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Likelihood ratio ordering of order statistics, mixtures and systems
Journal of Statistical Planning and Inference 138 (2008) 1242 – 1257 www.elsevier.com/locate/jspi
Likelihood ratio ordering of order statistics, mixtures and systems夡 Jorge Navarro∗ Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain Received 20 February 2007; received in revised form 24 April 2007; accepted 27 April 2007 Available online 7 May 2007
Abstract Let X=(X1 , X2 , . . . , Xn ) be an exchangeable random vector, and denote X1:i =min{X1 , X2 , . . . , Xi } and Xi:i =max{X1 , X2 , . . . , Xi }, 1 i n. These order statistics represent the lifetimes of the series and the parallel systems, respectively, with component lifetimes Xi . In this paper we obtain conditions under which X1:i (or Xi:i ) decreases (increases) in i in the likelihood ratio (lr) order. An even more general result involving general (that is, not necessary exchangeable) random vectors is also derived for general series (or parallel) systems. We show that the series (parallel) systems are not necessarily lr-ordered even if the components are independent. The likelihood ratio order can be characterized in terms of Glaser’s function, defined by (t) = −f (t)/f (t) where f is the density function. This function is also a very useful tool to study the shape of hazard (or failure) rate and the mean residual life functions (see Glaser, R.E., 1980. Bathtub and related failure rate characterizations. J. Amer. Statist. Assoc. 75 (371), 667–672). It is also useful to study the likelihood ratio ordering and the increasing (or decreasing) likelihood ratio ILR (DLR) class. In this paper we also study properties of Glaser’s function of mixtures. Specifically, we study ordering properties, monotonicity and the limiting behaviour. We show that, under some conditions, the limiting behaviour is similar to that of the strongest member (in the likelihood ratio order) of the mixture. We also consider the case of finite negative mixtures (i.e. mixtures which have some negative coefficients) which is applied to study Glaser’s function of general coherent systems and order statistics and, in particular, the likelihood ratio ordering of coherent systems. The results are illustrated through a series of examples. © 2007 Elsevier B.V. All rights reserved. MSC: 60E15; 60K10 Keywords: Likelihood ratio order; ILR; DLR; Mixture; Hazard rate; Order statistics; k-Out-of-n system; Coherent system
1. Introduction The failure (hazard) rate and the mean residual life functions are very important in reliability and survival theories because they describe the aging process. It is well known that both of them uniquely determine the distribution function, i.e. they have all the information about the model. However, in several models (including mixture models), it is not easy to determine the shape of these functions.
夡
Partially supported by Ministerio de Ciencia y Tecnología under Grant MTM2006-12834 and Fundación Seneca under Grant 00698/PI/04.
∗ Fax: +34 968364 182.
E-mail address: [email protected]. 0378-3758/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2007.04.022
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For that reason, Glaser (1980) introduced the function (t) = −f (t)/f (t), where f is the density function, as a useful tool to determine the shape of the hazard rate. For example, he proved that if is increasing, then the hazard rate h is increasing. Thus, one can use function to prove that the hazard rate of normal and gamma models are increasing. Note that the hazard rate functions of these models do not have explicit expressions. A discrete version of Glaser’s function (called ‘equilibrium rate’) was defined in Shanthikumar and Yao (1986a, b, 1988). Glaser’s results were extended by Gupta and Warren (2001) and Navarro and Hernandez (2004). Moreover, can also be used to determine the shape of the mean residual life function and to study the likelihood ratio order and the increasing and decreasing likelihood ratio (ILR and DLR) classes (see Mi, 2004; Navarro and