Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables

Journal of Multivariate Analysis 100 (2009) 952–962

Contents lists available at ScienceDirect

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

Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variablesI Peng Zhao a , Xiaohu Li a,∗ , N. Balakrishnan b a

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

b

Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1

article

info

Article history: Received 5 February 2008 Available online 24 September 2008 AMS subject classifications: primary 60E15 secondary 60K10 Keywords: Majorization order Weakly majorization order p-larger order Hazard rate order

a b s t r a c t Let X1 , . . . , Xn be independent exponential random variables with respective hazard rates λ1 , . . . , λn , and let Y1 , . . . , Yn be independent exponential random variables with common hazard rate λ. This paper proves that X2:n , the second order statistic of X1 , . . . , Xn , is larger than Y2:n , the second order statistic of Y1 , . . . , Yn , in terms of the likelihood ratio order if and only if



1

λ≥

2Λ1 +

Λ3 − Λ1 Λ2



Λ21

2n − 1 − Λ2 Pn k with Λk = i=1 λi , k = 1, 2, 3. Also, it is shown that X2:n is smaller than Y2:n in terms of the likelihood ratio order if and only if n P

λ≤

λi − max λi 1≤i≤n

i=1

n−1

.

These results form nice extensions of those on the hazard rate order in Pˇaltˇanea [E. Pˇaltˇanea, On the comparison in hazard rate ordering of fail-safe systems, Journal of Statistical Planning and Inference 138 (2008) 1993–1997]. © 2008 Elsevier Inc. All rights reserved.

1. Introduction Order statistics play an important role in statistics, reliability theory, and many applied areas. Let X1:n ≤ X2:n ≤ · · · ≤ Xn:n denote the order statistics of random variables X1 , X2 , . . . , Xn . It is well known that the kth order statistic Xk:n is the lifetime of an (n − k + 1)-out-of-n system, which is a very popular structure of redundancy in fault-tolerant systems and has been applied in industrial and military systems. In particular, the lifetimes of parallel and series systems correspond to the order statistics, Xn:n and X1:n . Order statistics have been extensively investigated 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 expression of the distribution in the non-i.i.d case, only limited results are found in the literature. One may refer to [1–3] for comprehensive discussions on this topic, and the recent review article of [4] for results on the independent and nonidentically distributed (i.ni.d) case. Due to the nice mathematical form and the unique memoryless property, the exponential distribution has widely been applied in statistics, reliability, operations research, life testing and other areas. Readers may refer to [5,6] for an encyclopedic I Supported by the National Natural Science Foundation of China (10771090).

∗

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

0047-259X/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jmva.2008.09.010

P. Zhao et al. / Journal of Multivariate Analysis 100 (2009) 952–962

953

treatment to developments on the exponential distribution. Here, we will focus on the second order statistic, which characterizes the lifetime of the (n − 1)-out-of-n system (referred as the fail-safe system) in reliability theory and yields the winner’s price for the bid in the second price reverse auction; see, for example, [7,8]. Pledger and Proschan [9] were among the first to compare stochastically the order statistics of non-i.i.d exponential random variables with corresponding order statistics from i.i.d exponential random variables. Since then, many researchers followed up this topic including [10–18]. It is also worth noting that some stochastic comparison results on order statistics from two i.ni.d. (not necessarily exponential) samples are closely related to the subject matter of this paper. For instance, [19] proved that the order statistics from two different collections of random variables are ordered in the likelihood ratio order if the underlying random variables are so ordered. Similar results were also given by [20]. For ease of reference, let us first recall some stochastic orders which are closely related to the main results to be developed here in this paper. Throughout, the term increasing stands for monotone non-decreasing and the term decreasing stands for monotone non-increasing. Definition 1.1. For two random variables X and Y with their densities f , g and distribution functions F , G, let F¯ = 1 − F and ¯ = 1 − G. As the ratios in the statements below are well defined, X is said to be smaller than Y in the: G (i) likelihood ratio order (denoted by X ≤lr Y ) if g (x)/f (x) is increasing in x; ¯ (x)/F¯ (x) is increasing in x; (ii) hazard rate order (denoted by X ≤hr Y ) if G ¯ (x) ≥ F¯ (x). (iii) stochastic order (denoted by X ≤st Y ) if G For a comprehensive discussion on stochastic orders, one may refer to [21,22]. Suppose X1 , . . . , Xn are independent exponential random variables with Xi having hazard rate λi , i = 1, . . . , n. Let Y1 , . . . , Yn be a random sample of size n from an exponential distribution with common hazard rate λ. Bon and Pˇaltˇanea [16] showed, for 1 ≤ k ≤ n,

" Xk:n ≥st Yk:n ⇐⇒ λ ≥

# 1k

 n −1

X

k

λi1 . . . λik

.

