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MRL ordering of parallel systems with two heterogeneous components
Journal of Statistical Planning and Inference 141 (2011) 631–638
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MRL ordering of parallel systems with two heterogeneous components Peng Zhao a,, N. Balakrishnan b,1 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 in fo
abstract
Article history: Received 27 August 2009 Received in revised form 6 July 2010 Accepted 13 July 2010 Available online 27 July 2010
In this paper, we investigate ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components in terms of the mean residual life order. We establish, among others, that the reciprocal majorization order between parameter vectors implies the mean residual life order between the lifetimes of two parallel systems. We then extend this result to the proportional hazard rate models. & 2010 Elsevier B.V. All rights reserved.
Keywords: Hazard rate order Likelihood ratio order Mean residual life order Majorization p-Larger order Reciprocal majorization order Parallel system
1. Introduction Let X1:n rX2:n r rXn:n denote the order statistics of random variables X1,X2,y,Xn. Then, the k-th order statistic Xk:n is just the lifetime of an (n k+1)-out-of-n system, which is a very popular structure of redundancy in fault-tolerant systems that have been studied extensively. For example, Xn:n and X1:n correspond to the lifetimes of parallel and series systems, respectively. The study of order statistics have been predominantly on the case when the observations are independent and identically distributed (i.i.d.). The case when observations are non-i.i.d., however, often arises in a natural way in many practical situations. Due to the complexity of the distribution theory in this case, only limited results are available in the literature; see, for example, David and Nagaraja (2003) and Balakrishnan and Rao (1998a, 1998b) for comprehensive discussions on this topic, and the recent review article of Balakrishnan (2007) for results on the independent and nonidentically distributed (i.ni.d.) case developed using the theory of permanents. Due to the nice mathematical form and the memoryless property, the exponential distribution has been widely used in many applied fields, including reliability and survival analyses. We refer the readers to Barlow and Proschan (1975) and Balakrishnan and Basu (1995) for an encyclopedic treatment to developments on the exponential distribution. Pledger and Proschan (1971) were the first to compare stochastically the order statistics arising from i.ni.d. exponential random variables. Since then, many researchers have worked on this topic, including Proschan and Sethuraman (1976), Kochar and
Corresponding author.
E-mail addresses: [email protected] (P. Zhao), [email protected] (N. Balakrishnan). Visiting Professor at King Saud Univeristy (Saudi Arabia) and National Central University (Taiwan).
1
0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.07.013
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P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 141 (2011) 631–638
Rojo (1996), Dykstra et al. (1997), Khaledi and Kochar (2000, 2002), Bon and Paˇltaˇnea (2006), Kochar and Xu (2007), Paˇltaˇnea (2008), Zhao and Balakrishnan (2009b, 2010), Zhao et al. (2008, 2009), and Joo and Mi (2010). Let X1, y,Xn be independent exponential random variables with Xi having hazard rate li for i= 1,y,n. Let X*1, y, X*n be another set of independent exponential random variables with X*i having hazard rate li . Then, Pledger and Proschan (1971) showed that, for 1 rk r n, m
ðl1 , . . . , ln Þjðl1 , . . . , ln Þ¼)Xk:n Z st Xk:n :
ð1Þ
Formal definitions of these and other orderings involved in this study are provided in Section 2. Proschan and Sethuraman (1976) strengthened the result in (1) from componentwise stochastic order to multivariate stochastic order, while Khaledi and Kochar (2000) improved (1) for the case when k= n as p
ðl1 , . . . , ln Þjðl1 , . . . , ln Þ¼)Xn:n Z st Xn:n :
ð2Þ
Boland et al. (1994) showed, by means of a counterexample, that (1) cannot be strengthened from stochastic order to hazard rate order even for parallel systems with three independent exponential components; but, they established for the case when n = 2 that m
ðl1 , l2 Þjðl1 , l2 Þ¼)X2:2 Z hr X2:2 :
ð3Þ
Dykstra et al. (1997) further improved (3) from hazard rate order to likelihood ratio order as m
: ðl1 , l2 Þjðl1 , l2 Þ¼)X2:2 Z lr X2:2
ð4Þ
Joo and Mi (2010) recently provided some conditions under which the hazard rate order in (3) holds. Specifically, they proved that, under the condition 0 o l1 r l1 r l2 r l2 , w
ðl1 , l2 Þjðl1 , l2 Þ¼)X2:2 Z hr X2:2 :
ð5Þ
Zhao and Balakrishnan (2010) established two characterization results for the likelihood ratio order [reversed hazard rate order] and hazard rate order [stochastic order] as follows. Under the condition 0 o l1 r l1 r l2 r l2 , we have w
ðl1 , l2 Þjðl1 , l2 Þ()X2:2 Z lr ½ Z rh X2:2
ð6Þ
and p
: ðl1 , l2 Þjðl1 , l2 Þ()X2:2 Z hr ½ Z st X2:2
Here, we consider the mean residual life order and prove that, under the condition rm
ðl1 , l2 Þjðl1 , l2 Þ¼)X2:2 Z mrl X2:2 :
ð7Þ 0 o l1 r l1 r l2 r l2 ,
ð8Þ
It should be noted that the result in (8) forms a nice complement to those in (5)–(7). We also extend the result in (8) to proportional hazard rate models. 2. Definitions In this section, we first recall some notions of stochastic orders, and majorization and related orders. Throughout this paper, the term increasing is used for monotone non-decreasing and decreasing is used for monotone non-increasing. 2.1. Stochastic orders
Definition 2.1. For two random variables X and Y with densities fX and fY, and distribution functions FX and FY, respectively, let F X ¼ 1FX and F Y ¼ 1FY be the corresponding survival functions. Then: (i) (ii) (iii) (iv) (v)
X is said to be smaller than Y in the likelihood ratio order (denoted by X r lr Y) if fY(x)/fX(x) is increasing in x; X is said to be smaller than Y in the hazard rate order (denoted by X r hr Y) if F Y ðxÞ=F X ðxÞ is increasing in x; X is said to be smaller than Y in the reversed hazard rate order (denoted by X r rh Y) if FY(x)/FX(x) is increasing in x; X is said to be smaller than Y in the stochastic order (denoted by X r st Y) if F Y ðxÞ ZF X ðxÞ; X is said to be smaller than Y in the mean residual life order (denoted by X r mrl Y) if EX t r EY t , where Xt ¼ ðXtjX 4tÞ is the residual life at age t 4 0 of the random lifetime X.