Hernandez, 2004). The mixtures of distributions are common models in reliability since they represent populations with different kinds of units. They can also be used to represent the distributions of coherent system lifetimes. Even if the shapes of the hazard rate functions of the members in the mixture are known, it is not easy to determine the shape of the hazard rate of the mixture. The same holds for the mean residual life. There exist several papers studying properties for the hazard rate and mean residual life functions of mixtures and systems (see, e.g., Block and Joe, 1997; Shaked and Spizzichino, 2001; Finkelstein, 2001; Navarro and Hernandez, 2004, 2007; Finkelstein and Esaulova, 2006a, b; Navarro and Shaked, 2006; Navarro and Eryilmaz, 2007 and the references given there). The purpose of this paper is to study properties for Glaser’s function of order statistics, systems and mixtures and apply them to study likelihood ratio ordering properties. We can also derive some properties for the hazard rate and the mean residual life functions. We also obtain properties for negative mixtures (i.e. mixtures with some negative coefficients) which can be applied to obtain properties of coherent systems, and in particular, for k-out-of-n systems (order statistics), by using that their distributions are negative mixtures of the distributions of series systems (see Navarro and Shaked, 2006; Navarro et al., 2007). The negative mixtures are also used to obtain the distribution of the MLEs of exponential parameters under step-stress models (see Balakrishnan and Xie, 2007a, b and Balakrishnan et al., 2007) or progressively censored order statistics (see Balakrishnan and Cramer, 2007). The paper is organized as follows. In Section 2, we introduce the basic definitions and known properties used in the paper. The main results are given in Section 3 which is devoted to study the likelihood ratio ordering of series and parallel systems. In Section 4, we study Glaser’s functions of generalized mixtures (i.e. mixtures which can have some negative weights). Finally, in Section 5, we apply the results on series systems and negative mixtures to obtain properties for order statistics and general coherent systems. Throughout the paper, ‘increasing’ stands for ‘non-decreasing’ and ‘decreasing’ stands for ‘non-increasing’. 2. Definitions and basic properties In this section we give basic definitions and known properties that we need to obtain the results included in this paper. We shall suppose that all the random variables considered in the paper have support (0, ∞) and differentiable density functions with continuous derivative in (0, ∞). However, the results in the paper can be easily extended to random variables with common support (, ∞) where −∞ < ∞. If X is a random variable with density function f satisfying these assumptions, Glaser’s function of X is defined by (t) = −f (t)/f (t) for t such that f (t) > 0. The random variable X belongs to increasing (decreasing) likelihood ratio ILR (DLR) class if f is log-concave (logconvex). A random variable X is less than or equal in the likelihood ratio order (denoted by X lr Y ) than another random variable Y if fX (t)/fY (t) decreases in t (see Shaked and Shanthikumar, 2007). Then, it is easy to obtain the following basic properties of . The proof of the first property is obtained by a straightforward integration of . Lemma 2.1. Under the preceding assumptions: (1) (2) (3) (4) (5)
t If t0 > 0 and f (t0 ) > 0, then f (t) = c exp(− t0 (x) dx) for t > 0. X is ILR (DLR) if, and only if, is increasing (decreasing) for t > 0. X lr Y if, and only t if, X Y . ∞ If f (t0 ) > 0, then t0 (x) dx is finite for all t > 0 and t0 (x) dx = ∞. If g(t) is a continuous function for t ∈ (0, ∞) satisfying the properties given in the preceding item, then g is Glaser’s function of a random variable X with support (0, ∞).