(1.1)

1≤i1 <···
Recently, [18] further proved

ˇ= X2:n ≥hr Y2:n ⇐⇒ λ ≥ λ

s  n −1 X 2

λi λj

(1.2)

1≤i
and n P

X2:n ≤hr Y2:n ⇐⇒ λ ≤

λi − max λi 1≤i≤n

i=1

.

n−1

(1.3)

These two results partially improve (1.1) in the case with k = 2. This paper investigates the likelihood ratio order instead. In this regard, it is proved that X2:n ≥lr Y2:n with Λk =

Pn

i =1

⇐⇒ λ ≥ λˆ =

1



2n − 1

2Λ1 +

Λ3 − Λ1 Λ2 Λ21 − Λ2

 (1.4)

λki , k = 1, 2, 3. Also, n P

X2:n ≤lr Y2:n ⇐⇒ λ ≤

i =1

λi − max λi 1≤i≤n

n−1

.

(1.5)

These two results form nice extensions of (1.2) and (1.3). Some examples of special cases are given for illustrating the performance of our main results. 2. Preliminaries This section presents some useful lemmas, which are not only helpful for proving our main results in what follows, but are also of independent interest. Definition 2.1. Suppose two vectors x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ). Let x(1) ≤ · · · ≤ x(n) and y(1) ≤ · · · ≤ y(n) be the increasing arrangements of their components, respectively.

954

P. Zhao et al. / Journal of Multivariate Analysis 100 (2009) 952–962 m

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

j X

x(i) ≤

i=1

and

y(i) ,

for j = 1, . . . , n − 1,

i=1

Pn

Pn

x(i) =

i=1

i =1

y(i) ;

w

n

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

j X

x(i) ≤

i=1

y(i) ,

for j = 1, . . . , n;

i=1 p

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

x(i) ≤

j Y

i=1

y(i) ,

for j = 1, . . . , n.

i=1 m

p

w

w

Clearly, x  y implies x  y, and x  y is equivalent to log(x)  log(y). Here log(x) is the vector of logarithms of the m

p

coordinates of x. Khaledi and Kochar [14] proved that x  y implies x  y for x, y ∈ Rn+ . The converse is, however, not true. Those functions that preserve the majorization ordering are said to be Schur-convex. For more discussions on majorization orders and their applications, see [23,13,15]. Boland et al. [24] proved that, for X1 , X2 , Y1 , Y2 , independent exponential random variables with respective hazard rates λ1 , λ2 , λ∗1 , λ∗2 , m

(λ1 , λ2 )  (λ∗1 , λ∗2 ) H⇒ X1 + X2 ≥lr Y1 + Y2 .

(2.1)

As the first result, Theorem 2.2 presents a characterization of X1 + X2 ≥lr Y1 + Y2 and hence improves (2.1). It should be remarked here that [25] built this result in a completely different but lengthy way, and also that it is only a one-way implication. Here, we present a simple and brief proof and summarize it in terms of the weakly majorization order, and moreover it is a two-way implication result. Theorem 2.2. For independent exponential random variables X1 , X2 , Y1 , Y2 with respective hazard rates λ1 , λ2 , λ∗1 , λ∗2 , w

X1 + X2 ≥lr Y1 + Y2 ⇐⇒ (λ1 , λ2 )  (λ∗1 , λ∗2 ). Proof. Without loss of generality, let us assume λ1 ≤ λ2 and λ∗1 ≤ λ∗2 . λ +λ 2 ⇐H If λ∗1 ≥ λ1 +λ , let Z and W be two independent exponential random variables with common hazard rate 1 2 2 . 2 λ1 +λ2

≤ λ∗1 ≤ λ∗2 . Since the exponential distribution has log-concave density, from Theorem 1.C.9 of   m λ1 +λ2 λ1 +λ2 [21], it follows that Z + W ≥lr Y1 + Y2 . On the other hand, we also have  (λ1 , λ2 ). By Example 1.C.50 of , 2 2 [21], we have X1 + X2 ≥lr Z + W . Now, we reach the required result by transitivity. λ +λ If λ∗1 < 1 2 2 , it is easy to observe that Obviously, we have

