random variable with hazard rate a. Let Y be a mixture of exponential random variables, with   distribution function FY = ni=1 pi Fai such that pi > 0 and ni=1 pi = 1, where Fai is the distribution function of an exponential random variable with hazard rate ai > 0. Then, X  mrl Y ⇐⇒ a  

1

n pi i=1 ai

(10)

and X  mrl Y ⇐⇒ a  min ai . 1in

(11)

Proof. Based on the fact that, with the assumption of finite means, the class of distributions with IMRL is closed under mixtures (see Marshall and Olkin, 2007, D.5. Proposition, p. 172), the mean residual life function of Y, given by

Y (t) =

1 F¯ Y (t)

 t

∞

n pi −a t e i i=1 ¯FY (u) du =  ai , n −ai t i=1 pi e

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P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 139 (2009) 3027 -- 3037

is increasing in t  0. Thus, we have 1 min1  i  n ai

= lim Y (t)  Y (t)  Y (0) = t→∞

n  pi . ai i=1

If X  mrl Y, it is easy to see that n  pi 1 = Y (0)  X (0) = , ai a i=1

which readily yields a 

1 n pi i=1 ai

.

(12)

Conversely, if (12) holds, then

X (t) =

1 1  = Y (0)  Y (t), a n pi i=1 ai

which means that X  mrl Y. The equivalence in (11) can be established in a similar manner.



3. Main results Theorem 3.1. Let X1 , . . . , Xn be independent exponential random variables with respective hazard rates 1 , . . . , n , and Y1 , . . . , Yn be an independent random sample from an exponential population with common hazard rate . Then, X2:n  mrl Y2:n ⇐⇒   mrl =

 n(n − 1)

2n − 1 n 1 n=1

i

−

n−1

.

(13)



Proof (Necessity). Suppose   mrl . Kochar and Korwar (1996) pointed out that the first sample spacing X1:n is independent of the second sample spacing X2:n − X1:n . Let T1 be an exponential random variable with hazard rate , and T2 be a mixture of  d exponential random variables with distribution function FT2 = ni=1 (i / )Fi , which is independent of T1 . Then, X2:n = T1 + T2 . d

Likewise, Y2:n = U1 + U2 , where U1 and U2 are independent exponential random variables with hazard rates n and (n − 1), respectively. Consider another exponential random variable T3 with hazard rate 1 n i=1

i i

that is independent of T1 . From the equivalence in (10) of Lemma 2.6, it follows immediately that T2  mrl T3 . Since T1 is an IFR random variable, upon applying Lemma 2.A.8 of Shaked and Shanthikumar (2007), we also have T1 + T2  mrl T1 + T3 . Now, for proving the desired result, we need to show that T1 + T3  mrl U1 + U2 . Due to Lemma 2.4, it is then enough to prove the following two statements: ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ 1 min ,  min{n, (n − 1)}, ⎪ n i ⎪ ⎪ ⎪ ⎩ ⎭ i=1

(14)

i

and 1



+

n  i i=1

i



1 1 + . n (n − 1)

(15)

P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 139 (2009) 3027 -- 3037

3033

Note that 1

i i=1 i

n

=

n

1

i=1

i

1 = n−1 1 − −





1 . 2n − 1 1 − n(n − 1)mrl nam

Upon using Lemma 2.5, we also have 1  (n − 1)mrl  (n − 1). 2n − 1 1 − n(n − 1)mrl nam Thus, the inequality in (14) is valid. Next, we observe that 1



+

n  i i=1

=

i

n  1

i

i=1

−

n−1





2n − 1 1 1  + , n(n − 1)mrl n (n − 1)

which is precisely the inequality in (15). (Sufficiency). Suppose X2:n  mrl Y2:n . The mean residual life function of X2:n is given by

X2:n (t) =



1 F¯ X2:n (t)

n 1 − t n − 1 −t e i − e i=1  ¯FX (u) du =  i 2:n n −i t − (n − 1)e−t i=1 e

∞

t

for t  0.