It is known that hazard rate order implies both usual stochastic order and mean residual life order, but neither usual stochastic order nor mean residual life order implies the other; see Shaked and Shanthikumar (2007).
P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 141 (2011) 631–638
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2.2. Majorization and related orders The notion of majorization is quite useful in establishing various inequalities. Let xð1Þ r rxðnÞ be the increasing arrangement of the components of the vector x= (x1,y,xn). m
Definition 2.2. (i) A vector x ¼ ðx1 , . . . ,xn Þ 2 Rn is said to majorize another vector y ¼ ðy1 , . . . ,yn Þ 2 Rn (written as xjy) if j X
xðiÞ r
i¼1
j X
yðiÞ
for j ¼ 1, . . . ,n1,
i¼1
P P and ni¼ 1 xðiÞ ¼ ni¼ 1 yðiÞ . w (ii) A vector x 2 Rn is said to weakly majorize another vector y 2 Rn (written as xjy) if j X
xðiÞ r
i¼1
j X
yðiÞ
for j ¼ 1, . . . ,n:
i¼1 p
(iii) A vector x 2 Rnþ is said to be p-larger than another vector y 2 Rnþ (written as xjy) if j Y
xðiÞ r
i¼1
j Y
yðiÞ
for j ¼ 1, . . . ,n:
i¼1 m
p
w
w
Obviously, xjy implies xjy, and xjy is equivalent to logðxÞjlogðyÞ, where log(x) is the vector of logarithms of the p
m
coordinates of x. Khaledi and Kochar (2002) showed that xjy implies xjy for any x,y 2 Rnþ . The converse is, however, not p
true. For example, ð1,5:5Þjð2,3Þ, but clearly the majorization order does not hold between these two vectors. For more details on majorization and p-larger orders and their applications, one may refer to Marshall and Olkin (1979), Bon and Paˇltaˇnea (1999), and Khaledi and Kochar (2002). Zhao and Balakrishnan (2009a) recently introduced a new partial order, called as reciprocal majorization order. rm
Definition 2.3. The vector x 2 Rnþ is said to reciprocal majorize another vector y 2 Rnþ (written as xjy) if j j X X 1 1 Z x y i ¼ 1 ðiÞ i ¼ 1 ðiÞ
for j =1,y, n. It is known from Kochar and Xu (2009) that w
p
rm
xjy¼)xjy¼)xjy p
rm
for any two non-negative vectors x and y. On the other hand, the j order does not imply the j order. For example, rm
p
rm
from the definition of the j order, it follows that ð1,4Þjð43 ,2Þ, but clearly the j order does not hold between these two vectors. 3. Mean residual life ordering Lemma 3.1 (Marshall and Olkin, 1979, p. 57). Let I R be an open interval and let f : In -R be continuously differentiable. Then, f is Schur-convex [Schur-concave] on In if and only if f is symmetric on In and for all iaj, @ @ fðzÞ fðzÞ Z ½ r 0 for all z 2 In , ðzi zj Þ @zi @zj where ð@=@zi ÞfðzÞ denotes the partial derivative of fðzÞ with respect to its i-th argument. Theorem 3.2. Let (X1, X2) be a vector of independent exponential random variables with respective hazard rates l1 and l2 , and (X*1,X*2) be another vector of independent exponential random variables with respective hazard rates l1 and l2 . If minðl1 , l2 Þ rminðl1 , l2 Þ, l1 þ l2 4 l1 þ l2 and 1=l1 þ 1=l2 ¼ 1=l1 þ1=l2 , then X2:2 Z mrl X2:2 .
Proof. Without loss of generality, let us assume that 0 o l1 r l2 and 0 o l1 r l2 so that l1 r l1 r l2 r l2 . The MRL function of X2:2 can be written as 1
jX2:2 ðtÞ ¼
l1
el1 t þ
1
l2
el2 t
1
l1 þ l2
eðl1 þ l2 Þt
el1 t þel2 t eðl1 þ l2 Þt
,
t 2 Rþ ,
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and now we need to show, under the condition l1 r l1 , l1 þ l2 4 l1 þ l2 , and 1=l1 þ 1=l2 ¼ 1=l1 þ 1=l2 , that 1
l1
el1 t þ
1
l2
1
el2 t
l1 þ l2
1
eðl1 þ l2 Þt r
el1 t þ el2 t eðl1 þ l2 Þt
l1
1
el1 t þ
l2
el2 t
1
l1 þ l2
eðl1 þ l2 Þt
el1 t þel2 t eðl1 þ l2 Þt
for all t 2 R þ . If t= 0, the above inequality becomes 1
l1
1
þ
l2
1
l1 þ l2
Z
1
l1
þ
1
l2
1
l1 þ l2
,
which is seen to be true from the assumptions. In what follows, we assume that t 4 0, in which case it is equivalent to showing 1 x 1 x 1 1 x1 1 x2 1 e 1 þ e 2 eðx1 þ x2 Þ e þ e eðx1 þ x2 Þ x1 x2 x1 þ x2 x1 x2 x1 þ x2 r ex1 þ ex2 eðx1 þ x2 Þ ex1 þ ex2 eðx1 þ x2 Þ under the condition 0 ox1 rx1 and 1=x1 þ 1=x2 ¼ 1=x1 þ 1=x2 . Denote y1 ¼ 1=x1 ,y2 ¼ 1=x2 ,y1 ¼ 1=x1 and y2 ¼ 1=x2 ; we then have y1 Z y1 Z y2 Zy2 4 0 and m