λ1 ≤ λ∗1 <

2

λ1 + λ2 2

< λ1 + λ2 − λ∗1 ≤ min(λ2 , λ∗2 ).

m

Since (λ1 , λ2 )  (λ∗1 , λ1 + λ2 − λ∗1 ), we have X1 + X2 ≥lr X10 + X20 , where X10 and X20 are independent exponentials having respective hazard rates λ∗1 and λ1 + λ2 − λ∗1 . On the other hand, as discussed above, it also holds that X10 + X20 ≥lr Y1 + Y2 . Hence, we reach the required result again. H⇒ Denote by f(λ1 ,λ2 ) and f(λ∗ ,λ∗ ) the density functions of X1 + X2 and Y1 + Y2 , respectively. If λ1 > λ∗1 , then 1

lim

f(λ1 ,λ2 ) (t )

t →∞ f(λ∗ ,λ∗ ) 1 2

(t )

=

2

λ1 λ2 λ −λ lim 2∗ ∗ 1 t →∞ λ1 λ2 λ∗2 −λ∗1


· e−λ1 t − e−λ2 t  ∗  ∗ · e−λ1 t − e−λ2 t

λ1 λ2 (λ∗2 + λ∗1 ) 1 − e(λ1 −λ2 )t (λ∗ −λ1 )t · ·e 1 ∗ ∗ t →∞ λ∗ λ∗ (λ2 − λ1 ) 1 − e(λ1 −λ2 )t 1 2 = 0. = lim

This contradicts the fact that

f(λ ,λ ) (t ) 1 2 f(λ∗ ,λ∗ ) (t ) 1

2

is increasing in t > 0. Hence, we conclude that λ1 ≤ λ∗1 .

P. Zhao et al. / Journal of Multivariate Analysis 100 (2009) 952–962

955

On the other hand, by Taylor’s expansion at the origin, we have, for t > 0, f(λ1 ,λ2 ) (t ) = λ1 λ2

 t−

λ1 + λ2 2

t

2




+ o t2 ,

and its derivative 0 f(λ (t ) = λ1 λ2 − λ1 λ2 (λ1 + λ2 )t + o(t ). 1 ,λ2 )

As a result, 0 f(λ (t ) λ1 λ2 − λ1 λ2 (λ1 + λ2 )t + o(t ) 1 1 ,λ2 ) = = − (λ1 + λ2 ) + o(1). f(λ1 ,λ2 ) (t ) λ1 λ2 t + o(t ) t

Similarly, 0 f(λ ∗ ,λ∗ ) (t ) 1

2

f(λ∗ ,λ∗ ) 1

2

=

(t )

1 t

− (λ∗1 + λ∗2 ) + o(1).

By X1 + X2 ≥lr Y1 + Y2 , we have, for t > 0, 0 0 f(λ ∗ ,λ∗ ) (t ) (t ) f(λ 1 ,λ2 ) ≥ 1 2 , f(λ1 ,λ2 ) (t ) f(λ∗ ,λ∗ ) (t ) 1

2

and this implies λ1 + λ2 ≤ λ∗1 + λ∗2 .



The following lemma will be used to prove Lemma 2.4. Lemma 2.3 ([23], p. 57). Let I ⊂ R be an open interval. A continuously differentiable function ϕ : I n → R is Schur-convex on I n if and only if, for all i 6= j and z ∈ I n ,

  (zi − zj ) ϕ(i) (z ) − ϕ(j) (z ) ≥ 0, where ϕ(i) (z ) is the partial derivative with respect to its ith argument. TheP lemma below will be useful to prove Pnthe main result in the next section. For brevity, let us first introduce the notation n k λ , k = 1 , 2 , 3 and Λ ( m ) = i =1 i i=1 λi − λm , m = 1, . . . , n.

Λk =

Lemma 2.4. Let λ = (λ1 , . . . , λn ) be a positive vector. Then

¯= where λ

1 n



1

λ¯ ≥ λˆ =

2n − 1

Pn

i=1

2Λ1 +

Λ3 − Λ1 Λ2



Λ21 − Λ2

,

λi is the arithmetic mean of λ0 s.