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The hazard rate h and the mean residual life m functions of a random variable X are defined by h(t) = f (t)/F (t) and ∞ 1 F (x) dx, (2.1) m(t) = E(X − t | X t) = F (t) t respectively, for t such that F (t) > 0, where F (t) = P (X t) is the reliability (or survival) function. When we consider m, we shall assume that it exists. The following lemma gives some relationships between functions , h and m. It is based on L’Hôpital’s rule and on properties given in Glaser (1980) and Navarro and Hernandez (2004). More properties are given in Gupta and Warren (2001) and Mi (2004). Lemma 2.2. Under the preceding assumptions: (1) (2) (3) (4)
If is increasing (decreasing), then h is increasing (decreasing) and m is decreasing (increasing). If limt→∞ (t) = where 0 ∞, then limt→∞ h(t) = . If limt→∞ h(t) = where 0 ∞, then limt→∞ m(t) = 1/. If limt→∞ (t) = where 0 ∞, then lim −
t→∞
log f (t) = . t
3. Likelihood ratio ordering of series and parallel systems Let (X1 , X2 , . . . , Xn ) be a random vector representing the lifetimes of n components in a system with absolutely continuous joint reliability (survival) function F (x1 , x2 , . . . , xn )=Pr(X1 x1 , X2 x2 , . . . , Xn xn ). Let X1:n X2:n · · · Xn:n be the corresponding order statistics which also represent the lifetimes of the k-out-of-n systems. In particular, X1:i = min(X1 , X2 , . . . , Xi ) represents the series system and Xi:i = min(X1 , X2 , . . . , Xi ) the parallel system for i = 1, 2, . . . , n. If ti = (t, t, . . . , t , 0, 0, . . . , 0) i times
n−i times
the reliability function of X1:i is F 1:i (t) = F (ti ). In general, the lifetime of the series system with components in P will be represented by YP = mini∈P Xi and its reliability function by F P (t). It is well known that the series systems are ordered in the usual stochastic order, i.e. X1:i st X1:i+1 . Hu et al. (2001) proved that they are lr-ordered when the components are independent and X1 lr X2 lr . . . lr Xn . Navarro and Shaked (2006) showed that the series systems X1:i and X1:i+1 are not necessarily ordered in the hazard rate (hr) order when the components are dependent. Therefore, as the likelihood ratio (lr) order is stronger than the hr-order, in general, the series systems are not likelihood ratio ordered. The purpose of this section is to obtain some conditions to have the lr-ordering of series systems. As a consequence, we will have that they are also hr and mrl ordered. Similar results are obtained for the parallel systems. Also, these results can be applied to obtain tail ordering properties of coherent systems and order statistics (see Section 5). We will say that F is exchangeable if F (x1 , x2 , . . . , xn ) = F (x(1) , x(2) , . . . , x(n) ) for any permutation . Clearly, if F is exchangeable and P has cardinality i, then YP =st X1:i , where =st denotes equality in law. We shall use the following notation: Di F (x1 , x2 , . . . , xn ) =
j F (x1 , x2 , . . . , xn ) jxi
and Di,j F (x1 , x2 , . . . , xn ) =
j j F (x1 , x2 , . . . , xn ). jxj jxi
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The main result of this section is given in the following proposition: Proposition 3.1. If F is exchangeable, twice differentiable and satisfies D1,1 F (tk ) D1 F (tk )
D1,1 F (tk+1 )
(3.1)
D1 F (tk+1 )
and (k − 1)
D1,2 F (tk ) D1 F (tk )
k
D1,2 F (tk+1 )
(3.2)
D1 F (tk+1 )
for all t 0, then X1:k lr X1:k+1 . Proof. The reliability function of X1:k is F 1:k (t) = F (tk ). Hence its density function is given by f1:k (t) = −
k
Di F (tk ).
(3.3)
i=1
Therefore, Glaser’s function 1:k of X1:k is given by k k (t) f1:k i=1 j =1 Di,j F (tk ) 1:k (t) = − =− . k f1:k (t) i=1 Di F (tk ) Using now that F is exchangeable, we have f1:k (t) = −kD 1 F (tk )
(3.4)
and 1:k (t) = −
(k − 1)D1,2 F (tk ) + D1,1 F (tk ) D1 F (tk )
.
Hence X1:k lr X1:k+1 if, and only if, (k − 1)D1,2 F (tk ) + D1,1 F (tk ) D1 F (tk )
kD 1,2 F (tk+1 ) + D1,1 F (tk+1 ) D1 F (tk+1 )
.
Finally, it is easy to check that this condition holds when (3.1) and (3.2) hold.
Condition (3.1) can be interpreted as follows. First, note that Glaser’s function of the random variable (X1 | X2 > t, . . . , Xk > t) at time t is (X1 |X2 >t,...,Xk >t) (t) = −
D1,1 F (tk ) D1 F (tk )
.
Hence (3.1) can also be written as (X1 |X2 >t,...,Xk >t) (t)(X1 |X2 >t,...,Xk+1 >t) (t).