ðy1 ,y2 Þjðy1 ,y2 Þ: Then, it suffices to show that the symmetrical differentiable function K : ð0,1Þ2 -ð0,1Þ, defined by y1 y2 ð1=y1 þ 1=y2 Þ y1 e1=y1 þ y2 e1=y2 e y1 þ y2 Kðy1 ,y2 Þ ¼ e1=y1 þ e1=y2 eð1=y1 þ 1=y2 Þ is Schur-convex. We first observe that h i2 @K ðy1 ,y2 Þ e1=y1 þ e1=y2 eð1=y1 þ 1=y2 Þ @y1 y22 y2 eð1=y1 þ 1=y2 Þ eð1=y1 þ 1=y2 Þ y1 ðy1 þ y2 Þ ðy1 þy2 Þ2 ½e1=y1 þe1=y2 eð1=y1 þ 1=y2 Þ " # 1 y1 y2 ð1=y1 þ 1=y2 Þ e 2 e1=y1 1=y21 eð1=y1 þ 1=y2 Þ y1 e1=y1 þy2 e1=y2 y1 þ y2 y1 " # 2 y2 eð1=y1 þ 1=y2 Þ ½e1=y1 þe1=y2 eð1=y1 þ 1=y2 Þ ¼ e1=y1 ðy1 þ y2 Þ2 " # y2 eð1=y1 þ 1=y2 Þ : þ 1 22 ð1e1=y2 Þ y1 þ y2 y1 ¼ ½e1=y1 þ 1=y1 e1=y1
Similarly, we have @K ðy1 ,y2 Þ½e1=y1 þ e1=y2 eð1=y1 þ 1=y2 Þ 2 @y2 " # y21 ð1=y1 þ 1=y2 Þ ½e1=y1 þe1=y2 eð1=y1 þ 1=y2 Þ ¼ e1=y2 e ðy1 þ y2 Þ2 " # y21 eð1=y1 þ 1=y2 Þ 1=y1 Þ : þ 1 2 ð1e y1 þ y2 y2 Thus, we find @K @K ðy1 ,y2 Þ ðy1 ,y2 Þ @y1 @y2 " sgn
1=y1
¼ ðe þ
1=y2
e
Þþ
y21 y22 ðy1 þy2 Þ2
# ð1=y1 þ 1=y2 Þ
e
½e1=y1 þ e1=y2 eð1=y1 þ 1=y2 Þ
" # y21 y22 eð1=y1 þ 1=y2 Þ 1=y1 1=y2 ð1e Þ ð1e Þ : 2 2 y1 þ y2 y2 y1
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For y1 Zy2 4 0, it can then be readily verified that ðe1=y1 e1=y2 Þ þ
y21 y22 ðy1 þ y2 Þ2
eð1=y1 þ 1=y2 Þ Z 0:
Since the function ð1ex Þ=x is decreasing in x 4 0 and so y2 ð1e1=y Þ is increasing in y4 0, it follows that y21 y2 1 ð1e1=y1 Þ 22 ð1e1=y2 Þ Z 2 ½y21 ð1e1=y1 Þy22 ð1e1=y2 Þ y1 y22 y1 Z0 for y1 Z y2 40. Thus, we obtain @K @K ðy1 ,y2 Þ ðy1 ,y2 Þ Z 0: ðy1 y2 Þ @y1 @y2 Upon applying Lemma 3.1 now, we can conclude that the function K(y1, y2) is Schur-convex, and hence the theorem follows. & Theorem 3.3. Let (X1, X2) be a vector of independent exponential random variables with respective hazard rates l1 and l2 , and (X*1,X*2) be another vector of independent exponential random variables with respective hazard rates l1 and l2 . Suppose
minðl1 , l2 Þ r minðl1 , l2 Þ rmaxðl1 , l2 Þ r maxðl1 , l2 Þ: Then, rm
ðl1 , l2 Þjðl1 , l2 Þ¼)X2:2 Z mrl X2:2 :
Proof. Without loss of generality, let us assume that l1 r l1 r l2 r l2 . Clearly, we have l1 r l1 r l2 r l2 and 1=l1 þ1=l2 Z1=l1 þ1=l2 . We now need to discuss two cases. (i) Case 1: l1 r l1 r l2 r l2 and l1 þ l2 r l1 þ l2 . which, in turn, implies that X2:2 Z mrl X2:2 . In this case, it follows from Joo and Mi (2010) that X2:2 Z hr X2:2 (ii) Case 2: l1 r l1 r l2 r l2 , l1 þ l2 4 l1 þ l2 , and 1=l1 þ 1=l2 Z1=l1 þ1=l2 . For the case when 1=l1 þ 1=l2 ¼ 1=l1 þ1=l2 , the result follows from Theorem 3.2. Next, let us assume that 1=l1 þ1=l2 41=l1 þ1=l2 . Let
lu ¼
1
l1
þ
1 1
l2
1
:
l2
We then have 1=lu þ 1=l2 ¼ 1=l1 þ 1=l2 and l1 o lu r l1 . Let Z2:2 be the lifetime of a parallel system consisting of two independent exponential components with respective hazard rates lu and l2 . From Theorem 3.2, it follows that . Also, we have X2:2 Z hr Z2:2 by Theorem 2.2 of Joo and Mi (2010). We thus obtain the desired result that Z2:2 Z mrl X2:2 . X2:2 Z mrl X2:2 Hence, the theorem. &
Remark 3.4. Suppose l1 r l1 r l2 r l2 . Then, Zhao and Balakrishnan (2010) established that l1 þ l2 r l1 þ l2 and l1 l2 r l1 l2 are necessary and sufficient conditions for X2:2 Z lr X2:2 and X2:2 Z hr X2:2 , respectively. It can be seen here that Theorem 3.3 provides a sufficient condition for the mean residual life order which includes the condition l1 l2 r l1 l2 as a p
rm
special case since ðl1 , l2 Þjðl1 , l2 Þ implies ðl1 , l2 Þjðl1 , l2 Þ. For example, let ðl1 , l2 Þ ¼ ð2,6:5Þ and ðl1 , l2 Þ ¼ ð3,4Þ. Then, we have l1 þ l2 4 l1 þ l2 , l1 l2 4 l1 l2 , and 1=l1 þ 1=l2 41=l1 þ1=l2 , and while X2:2 Z hr X2:2 does not hold, X2:2 Z mrl X2:2 does hold, as can be seen in Figs. 1 and 2. The following result is a direct consequence of Theorem 3.3, which has been given earlier by Zhao and Balakrishnan (2009b). Corollary 3.5. Let (X1, X2) be a vector of independent exponential random variables with respective hazard rates l1 and l2 , and (X*1,X*2) be another vector of independent exponential random variables with common hazard rate l. Then,
lZ
2 1
l1
þ
1
l2
()X2:2 Z mrl X2:2 :
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P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 141 (2011) 631–638
Fig. 1. Plot of the hazard rate functions of two lifetimes of parallel systems with exponential units, where h(t;a,b) denotes the hazard rate function with exponential parameter vector (a,b).