Proof. The right side of (2.2) may be rewritten as

λˆ =

1



2Λ1 +

Λ3 − Λ1 Λ2



Λ21 − Λ2  n P λ2i Λ(i)   1 2Λ1 − i=1 . = n   P 2n − 1 λi Λ(i) 2n − 1



i =1

To reach the required result, we need to prove

 λˆ =

n P

1

 2Λ1 − i=1 n P 2n − 1 

λ2i Λ(i) λi Λ(i)

i =1

which is in fact equivalent to n n X X λi 1 λi Λ(i) ≥ λi Λ(i). Λ n 1 i =1 i =1

  Λ1 ≤ ¯ = λ,  n

(2.2)

956

P. Zhao et al. / Journal of Multivariate Analysis 100 (2009) 952–962

Note that, for any i 6= j, sgn

λi Λ(i) − λj Λ(j) = (λi − λj )[Λ1 − (λi + λj )] = λi − λj , sgn

and so it suffices to prove that, under the restriction pi − pj = yi − yj for i 6= j, the function g (p1 , . . . , pn ) = Pn Schur-convex in (p1 , . . . , pn ), where i=1 pi = 1 and yi > 0 for i = 1, . . . , n. It is easy to verify that, for i 6= j,

Pn

i=1

pi yi is

 ∂ g (p1 , . . . , pn ) ∂ g (p1 , . . . , pn ) − = (pi − pj )(yi − yj ) ≥ 0. (pi − pj ) ∂ pi ∂ pj 

Then, the desired result follows immediately from Lemma 2.3.



3. Main results In this section, we present the main results. The first one is an analogue of Lemma 2.1 in [18]. Lemma 3.1. Let X be an exponential variable with Prandom Pn hazard rate a and Y be a mixture of exponential random variables, n with the distribution function FY = p F such that i a i =1 i=1 pi = 1 and Fai is the distribution function of a exponential random i variable with hazard rate ai > 0. Then n P

X ≤lr Y ⇐⇒ a ≥

pi a2i

i =1 n

P

(3.1) p i ai

i =1

and X ≥lr Y ⇐⇒ a ≤ min ai .

(3.2)

1≤i≤n

Proof. It is well known that log-convexity is preserved under mixing (see [26]), which implies that the ratio in t ≥ 0. Note that, for t ≥ 0 fY (t )

n P

0

∆Y (t ) =

fY ( t )

=−

pi a2i e−ai t

i =1 n

P

, pi ai e−ai t

i =1

and we have n P

− min ai = lim ∆Y (t ) ≥ ∆Y (t ) ≥ ∆Y (0) = − 1≤i≤n

t →∞

pi a2i

i =1 n

P

. p i ai

i =1

Assume X ≤lr Y , it holds that n P

−

pi a2i

i =1 n

P

= ∆Y (0) ≥ ∆X (0) = −a, p i ai

i =1

which implies the right-hand side of (3.1). Conversely, if the right-hand side of (3.1) holds, then n P

∆X (t ) = −a ≤ −

pi a2i

i =1 n

P

= ∆Y (0) ≤ ∆Y (t ), p i ai

i =1

which is actually an equivalent characterization for X ≤lr Y . Equivalence in (3.2) may be proved in a similar manner and hence is omitted.



fY0 (t ) fY (t )

is increasing

P. Zhao et al. / Journal of Multivariate Analysis 100 (2009) 952–962

957

Theorem 3.2. Let X1 , . . . , Xn be independent exponential random variables with respective hazard rates λ1 , . . . , λn , and Y1 , . . . , Yn be a random sample of size n from an exponential population with hazard rate λ. Then, X2:n ≥lr Y2:n if and only if

with Λk =



1

λ ≥ λˆ =

2n − 1

Pn

i =1

2Λ1 +

Λ3 − Λ1 Λ2



Λ21 − Λ2

λki , k = 1, 2, 3.

Proof. Sufficiency X2:n has its density function as, for t ≥ 0, fX2:n (t ) = −(n − 1)Λ1 e−Λ1 t +

n X

Λ(i)e−Λ(i)t .

i =1

Applying Taylor’s expansion at the origin, we get fX2:n (t ) = −

" n X

# Λ (i) − (n − 1)

Λ21

2

t+

i =1

1 2

" (n − 1)

Λ31

−

n X

# Λ (i) t 2 + o(t 2 ), 3

i =1

and the derivative of the density function fX02:n (t ) = −

" n X

#

"

Λ2 (i) − (n − 1)Λ21 + (n − 1)Λ31 −

i =1

n X

# Λ3 (i) t + o(t ).

i=1

Thus, fX2:n (t ) 0

∆X2:n (t ) =

fX2:n (t )

=

1 t

(n − 1)Λ31 − +

n P

Λ3 (i)

i =1 n P

+ o(1).