(16)

Upon using Taylor's expansion at the origin, we get

X2:n (t) =

n  1 i=1

i

−

n−1

+ o(1) for t > 0,



and similarly,

Y2:n (t) =

n n−1 2n − 1 − + o(1) = + o(1) for t > 0. (n − 1) n n(n − 1)

So, X2:n  mrl Y2:n implies that n  1 i=1

i

−

n−1



+ o(1) 

2n − 1 + o(1), n(n − 1)

which guarantees that



(2n − 1) n 1

 n(n − 1)

Hence, the theorem.

n=1

i

−

n−1

.





Remark 3.2. Note that the characterizations in (1), (3), and (13) based on hazard rate order, likelihood ratio order, and mean residual life order, respectively, are all under the same setup. Therefore, based on these three characterizations and the fact that the likelihood ratio order implies the hazard rate order which in turn implies the mean residual life order, we obtain the following interesting inequalities between different means:

mrl  hr  lr  am ,

(17)

where am is the arithmetic mean, while mrl , hr , and lr are the corresponding means associated with the mean residual life order, the hazard rate order, and the likelihood ratio order. For example, for the non-negative vector (1 , 2 , 3 ) = (1, 10, 40), we have

mrl ≈ 8.673,

hr ≈ 12.2474,

lr ≈ 15.5667,

am = 17,

which support the inequalities in (17). Theorem 3.3. Let X1 , . . . , Xn be independent exponential random variables with respective hazard rates 1 , . . . , n , and Y1 , . . . , Yn be an independent random sample from an exponential population with a common hazard rate . Then, X2:n  mrl Y2:n if and only if n  − max1  i  n i   ˆ = i=1 i . n−1

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P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 139 (2009) 3027 -- 3037

0.14 0.12 0.1

(t;3,5.5) (t;1,10,40)

0.08

(t;3,8.673)

0.06 (t;3,12.2474)

0.04

(t;3,15.5667)

0

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 1. Plots of mean residual life functions of X2:3 from three exponentials with hazards (1,10,40) and Y2:3 from i.i.d. exponentials with parameter ˆ = 5.5, mrl = 8.673, hr = 12.2474, and lr = 15.5667.

Proof. Paltanea (2008) established that X2:n  hr Y2:n if and only if   ˆ . Since the hazard rate order implies the mean residual life order, it suffices to prove that X2:n  mrl Y2:n implies   ˆ . Now, for Eq. (16), we have, for t > 0, n

1

i=1

X2:n (t) = n

i

e−i t −

− i t i=1 e

n−1



e−t

− (n − 1) e−t

n n − 1 −nt e−(n−1)t − e (n − 1) n  = Y2:n (t). −(n−1)  t −n ne − (n − 1) e t

Letting t → ∞ on both sides of the above inequality, we get lim X2:n (t) =

t→∞

1 1  = lim  (t), min1  i  n i (n − 1) t→∞ Y2:n

yielding the required result that



min1  i  n i = ˆ . n−1



Let (X1 , X2 , X3 ) be a vector of independent exponential random variables with hazard rate vector (1,10,40). Denote by

(t; 1, 10, 40) and r(t; 1, 10, 40) the corresponding mean residual life and hazard rate functions of the second order statistic X2:3 . Let (Y1 , Y2 , Y3 ) be a vector of independent and identical exponential random variables with common hazard rate , and denote by (t; 3, ) and r(t; 3, ) the corresponding mean residual life and hazard rate functions of Y2:3 . Fig. 1 presents the mean residual life functions of X2:3 and Y2:3 when parameter  is taken as ˆ = 5.5, mrl = 8.673, hr = 12.2474 and lr = 15.5667. It can be seen that the best bounds for (t; 1, 10, 40) are (t; 3, mrl ) and (t; 3, ˆ ), with the former being the best approximation near the origin and the latter having the same limit (on the right) as (t; 1, 10, 40). In Fig. 2, we have presented the corresponding hazard rate functions. Clearly, the best bounds for r(t; 1, 10, 40) are r(t; 3, hr ) and r(t; 3, ˆ ), but if  = mrl , we can not compare the hazard rates r(t; 1, 10, 40) and r(t; 3, mrl ). Finally, as a consequence of Theorems 3.1 and 3.3, we also obtain the following corollary, which provides a comparison of the second order statistics in terms of the mean residual life order for the case when both exponential samples are heterogeneous. Corollary 3.4. Let X1 , . . . , Xn be independent exponential random variables with respective hazard rates 1 , . . . , n , and Y1 , . . . , Yn be another independent set of exponential random variables with respective hazard rates 1 , . . . , n . If 2n − 1 n 1