Fig. 2. Plot of the mean residual life functions of two lifetimes of parallel systems with exponential units, where jðt; a,bÞ denotes the mean residual life function with exponential parameter vector (a,b).
4. Extension to the PHR model Independent random variables X1,y, Xn are said to follow the proportional hazard rate (PHR) model if, for i= 1,y, n, the survival function of Xi can be written as F i ðxÞ ¼ ½F ðxÞli , where F ðxÞ is the baseline survival function of some random variable X. Let r(t) be the survival function of the baseline distribution F. Then, the hazard rate of Xi can be written as F i ðxÞ ¼ eli RðxÞ Rx for i=1,y,n, where RðxÞ ¼ 0 rðtÞ dt is the cumulative hazard rate of X. Evidently, the case of exponential random variables with hazard rates l1 , . . . , ln is a special case of the above PHR model. The next result extends Theorem 3.3 to the PHR model.
P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 141 (2011) 631–638
637 l1
l2
Theorem 4.1. Let (X1, X2) be a vector of independent random variables with respective survival functions F and F , and l l (X*1,X*2) be another vector of independent random variables with respective survival functions F 1 and F 2 . Suppose
minðl1 , l2 Þ r minðl1 , l2 Þ rmaxðl1 , l2 Þ r maxðl1 , l2 Þ: Then, if the baseline distribution F is DFR (decreasing failure rate), we have rm
: ðl1 , l2 Þjðl1 , l2 Þ¼)X2:2 Z mrl X2:2
Proof. We may write the cumulative hazard function of F as HðxÞ ¼ logF ðxÞ, and then, for i= 1, 2, we have li
PðHðXi Þ 4xÞ ¼ PðXi 4H1 ðxÞÞ ¼ F ðF
1
ðex ÞÞ ¼ eli x ,
where H1 ¼ inffy : HðyÞ Zxg. Note that Xui ¼ HðXi Þ is in fact an exponential random variable with hazard rate li ,i ¼ 1,2. Similarly, Yui ¼ HðXi Þ is also an exponential random variable with hazard rate li . From Theorem 3.3, it then follows that Xu2:2 Z mrl Yu2:2 , i.e., HðX2:2 Þ Z mrl HðX2:2 Þ:
The DFR property of F implies the function H1 ðÞ is increasing and convex, and so it follows from Theorem 2.A.19 of Shaked and Shanthikumar (2007) that Þ ¼ X2:2 , X2:2 ¼ H1 HðX2:2 Þ Z mrl H1 HðX2:2
and from which the desired result follows.
&
If X1,y, Xn are independent Weibull random variables with common shape parameter a 40 and scale parameter vectors a a ðl1 , . . . , ln Þ, then they belong to the PHR model with a new parameter vector ðl1 , . . . , ln Þ and a baseline survival function t a F ðtÞ ¼ e . Since F is DFR if a 2 ð0,1Þ, we have the following corollary which is a direct consequence of Theorem 4.1. Corollary 4.2. Let (X1, X2) be a vector of independent Weibull random variables with common shape parameter a 4 0 and scale parameter vector ðl1 , l2 Þ, and (X*1,X*2) be another vector of independent Weibull random variables with common shape parameter a and scale parameter vector ðl1 , l2 Þ. Suppose
minðl1 , l2 Þ r minðl1 , l2 Þ rmaxðl1 , l2 Þ r maxðl1 , l2 Þ: Then, for a 2 ð0,1Þ, we have a
a rm
ðl1 , l2 Þjððl1 Þa ,ðl2 Þa Þ¼)X2:2 Z mrl X2:2 :
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Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
MRL ordering of parallel systems with two heterogeneous components Peng Zhao a,, N. Balakrishnan b,1 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 in fo
abstract
Article history: Received 27 August 2009 Received in revised form 6 July 2010 Accepted 13 July 2010 Available online 27 July 2010
In this paper, we investigate ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components in terms of the mean residual life order. We establish, among others, that the reciprocal majorization order between parameter vectors implies the mean residual life order between the lifetimes of two parallel systems. We then extend this result to the proportional hazard rate models. & 2010 Elsevier B.V. All rights reserved.