Λ2 (i) − (n − 1)Λ21

i=1

Likewise,

∆Y2:n (t ) =

fY02:n (t ) fY2:n (t )

=

1 t

− (2n − 1)λ + o(1).

Since X2:n ≥lr Y2:n implies ∆X2:n (t ) ≥ ∆Y2:n (t ) for all t ≥ 0, we have n P (n − 1)Λ31 − Λ3 (i)   Λ3 − Λ1 Λ2 1 i =1 ˆ   λ≥ 2 Λ + = λ. = 1 n P 2n − 1 Λ21 − Λ2 (2n − 1) (n − 1)Λ21 − Λ2 (i) i=1

Necessity. According to [27], 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 Λ1 , and T2 be a mixture of exponential random variables with distribution

Pn

λ

st

st

i function FT2 = i=1 Λ1 FΛ(i) , which is independent of T1 . Then, X2:n = T1 + T2 . Likewise, Y2:n = U1 + U2 , where U1 and U2 are independent exponential random variables with respective hazard rates nλ and (n − 1)λ. Consider the other exponential random variable T3 with hazard rate

n P

n P

λi Λ2 (i)

i=1 n

P

= λi Λ(i)

i=1

i=1 n

P i =1

λi Λ2 (i) Λ1 λi Λ(i) Λ1

,

which is independent of T1 . From equivalence (3.1) in Lemma 3.1, it follows immediately that T2 ≥lr T3 . Since T1 has a log-concave density, by Theorem 1.C.9 of [21], we have T1 + T2 ≥lr T1 + T3 . To reach the desired conclusion, we need to prove T1 + T3 ≥lr U1 + U2 .

958

P. Zhao et al. / Journal of Multivariate Analysis 100 (2009) 952–962

Due to Theorem 2.2, it suffices to show n P



λi Λ2 (i)

 Λ1 , i=1 n  P

λi Λ(i)

 w   (nλ, (n − 1)λ). 

i=1

Since n P

n P

λi Λ2 (i)

i =1 n

= Λ1 −

P

λi Λ(i)

i=1 n

P

i =1

λ2i < Λ1 , λi

i=1

it is enough to prove the following two statements: n P

λi Λ2 (i)

i =1 n

≤ (n − 1)λ

P

(3.3)

λi Λ(i)

i =1

and n P

Λ1 +

λi Λ2 (i)

i =1 n

P

≤ (2n − 1)λ.

(3.4)

λi Λ(i)

i=1

Note that

 λˆ =



1

2Λ1 +

2n − 1

Λ3 − Λ1 Λ2 Λ21 − Λ2

 =

n P

λi Λ2 (i)

 1 Λ1 + i=1 n P 2n − 1 

λi Λ(i)

  , 

(3.5)

i=1

and so n P

λi Λ2 (i)

i =1 n

P

= (2n − 1)λˆ − Λ1 ≤ (2n − 1)λˆ − nλˆ = (n − 1)λˆ ≤ (n − 1)λ, λi Λ(i)

i =1

and hence (3.3) holds by Lemma 2.4. By using (3.5), inequality (3.4) can be directly derived from Lemma 2.4.



As pointed out earlier in Section 1, Pˇaltˇanea [18] built characterization (1.2) on hazard rate order under the setup in Theorem 3.2. Since the likelihood ratio order implies the hazard rate order, it may be concluded that

¯ λ˜ ≤ λˇ ≤ λˆ ≤ λ, ˜ is the geometric mean of λ0 s. The following example is an illustration of the above assertion. where λ

(3.6)

Example 3.3. For an independent exponential random vector (X1 , X2 , X3 ) with hazard rate vector (λ1 , λ2 , λ3 ) (5.5, 5.5, 40), it can be easily evaluated that

λˆ ≈ 16.0719,

λ˜ ≈ 10.656,

λˇ ≈ 12.52,

=

λ¯ = 17.

Clearly, (3.6) holds. ˇ. Consider independent and identical exponential random variables (Y1 , Y2 , Y3 ) with the common hazard rate λ = λ By equivalence (1.2), we have X2:3 ≥hr Y2:3 . However, with fX2:3 and fY2:3 denoting the density functions of X2:3 and Y2:3 , respectively, we have fX2:3 (0.02) fY2:3 (0.02) Thus, the ratio

fX fY

≈ 0.8604 > 0.788055 ≈

2:3

2:3

(x) (x)

fX2:3 (0.04) fY2:3 (0.04)

.

is not increasing with respect to x ≥ 0, which reveals X2:3 6≥lr Y2:3 .