 n(n − 1)

n=1

i

−

n−1





n

i=1 i

− max1  i  n i , n−1

then X2:n  mrl Y2:n . Proof. Let Z1 , . . . , Zn be a random sample of size n from an exponential population with common hazard rate , where n  − max1  i  n i 2n − 1      i=1 i . n 1 n−1 n−1 n(n − 1) − n=1

i



P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 139 (2009) 3027 -- 3037

3035

30 r(t;3,15.5667)

25

r(t;3,12.2474)

20 r(t;3,8.673)

15 r(t;1,10,40)

10 r(t;3,5.5)

5

0

0.1

0.2

0.3

0.4

Fig. 2. Plots of hazard rate functions of X2:3 from three exponentials with hazards (1,10,40) and Y2:3 from i.i.d. exponentials with parameter ˆ = 5.5, mrl = 8.673, hr = 12.2474, and lr = 15.5667.

From Theorems 3.1 and 3.3, it then follows that X2:n  mrl Z2:n  mrl Y2:n , which immediately gives the required result.



4. Some examples In this section, we present some special examples in order to illustrate the performance of the results established in Section 3. Let X1 , . . . , Xn be independent exponential random variables with respective hazard rates 1 , . . . , n , and Y1 , . . . , Yn be an independent random sample from an exponential population with common hazard rate . Example 4.1. The case of two i.i.d. exponential samples is the simplest one. Suppose 1 = · · · = n = . In this case, it is easy to check that

mrl = n(n − 1)

2n − 1  n 1 i=1

i

−

n−1

 =



and

ˆ =

n

i=1 i

− max1  i  n i = . n−1

From Theorems 3.1 and 3.3, it then follows immediately that X2:n  mrl Y2:n ⇐⇒    and X2:n  mrl Y2:n ⇐⇒   . Example 4.2. The case when 1 = · · · = n−1 =  and n ≡ 0   is referred to as a single-outlier model; see Balakrishnan (2007). In this case, it is easy to see (after some simplifications) that

mrl = n(n − 1) =

2n − 1 n 1



i=1

i

−

n−1





(2n − 1){(n − 2) + 0 }{(n − 1) + 0 } 2

n{(n − 1)2  + ((n − 2) + 0 )((n − 1) + 0 )}

;

then, from Theorem 3.1, we have X2:n  mrl Y2:n ⇐⇒   mrl =

(2n − 1){(n − 2) + 0 }{(n − 1) + 0 } 2

n{(n − 1)2  + ((n − 2) + 0 )((n − 1) + 0 )}

.

3036

P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 139 (2009) 3027 -- 3037

Upon utilizing Theorem 3.3, if  > 0 , we have (n − 2) + 0 X2:n  mrl Y2:n ⇐⇒   ˆ = , n−1 and if  < 0 , we have X2:n  mrl Y2:n ⇐⇒   ˆ = . Example 4.3. For n = 2m + 1, let X1 , . . . , Xn be independent exponential random variables with X1 , . . . , Xm having respective hazard rates (1 − i/n), i = 1, . . . , m, Xm+1 having hazard rate , and Xm+2 , . . . , Xn having respective hazard rates {1 + (l − (m + 1))/n}, l = m + 2, . . . , n. In this case, after some calculations, we obtain

mrl =

2n − 1 n 1

 n(n − 1)

= n(n − 1)2

i=1

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

i

m i=1 

−

n−1





(2n − 1)

i n−1+ n

2  n−1−

i n

+

⎫ ⎪ ⎪ 1⎬ n⎪ ⎪ ⎭

and 

ˆ = 1 −

m . n(n − 1)

From Theorems 3.1 and 3.3 once again, we get X2:n  mrl Y2:n ⇐⇒   mrl = n(n − 1)2

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(2n − 1) m i=1 

i n−1+ n

2  n−1−

i n

+

⎫ ⎪ ⎪ 1⎬ n⎪ ⎪ ⎭

and  X2:n  mrl Y2:n ⇐⇒   ˆ = 1 −

m . n(n − 1)

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