Keywords: Hazard rate order Likelihood ratio order Mean residual life order Majorization p-Larger order Reciprocal majorization order Parallel system
1. Introduction Let X1:n rX2:n r rXn:n denote the order statistics of random variables X1,X2,y,Xn. Then, the k-th order statistic Xk:n is just the lifetime of an (n k+1)-out-of-n system, which is a very popular structure of redundancy in fault-tolerant systems that have been studied extensively. For example, Xn:n and X1:n correspond to the lifetimes of parallel and series systems, respectively. The study of order statistics have been predominantly on the case when the observations are independent and identically distributed (i.i.d.). The case when observations are non-i.i.d., however, often arises in a natural way in many practical situations. Due to the complexity of the distribution theory in this case, only limited results are available in the literature; see, for example, David and Nagaraja (2003) and Balakrishnan and Rao (1998a, 1998b) for comprehensive discussions on this topic, and the recent review article of Balakrishnan (2007) for results on the independent and nonidentically distributed (i.ni.d.) case developed using the theory of permanents. Due to the nice mathematical form and the memoryless property, the exponential distribution has been widely used in many applied fields, including reliability and survival analyses. We refer the readers to Barlow and Proschan (1975) and Balakrishnan and Basu (1995) for an encyclopedic treatment to developments on the exponential distribution. Pledger and Proschan (1971) were the first to compare stochastically the order statistics arising from i.ni.d. exponential random variables. Since then, many researchers have worked on this topic, including Proschan and Sethuraman (1976), Kochar and
Corresponding author.
E-mail addresses: [email protected] (P. Zhao), [email protected] (N. Balakrishnan). Visiting Professor at King Saud Univeristy (Saudi Arabia) and National Central University (Taiwan).
1
0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.07.013
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Rojo (1996), Dykstra et al. (1997), Khaledi and Kochar (2000, 2002), Bon and Paˇltaˇnea (2006), Kochar and Xu (2007), Paˇltaˇnea (2008), Zhao and Balakrishnan (2009b, 2010), Zhao et al. (2008, 2009), and Joo and Mi (2010). Let X1, y,Xn be independent exponential random variables with Xi having hazard rate li for i= 1,y,n. Let X*1, y, X*n be another set of independent exponential random variables with X*i having hazard rate li . Then, Pledger and Proschan (1971) showed that, for 1 rk r n, m
ðl1 , . . . , ln Þjðl1 , . . . , ln Þ¼)Xk:n Z st Xk:n :
ð1Þ
Formal definitions of these and other orderings involved in this study are provided in Section 2. Proschan and Sethuraman (1976) strengthened the result in (1) from componentwise stochastic order to multivariate stochastic order, while Khaledi and Kochar (2000) improved (1) for the case when k= n as p
ðl1 , . . . , ln Þjðl1 , . . . , ln Þ¼)Xn:n Z st Xn:n :
ð2Þ
Boland et al. (1994) showed, by means of a counterexample, that (1) cannot be strengthened from stochastic order to hazard rate order even for parallel systems with three independent exponential components; but, they established for the case when n = 2 that m
ðl1 , l2 Þjðl1 , l2 Þ¼)X2:2 Z hr X2:2 :
ð3Þ
Dykstra et al. (1997) further improved (3) from hazard rate order to likelihood ratio order as m
: ðl1 , l2 Þjðl1 , l2 Þ¼)X2:2 Z lr X2:2
ð4Þ
Joo and Mi (2010) recently provided some conditions under which the hazard rate order in (3) holds. Specifically, they proved that, under the condition 0 o l1 r l1 r l2 r l2 , w
ðl1 , l2 Þjðl1 , l2 Þ¼)X2:2 Z hr X2:2 :
ð5Þ
Zhao and Balakrishnan (2010) established two characterization results for the likelihood ratio order [reversed hazard rate order] and hazard rate order [stochastic order] as follows. Under the condition 0 o l1 r l1 r l2 r l2 , we have w
ðl1 , l2 Þjðl1 , l2 Þ()X2:2 Z lr ½ Z rh X2:2
ð6Þ
and p
: ðl1 , l2 Þjðl1 , l2 Þ()X2:2 Z hr ½ Z st X2:2
Here, we consider the mean residual life order and prove that, under the condition rm
ðl1 , l2 Þjðl1 , l2 Þ¼)X2:2 Z mrl X2:2 :
ð7Þ 0 o l1 r l1 r l2 r l2 ,
ð8Þ
It should be noted that the result in (8) forms a nice complement to those in (5)–(7). We also extend the result in (8) to proportional hazard rate models. 2. Definitions In this section, we first recall some notions of stochastic orders, and majorization and related orders. Throughout this paper, the term increasing is used for monotone non-decreasing and decreasing is used for monotone non-increasing. 2.1. Stochastic orders
Definition 2.1. For two random variables X and Y with densities fX and fY, and distribution functions FX and FY, respectively, let F X ¼ 1FX and F Y ¼ 1FY be the corresponding survival functions. Then: (i) (ii) (iii) (iv) (v)
X is said to be smaller than Y in the likelihood ratio order (denoted by X r lr Y) if fY(x)/fX(x) is increasing in x; X is said to be smaller than Y in the hazard rate order (denoted by X r hr Y) if F Y ðxÞ=F X ðxÞ is increasing in x; X is said to be smaller than Y in the reversed hazard rate order (denoted by X r rh Y) if FY(x)/FX(x) is increasing in x; X is said to be smaller than Y in the stochastic order (denoted by X r st Y) if F Y ðxÞ ZF X ðxÞ; X is said to be smaller than Y in the mean residual life order (denoted by X r mrl Y) if EX t r EY t , where Xt ¼ ðXtjX 4tÞ is the residual life at age t 4 0 of the random lifetime X.
It is known that hazard rate order implies both usual stochastic order and mean residual life order, but neither usual stochastic order nor mean residual life order implies the other; see Shaked and Shanthikumar (2007).