P. Zhao et al. / Journal of Multivariate Analysis 100 (2009) 952–962

959

Theorem 3.4. Let X1 , . . . , Xn be independent exponential random variables with respective hazard rates λ1 , . . . , λn , and Y1 , . . . , Yn be a random sample of size n from an exponential population with hazard rate λ. Then, X2:n ≤lr Y2:n if and only if n P

λ≤

λi − max λi 1≤i≤n

i=1

.

n−1

Proof. Pˇaltˇanea [18] proved that X2:n ≤hr Y2:n if and only if λ ≤

Pn

i=1

λi −max1≤i≤n λi n −1

=

min1≤i≤n Λ(i) n−1

with Λ(m) =

Pn

λ −

i =1 i min1≤i≤n Λ(i) n −1

λm , m = 1, . . . , n. Since the likelihood ratio order implies the hazard rate order, it suffices to prove that λ ≤ implies X2:n ≤lr Y2:n . Suppose T1 is exponential with hazard rate Λ1 , T2 is a mixture of exponential random variables with distribution function Pn λi FT2 = i=1 Λ1 FΛ(i) , and T4 is an exponential random variable with hazard rate min1≤i≤n Λ(i), both of which are independent of T1 . According to (3.2), it follows from λ ≤

min1≤i≤n Λ(i) n −1

that T2 ≤lr T4 . Also, by Theorem 1.C.9 of [21], we have

st

X2:n = T1 + T2 ≤lr T1 + T4 . Suppose U1 and U2 are independent exponential random variables with respective hazard rates nλ and (n − 1)λ. Now, we only need to prove st

T1 + T4 ≤lr U1 + U2 = Y2:n . By Theorem 2.2 again, this inequality is equivalent to



 w Λ1 , min Λ(i)  (nλ, (n − 1)λ).

(3.7)

1≤i≤n

It is easy to verify that

  min Λ1 , min Λ(i) = min Λ(i) ≥ (n − 1)λ = min{nλ, (n − 1)λ} 1≤i≤n

1≤i≤n

and

Λ1 + min Λ(i) = 1≤i≤n

1

n X

n − 1 i=1

Λ(i) + min Λ(i) ≥ (2n − 1) 1≤i≤n

which guarantees inequality (3.7). This completes the proof.

min Λ(i)

1≤i≤n

n−1

≥ (2n − 1)λ,



As a consequence of Theorems 3.2 and 3.4, we immediately obtain the following corollary, which compares the corresponding second order statistics in terms of the likelihood ratio order for the case when both exponential samples are heterogeneous. Corollary 3.5. Let X1 , . . . , Xn be independent exponential random variables with respective hazard rates λ1 , . . . , λn , and Y1 , . . . , Yn be another set of exponential random variables with respective hazard rates µ1 , . . . , µn . If



1 2n − 1 with Λk =

Pn

2Λ1 +

i =1

Λ3 − Λ1 Λ2 Λ21

n P

 ≤

− Λ2

µi − max µi 1≤i≤n

i =1

n−1

λ k = 1, 2, 3, then k i,

X2:n ≥lr Y2:n . Proof. Let Z1 , . . . , Zn be a random sample of size n from an exponential population with hazard rate λ such that 1 2n − 1



2Λ1 +

Λ3 − Λ1 Λ2 Λ21

− Λ2

n P



≤λ≤

From Theorems 3.2 and 3.4, it follows that X2:n ≥lr Z2:n ≥lr Y2:n . 

i=1

µi − max µi 1≤i≤n

n−1

.