P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 141 (2011) 631–638
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2.2. Majorization and related orders The notion of majorization is quite useful in establishing various inequalities. Let xð1Þ r rxðnÞ be the increasing arrangement of the components of the vector x= (x1,y,xn). m
Definition 2.2. (i) A vector x ¼ ðx1 , . . . ,xn Þ 2 Rn is said to majorize another vector y ¼ ðy1 , . . . ,yn Þ 2 Rn (written as xjy) if j X
xðiÞ r
i¼1
j X
yðiÞ
for j ¼ 1, . . . ,n1,
i¼1
P P and ni¼ 1 xðiÞ ¼ ni¼ 1 yðiÞ . w (ii) A vector x 2 Rn is said to weakly majorize another vector y 2 Rn (written as xjy) if j X
xðiÞ r
i¼1
j X
yðiÞ
for j ¼ 1, . . . ,n:
i¼1 p
(iii) A vector x 2 Rnþ is said to be p-larger than another vector y 2 Rnþ (written as xjy) if j Y
xðiÞ r
i¼1
j Y
yðiÞ
for j ¼ 1, . . . ,n:
i¼1 m
p
w
w
Obviously, xjy implies xjy, and xjy is equivalent to logðxÞjlogðyÞ, where log(x) is the vector of logarithms of the p
m
coordinates of x. Khaledi and Kochar (2002) showed that xjy implies xjy for any x,y 2 Rnþ . The converse is, however, not p
true. For example, ð1,5:5Þjð2,3Þ, but clearly the majorization order does not hold between these two vectors. For more details on majorization and p-larger orders and their applications, one may refer to Marshall and Olkin (1979), Bon and Paˇltaˇnea (1999), and Khaledi and Kochar (2002). Zhao and Balakrishnan (2009a) recently introduced a new partial order, called as reciprocal majorization order. rm
Definition 2.3. The vector x 2 Rnþ is said to reciprocal majorize another vector y 2 Rnþ (written as xjy) if j j X X 1 1 Z x y i ¼ 1 ðiÞ i ¼ 1 ðiÞ
for j =1,y, n. It is known from Kochar and Xu (2009) that w
p
rm
xjy¼)xjy¼)xjy p
rm
for any two non-negative vectors x and y. On the other hand, the j order does not imply the j order. For example, rm
p
rm
from the definition of the j order, it follows that ð1,4Þjð43 ,2Þ, but clearly the j order does not hold between these two vectors. 3. Mean residual life ordering Lemma 3.1 (Marshall and Olkin, 1979, p. 57). Let I R be an open interval and let f : In -R be continuously differentiable. Then, f is Schur-convex [Schur-concave] on In if and only if f is symmetric on In and for all iaj, @ @ fðzÞ fðzÞ Z ½ r 0 for all z 2 In , ðzi zj Þ @zi @zj where ð@=@zi ÞfðzÞ denotes the partial derivative of fðzÞ with respect to its i-th argument. Theorem 3.2. Let (X1, X2) be a vector of independent exponential random variables with respective hazard rates l1 and l2 , and (X*1,X*2) be another vector of independent exponential random variables with respective hazard rates l1 and l2 . If minðl1 , l2 Þ rminðl1 , l2 Þ, l1 þ l2 4 l1 þ l2 and 1=l1 þ 1=l2 ¼ 1=l1 þ1=l2 , then X2:2 Z mrl X2:2 .
Proof. Without loss of generality, let us assume that 0 o l1 r l2 and 0 o l1 r l2 so that l1 r l1 r l2 r l2 . The MRL function of X2:2 can be written as 1
jX2:2 ðtÞ ¼
l1
el1 t þ
1
l2
el2 t
1
l1 þ l2
eðl1 þ l2 Þt
el1 t þel2 t eðl1 þ l2 Þt
,
t 2 Rþ ,
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and now we need to show, under the condition l1 r l1 , l1 þ l2 4 l1 þ l2 , and 1=l1 þ 1=l2 ¼ 1=l1 þ 1=l2 , that 1
l1
el1 t þ
1
l2
1
el2 t
l1 þ l2
1
eðl1 þ l2 Þt r
el1 t þ el2 t eðl1 þ l2 Þt
l1
1
el1 t þ
l2
el2 t
1
l1 þ l2
eðl1 þ l2 Þt
el1 t þel2 t eðl1 þ l2 Þt
for all t 2 R þ . If t= 0, the above inequality becomes 1
l1
1
þ
l2
1
l1 þ l2
Z
1
l1
þ
1
l2
1
l1 þ l2
,
which is seen to be true from the assumptions. In what follows, we assume that t 4 0, in which case it is equivalent to showing 1 x 1 x 1 1 x1 1 x2 1 e 1 þ e 2 eðx1 þ x2 Þ e þ e eðx1 þ x2 Þ x1 x2 x1 þ x2 x1 x2 x1 þ x2 r ex1 þ ex2 eðx1 þ x2 Þ ex1 þ ex2 eðx1 þ x2 Þ under the condition 0 ox1 rx1 and 1=x1 þ 1=x2 ¼ 1=x1 þ 1=x2 . Denote y1 ¼ 1=x1 ,y2 ¼ 1=x2 ,y1 ¼ 1=x1 and y2 ¼ 1=x2 ; we then have y1 Z y1 Z y2 Zy2 4 0 and m
ðy1 ,y2 Þjðy1 ,y2 Þ: Then, it suffices to show that the symmetrical differentiable function K : ð0,1Þ2 -ð0,1Þ, defined by y1 y2 ð1=y1 þ 1=y2 Þ y1 e1=y1 þ y2 e1=y2 e y1 þ y2 Kðy1 ,y2 Þ ¼ e1=y1 þ e1=y2 eð1=y1 þ 1=y2 Þ is Schur-convex. We first observe that h i2 @K ðy1 ,y2 Þ e1=y1 þ e1=y2 eð1=y1 þ 1=y2 Þ @y1 y22 y2 eð1=y1 þ 1=y2 Þ eð1=y1 þ 1=y2 Þ y1 ðy1 þ y2 Þ ðy1 þy2 Þ2 ½e1=y1 þe1=y2 eð1=y1 þ 1=y2 Þ " # 1 y1 y2 ð1=y1 þ 1=y2 Þ e 2 e1=y1 1=y21 eð1=y1 þ 1=y2 Þ y1 e1=y1 þy2 e1=y2 y1 þ y2 y1 " # 2 y2 eð1=y1 þ 1=y2 Þ ½e1=y1 þe1=y2 eð1=y1 þ 1=y2 Þ ¼ e1=y1 ðy1 þ y2 Þ2 " # y2 eð1=y1 þ 1=y2 Þ : þ 1 22 ð1e1=y2 Þ y1 þ y2 y1 ¼ ½e1=y1 þ 1=y1 e1=y1