960

P. Zhao et al. / Journal of Multivariate Analysis 100 (2009) 952–962

Upon applying an argument similar to that used by Pˇaltˇanea [18] to prove (1.2) and (1.3), an analogue of Corollary 3.5 on the hazard rate order can also be established. Corollary 3.6. Let X1 , . . . , Xn be independent exponential random variables with respective hazard rates λ1 , . . . , λn , and Y1 , . . . , Yn be another set of exponential random variables with respective hazard rates µ1 , . . . , µn . If

v P n u P u λλ µi − max µi u 1≤i
Under the setup of the above corollary, Dykstra et al. [12] showed with the help of a counterexample that (λ1 , . . . , λn )  (µ1 , . . . , µn ) does not imply X2:n ≥hr Y2:n , and so is the likelihood ratio order between X2:n and Y2:n . Here, Corollaries 3.5 and 3.6 provide, respectively, the sufficient conditions for the likelihood ratio order and the hazard rate order between X2:n and Y2:n . 4. Examples In order to illustrate the performance of our main results established in Section 3, we present here some interesting special cases. Let X1 , . . . , Xn be independent exponential random variables with respective hazard rates λ1 , . . . , λn , and Y1 , . . . , Yn be a random sample of size n from an exponential population with hazard rate µ. For the sake of convenience, let us denote n P

λ` =

λi − max λi 1≤i≤n

i =1

.

n−1

Example 4.1. Suppose that λ1 = · · · = λn = λ. In this case, it is easy to check

λˆ =



1

2Λ31 − 3Λ1 Λ2 + Λ3

=

Λ21 − Λ2

2n − 1

1

n(n − 1)(2n − 1)λ3

2n − 1

n(n − 1)λ2



=λ

and n P

λ` =

λi − max λi 1≤i≤n

i =1

n−1

= λ.

Due to Theorems 3.2 and 3.4, it immediately follows that X2:n ≥lr Y2:n ⇐⇒ µ ≥ λ and X2:n ≤lr Y2:n ⇐⇒ µ ≤ λ, which actually are well known in the literature.  Example 4.2. Suppose that λ1 = · · · = λn−1 = λ and λn = λ0 6= λ. This case is of special interest in the modelling of a single outlier. Note that

Λ1 = (n − 1)λ + λ0 ,

Λ2 = (n − 1)λ2 + λ20 ,

Λ3 = (n − 1)λ3 + λ30 .

After some simplifications, it is easy to see that

λˆ = =

1 2n − 1



2Λ31 − 3Λ1 Λ2 + Λ3



Λ21 − Λ2

(n − 2)(2n − 3)λ3 + 3(2n − 3)λ2 λ0 + 3λλ20 , 2(2n − 1)λλ0

and from Theorem 3.2, we have

ˆ= X2:n ≥lr Y2:n ⇐⇒ µ ≥ λ

(n − 2)(2n − 3)λ3 + 3(2n − 3)λ2 λ0 + 3λλ20 . 2(2n − 1)λλ0

On the other hand, by Theorem 3.4, if λ > λ0 , then

`= X2:n ≤lr Y2:n ⇐⇒ µ ≤ λ

(n − 2)λ + λ0 , n−1

P. Zhao et al. / Journal of Multivariate Analysis 100 (2009) 952–962

961

and if λ < λ0 , then

` = λ. X2:n ≤lr Y2:n ⇐⇒ µ ≤ λ



Example 4.3. For n = 2m + 1, let X1 , . . . , Xn be independent exponential random variables with X1 , . . . , Xm having respective hazard rates (1 − ni )λ, i = 1, . . . , m, Xm+1 having its hazard rate λ, and Xm+2 , . . . , Xn having respective hazard

h

rates 1 +

l−(m+1) n

i

Λ1 = nλ,

λ, l = m + 2, . . . , n. In this case, it can be readily calculated that m X i2

Λ2 = nλ2 + 2λ2

i =1

n

, 2

Λ3 = nλ3 + 6λ3

m X i2 i =1

n2

.

We then get

λˆ =



1

2Λ31 − 3Λ1 Λ2 + Λ3

Λ21 − Λ2

2n − 1



2n − 3n + n − 6(n − 1) 3



2

m P i=1

=



(2n − 1) n2 − n − 2

m P i =1

i2 n2

i2 n2



λ



and

 λ` = 1 −

m n(n − 1)



λ.

Upon applying Theorems 3.2 and 3.4 once again, we arrive at



2n − 3n + n − 6(n − 1) 3

2

m P i=1

ˆ= X2:n ≥lr Y2:n ⇐⇒ µ ≥ λ



(2n − 1) n2 − n − 2

m P i=1

i2 n2

i2 n2



λ



and



` = 1− X2:n ≤lr Y2:n ⇐⇒ µ ≤ λ

m n(n − 1)



λ. 