Similarly, we have @K ðy1 ,y2 Þ½e1=y1 þ e1=y2 eð1=y1 þ 1=y2 Þ 2 @y2 " # y21 ð1=y1 þ 1=y2 Þ ½e1=y1 þe1=y2 eð1=y1 þ 1=y2 Þ ¼ e1=y2 e ðy1 þ y2 Þ2 " # y21 eð1=y1 þ 1=y2 Þ 1=y1 Þ : þ 1 2 ð1e y1 þ y2 y2 Thus, we find @K @K ðy1 ,y2 Þ ðy1 ,y2 Þ @y1 @y2 " sgn
1=y1
¼ ðe þ
1=y2
e
Þþ
y21 y22 ðy1 þy2 Þ2
# ð1=y1 þ 1=y2 Þ
e
½e1=y1 þ e1=y2 eð1=y1 þ 1=y2 Þ
" # y21 y22 eð1=y1 þ 1=y2 Þ 1=y1 1=y2 ð1e Þ ð1e Þ : 2 2 y1 þ y2 y2 y1
P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 141 (2011) 631–638
635
For y1 Zy2 4 0, it can then be readily verified that ðe1=y1 e1=y2 Þ þ
y21 y22 ðy1 þ y2 Þ2
eð1=y1 þ 1=y2 Þ Z 0:
Since the function ð1ex Þ=x is decreasing in x 4 0 and so y2 ð1e1=y Þ is increasing in y4 0, it follows that y21 y2 1 ð1e1=y1 Þ 22 ð1e1=y2 Þ Z 2 ½y21 ð1e1=y1 Þy22 ð1e1=y2 Þ y1 y22 y1 Z0 for y1 Z y2 40. Thus, we obtain @K @K ðy1 ,y2 Þ ðy1 ,y2 Þ Z 0: ðy1 y2 Þ @y1 @y2 Upon applying Lemma 3.1 now, we can conclude that the function K(y1, y2) is Schur-convex, and hence the theorem follows. & Theorem 3.3. Let (X1, X2) be a vector of independent exponential random variables with respective hazard rates l1 and l2 , and (X*1,X*2) be another vector of independent exponential random variables with respective hazard rates l1 and l2 . Suppose
minðl1 , l2 Þ r minðl1 , l2 Þ rmaxðl1 , l2 Þ r maxðl1 , l2 Þ: Then, rm
ðl1 , l2 Þjðl1 , l2 Þ¼)X2:2 Z mrl X2:2 :
Proof. Without loss of generality, let us assume that l1 r l1 r l2 r l2 . Clearly, we have l1 r l1 r l2 r l2 and 1=l1 þ1=l2 Z1=l1 þ1=l2 . We now need to discuss two cases. (i) Case 1: l1 r l1 r l2 r l2 and l1 þ l2 r l1 þ l2 . which, in turn, implies that X2:2 Z mrl X2:2 . In this case, it follows from Joo and Mi (2010) that X2:2 Z hr X2:2 (ii) Case 2: l1 r l1 r l2 r l2 , l1 þ l2 4 l1 þ l2 , and 1=l1 þ 1=l2 Z1=l1 þ1=l2 . For the case when 1=l1 þ 1=l2 ¼ 1=l1 þ1=l2 , the result follows from Theorem 3.2. Next, let us assume that 1=l1 þ1=l2 41=l1 þ1=l2 . Let
lu ¼
1
l1
þ
1 1
l2
1
:
l2
We then have 1=lu þ 1=l2 ¼ 1=l1 þ 1=l2 and l1 o lu r l1 . Let Z2:2 be the lifetime of a parallel system consisting of two independent exponential components with respective hazard rates lu and l2 . From Theorem 3.2, it follows that . Also, we have X2:2 Z hr Z2:2 by Theorem 2.2 of Joo and Mi (2010). We thus obtain the desired result that Z2:2 Z mrl X2:2 . X2:2 Z mrl X2:2 Hence, the theorem. &
Remark 3.4. Suppose l1 r l1 r l2 r l2 . Then, Zhao and Balakrishnan (2010) established that l1 þ l2 r l1 þ l2 and l1 l2 r l1 l2 are necessary and sufficient conditions for X2:2 Z lr X2:2 and X2:2 Z hr X2:2 , respectively. It can be seen here that Theorem 3.3 provides a sufficient condition for the mean residual life order which includes the condition l1 l2 r l1 l2 as a p
rm
special case since ðl1 , l2 Þjðl1 , l2 Þ implies ðl1 , l2 Þjðl1 , l2 Þ. For example, let ðl1 , l2 Þ ¼ ð2,6:5Þ and ðl1 , l2 Þ ¼ ð3,4Þ. Then, we have l1 þ l2 4 l1 þ l2 , l1 l2 4 l1 l2 , and 1=l1 þ 1=l2 41=l1 þ1=l2 , and while X2:2 Z hr X2:2 does not hold, X2:2 Z mrl X2:2 does hold, as can be seen in Figs. 1 and 2. The following result is a direct consequence of Theorem 3.3, which has been given earlier by Zhao and Balakrishnan (2009b). Corollary 3.5. Let (X1, X2) be a vector of independent exponential random variables with respective hazard rates l1 and l2 , and (X*1,X*2) be another vector of independent exponential random variables with common hazard rate l. Then,
lZ
2 1
l1
þ
1
l2
()X2:2 Z mrl X2:2 :
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P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 141 (2011) 631–638
Fig. 1. Plot of the hazard rate functions of two lifetimes of parallel systems with exponential units, where h(t;a,b) denotes the hazard rate function with exponential parameter vector (a,b).
Fig. 2. Plot of the mean residual life functions of two lifetimes of parallel systems with exponential units, where jðt; a,bÞ denotes the mean residual life function with exponential parameter vector (a,b).