Acknowledgments This work was initiated while Peng Zhao was visiting the Department of Mathematics and Statistics, McMaster University, Canada, and he is grateful to China Scholarship Council (CSC) for providing the financial support for this visit. The authors would also like to thank a referee for useful comments on the original version of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

H.A. David, H.N. Nagaraja, Order Statistics, 3rd ed., Wiley, Hoboken, New Jersey, 2003. N. Balakrishnan, C.R. Rao, Handbook of Statistics, in: Order Statistics: Theory and Methods, vol. 16, Elsevier, Amsterdam, 1998. N. Balakrishnan, C.R. Rao, Handbook of Statistics, in: Order Statistics: Applications, vol. 17, Elsevier, Amsterdam, 1998. N. Balakrishnan, Permanents, order statistics, outliers, and robustness, Revista Matemática Complutense 20 (2007) 7–107. R.E. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart and Winston Inc., New York, 1975. N. Balakrishnan, A.P. Basu, The Exponential Distribution: Theory, Methods and Applications, Gordon and Breach Publishers, Newark, New Jersey, 1995. A. Paul, G. Gutierrez, Mean sample spacings, sample size and variability in auction-theoretic framework, Operations Research Letters 32 (2004) 103–108. X. Li, A note on the expected rent in auction theory, Operations Research Letters 33 (2005) 531–534. P. Pledger, F. Proschan, Comparisons of order statistics and of spacings from heterogeneous distributions, in: J.S. Rustagi (Ed.), Optimizing Methods in Statistics, Academic Press, New York, 1971, pp. 89–113. F. Proschan, J. Sethuraman, Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability, Journal of Multivariate Analysis 6 (1976) 608–616. S.C. Kochar, J. Rojo, Some new results on stochastic comparisons of spacings from heterogeneous exponential distributions, Journal of Multivariate Analysis 59 (1996) 272–281. R. Dykstra, S.C. Kochar, J. Rojo, Stochastic comparisons of parallel systems of heterogeneous exponential components, Journal of Statistical Planning and Inference 65 (1997) 203–211. J.L. Bon, E. Pˇaltˇanea, Ordering properties of convolutions of exponential random variables, Lifetime Data Analysis 5 (1999) 185–192. B. Khaledi, S.C. Kochar, Some new results on stochastic comparisons of parallel systems, Journal of Applied Probability 37 (2000) 1123–1128.

962

P. Zhao et al. / Journal of Multivariate Analysis 100 (2009) 952–962

[15] B. Khaledi, S.C. Kochar, Stochastic orderings among order statistics and sample spacings, in: J.C. Misra (Ed.), Uncertainty and Optimality-Probability, Statistics and Operations Research, World Scientific Publishers, Singapore, 2002, pp. 167–203. [16] J.L. Bon, E. Pˇaltˇanea, Comparisons of order statistics in a random sequence to the same statistics with i.i.d. variables, ESAIM: Probability and Statistics 10 (2006) 1–10. [17] S.C. Kochar, M. Xu, Stochastic comparisons of parallel systems when components have proportional hazard rates, Probability in the Engineering and Informational Science 21 (2007) 597–609. [18] E. Pˇaltˇanea, On the comparison in hazard rate ordering of fail-safe systems, Journal of Statistical Planning and Inference 138 (2008) 1993–1997. [19] R.E. Lillo, A.K. Nanda, M. Shaked, Preservation of some likelihood ratio stochastic orders by order statistics, Statistics and Probability Letters 51 (2001) 111–119. [20] F. Belzunce, J.M. Ruiz, M.C. Ruiz, On preservation of some shifted and proportional orders by systems, Statistics and Probability Letters 60 (2002) 141–154. [21] M. Shaked, J.G. Shanthikumar, Stochastic Orders, Springer, New York, 2007. [22] A. Müller, D. Stoyan, Comparison Methods for Stochastic Models and Risks, Wiley, New York, 2002. [23] A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization and its Applications, Academic Press, New York, 1979. [24] P.J. Boland, E. EL-Neweihi, F. Proschan, Schur properties of convolutions of exponential and geometric random variables, Journal of Multivariate Analysis 48 (1994) 157–167. [25] K. Chang, Stochastic orders of the sums of two exponential random variables, Statistics and Probability Letters 51 (2001) 389–396. [26] M.Y. An, Logconcavity versus logconvexity: A complete characterization, Journal of Economic Theory 80 (1998) 350–369. [27] S.C. Kochar, R. Korwar, Stochastic orders for spacings of heterogeneous exponential random variables, Journal of Multivariate Analysis 57 (1996) 69–83.