4. Extension to the PHR model Independent random variables X1,y, Xn are said to follow the proportional hazard rate (PHR) model if, for i= 1,y, n, the survival function of Xi can be written as F i ðxÞ ¼ ½F ðxÞli , where F ðxÞ is the baseline survival function of some random variable X. Let r(t) be the survival function of the baseline distribution F. Then, the hazard rate of Xi can be written as F i ðxÞ ¼ eli RðxÞ Rx for i=1,y,n, where RðxÞ ¼ 0 rðtÞ dt is the cumulative hazard rate of X. Evidently, the case of exponential random variables with hazard rates l1 , . . . , ln is a special case of the above PHR model. The next result extends Theorem 3.3 to the PHR model.
P. Zhao, N. Balakrishnan / Journal of Statistical Planning and Inference 141 (2011) 631–638
637 l1
l2
Theorem 4.1. Let (X1, X2) be a vector of independent random variables with respective survival functions F and F , and l l (X*1,X*2) be another vector of independent random variables with respective survival functions F 1 and F 2 . Suppose
minðl1 , l2 Þ r minðl1 , l2 Þ rmaxðl1 , l2 Þ r maxðl1 , l2 Þ: Then, if the baseline distribution F is DFR (decreasing failure rate), we have rm
: ðl1 , l2 Þjðl1 , l2 Þ¼)X2:2 Z mrl X2:2
Proof. We may write the cumulative hazard function of F as HðxÞ ¼ logF ðxÞ, and then, for i= 1, 2, we have li
PðHðXi Þ 4xÞ ¼ PðXi 4H1 ðxÞÞ ¼ F ðF
1
ðex ÞÞ ¼ eli x ,
where H1 ¼ inffy : HðyÞ Zxg. Note that Xui ¼ HðXi Þ is in fact an exponential random variable with hazard rate li ,i ¼ 1,2. Similarly, Yui ¼ HðXi Þ is also an exponential random variable with hazard rate li . From Theorem 3.3, it then follows that Xu2:2 Z mrl Yu2:2 , i.e., HðX2:2 Þ Z mrl HðX2:2 Þ:
The DFR property of F implies the function H1 ðÞ is increasing and convex, and so it follows from Theorem 2.A.19 of Shaked and Shanthikumar (2007) that Þ ¼ X2:2 , X2:2 ¼ H1 HðX2:2 Þ Z mrl H1 HðX2:2
and from which the desired result follows.
&
If X1,y, Xn are independent Weibull random variables with common shape parameter a 40 and scale parameter vectors a a ðl1 , . . . , ln Þ, then they belong to the PHR model with a new parameter vector ðl1 , . . . , ln Þ and a baseline survival function t a F ðtÞ ¼ e . Since F is DFR if a 2 ð0,1Þ, we have the following corollary which is a direct consequence of Theorem 4.1. Corollary 4.2. Let (X1, X2) be a vector of independent Weibull random variables with common shape parameter a 4 0 and scale parameter vector ðl1 , l2 Þ, and (X*1,X*2) be another vector of independent Weibull random variables with common shape parameter a and scale parameter vector ðl1 , l2 Þ. Suppose
minðl1 , l2 Þ r minðl1 , l2 Þ rmaxðl1 , l2 Þ r maxðl1 , l2 Þ: Then, for a 2 ð0,1Þ, we have a
a rm
ðl1 , l2 Þjððl1 Þa ,ðl2 Þa Þ¼)X2:2 Z mrl X2:2 :
Acknowledgements Authors would like to thank the anonymous referee for his/her insightful comments and suggestions, which resulted in an improvement in the presentation of this manuscript. Peng Zhao’s work was supported by National Natural Science Foundation of China (TY10926092), the Research Fund for the Doctoral Program of Higher Education (20090211120019), and the Fundamental Research Funds for the Central Universities (lzujbky-2010-64) and N. Balakrishnan’s work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. References Balakrishnan, N., 2007. Permanents, order statistics, outliers, and robustness. Revista Matema´tica Complutense 20, 7–107. Balakrishnan, N., Basu, A.P. (Eds.), 1995. The Exponential Distribution: Theory Methods and Applications. Gordon and Breach, Newark, New Jersey. Balakrishnan, N., Rao, C.R., (Eds.), 1998a. Handbook of Statistics. Order Statistics: Theory and Methods, vol. 16. Elsevier, Amsterdam. Balakrishnan, N., Rao, C.R., (Eds.), 1998b. Handbook of Statistics. Order Statistics: Applications, vol. 17. Elsevier, Amsterdam. Barlow, R.E., Proschan, F., 1975. Statistical Theory of Reliability and Life Testing: Probability Models. To Begin With: Silver Spring, Maryland. Boland, P.J., EL-Neweihi, E., Proschan, F., 1994. Schur properties of convolutions of exponential and geometric random variables. Journal of Multivariate Analysis 48, 157–167. Bon, J.L., Paˇltaˇnea, E., 1999. Ordering properties of convolutions of exponential random variables. Lifetime Data Analysis 5, 185–192. Bon, J.L., Paˇltaˇnea, E., 2006. Comparisons of order statistics in a random sequence to the same statistics with i.i.d. variables. ESAIM: Probability and Statistics 10, 1–10. David, H.A., Nagaraja, H.N., 2003. Order Statistics, third ed. John Wiley & Sons, Hoboken, NJ. Dykstra, R., Kochar, S.C., Rojo, J., 1997. Stochastic comparisons of parallel systems of heterogeneous exponential components. Journal of Statistical Planning and Inference 65, 203–211. Joo, S., Mi, J., 2010. Some properties of hazard rate functions of systems with two components. Journal of Statistical Planning and Inference 140, 444–453. Khaledi, B., Kochar, S.C., 2000. Some new results on stochastic comparisons of parallel systems. Journal of Applied Probability 37, 283–291. Khaledi, B., Kochar, S.C., 2002. Stochastic orderings among order statistics and sample spacings. In: Misra, J.C. (Ed.), Uncertainty and Optimality—Probability, Statistics and Operations Research. World Scientific Publishers, Singapore, pp. 167–203.
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