(3.5)
This condition is similar to that used in Navarro and Shaked (2006) to obtain the hazard rate ordering of series systems. It is not clear the interpretation of (3.2). In particular, when k = 1, (3.2) can be written as D1,2 F (t2 )0 since D1 F 0. Moreover, in this case, we have D1,2 F (t2 ) = D1,2 F 1,2 (t, t) = f1,2 (t, t) 0,
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where F 1,2 (x1 , x2 )=Pr(X1 x1 , X2 x2 ) and f1,2 (x1 , x2 )=D1,2 F 1,2 (x1 , x2 ) are the reliability and density functions, respectively, of the random vector (X1 , X2 ). Hence, (3.2) always holds for k = 1. We do not know if (3.2) always holds for any k. If the components are independent and identically distributed (i.i.d.), then it is easy to see that (3.1) (or equivalently (3.5)) and (3.2) always holds and hence X1:k lr X1:k+1 . This result can also be obtained from Lillo et al. (2001) or Hu et al. (2001, Theorem 2.2) (see also Shaked and Shanthikumar, 2007, Theorem 1.C.31). The next proposition gives a sufficient condition to have (3.2). Proposition 3.2. If F is exchangeable, twice differentiable and satisfies D1,2 F (tk )D1,2 F (tk+1 )
(3.6)
for all t 0, then (3.2) holds for all t 0. Proof. Firstly, note that the joint reliability function can be written as F (x1 , . . . , xn ) = F 3,...,n (x3 , . . . , xn ) Pr(X1 x1 , X2 x2 | X3 x3 , . . . , Xn xn ),
(3.7)
where F 3,...,n (x3 , . . . , xn ) = Pr(X3 x3 , . . . , Xn xn ) is the reliability function of the random vector (X3 , . . . , Xn ). Therefore, D1,2 F (x1 , . . . , xn ) = F 3,...,n (x3 , . . . , xn )g(x1 , x2 | x3 , . . . , xn ) 0,
(3.8)
where g(x1 , x2 | x3 , . . . , xn ) is the density function of the random vector (X1 , X2 | X3 x3 , . . . , Xn xn ). Analogously, we can obtain that D1,k+1 F (x1 , . . . , xn ) 0
(3.9)
for k = 2, 3, . . . , n − 1. Hence D1 F (x1 , . . . , xn ) is increasing in xk+1 and 0 − (k − 1)
1 D1 F (tk )
−k
1 D1 F (tk+1 )
holds for all t 0 and for k = 2, 3, . . . , n − 1. Finally, if (3.6) holds, from (3.8) and (3.10), we obtain (3.2).
(3.10)
The next example shows that (3.6) is not necessary for (3.2) to hold. Example 3.3. Let (X1 , X2 , . . . , Xn ) have the following Farlie–Gumbel–Morgenstern exchangeable distribution with standard exponential marginals; that is, suppose that the reliability (survival) function of (X1 , X2 , . . . , Xn ) is given by
n n F (x1 , x2 , . . . , xn ) = e− i=1 xi 1 + (1 − e−xi ) , (x1 , x2 , . . . , xn ) (0, 0, . . . , 0), i=1
where ||1. It is not hard to verify that (3.1) and (3.2) hold for k = 1, 2, . . . , n − 2 since D1 F (tk ) = −F (tk ), and D1,1 F (tk ) = D1,2 F (tk ) = −D1 F (tk ) = F (tk ), for k = 1, 2, . . . , n − 1 when n 3. Therefore, X1:k lr X1:k+1 for k = 1, 2, . . . , n − 2 and n 3. However, note that (3.6) does not hold. Moreover, a tedious calculation (which can be obtained from the author) proves that (3.1) and (3.2) hold when 0 and therefore, X1:1 lr X1:2 · · · lr X1:n . The next proposition gives a necessary and sufficient condition to have the lr-ordering of series systems.
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Proposition 3.4. If F is exchangeable and differentiable, then X1:k lr X1:k+1 if, and only if, D1 F (tk ) D1 F (tk+1 )
is increasing in t
(3.11)
for all t 0. Proof. From the definition we have that X1:k lr X1:k+1 if, and only if, f1:k (t)/f1:k+1 (t) is increasing in t. Therefore, the stated result is obtained from (3.4). Similar results can be obtained when the joint distribution of the lifetimes of the components is not necessarily exchangeable for the general series systems YP = mini∈P Xi and YQ = mini∈Q Xi when P ⊂ Q by using that the density function fP of YP can be written as fP (t) = − Di F (tP ), i∈P
where tP = (z1 , z2 , . . . , zn ) with zi = t if i ∈ P and zi = 0 if i ∈ / P . For example, we have the following result: Proposition 3.5. If F is differentiable and P ⊂ Q, then YP lr YQ if, and only if, Di F (tP ) is increasing in t i∈P i∈Q Di F (tQ )
(3.12)
for all t 0. In particular, if the components are independent with reliability functions F i , i =1, 2, . . . , n, then (3.12) is equivalent to
i∈Q hi (t)
F i (t)
i∈Q−P
i∈P
hi (t)
is decreasing in t,
(3.13)
where hi (t) = fi (t)/F i (t) is the hazard function of Xi . Note that (3.13) holds when the components satisfy the proportional hazard rate model (PHR), i.e. when hi (t) = i h1 (t) for all t 0 and i = 2, 3, . . . , n and, in particular, when they are identically distributed. However, (3.13) is not necessarily true. For example, if Q = P ∪ {i} and the components in P are exponential, i.e. hj (t) = j for t 0 and j ∈ P , then (3.13) is equivalent to fi (t) F i (t) + is decreasing in t j ∈P j for all t 0. Differentiating, (3.14) holds if, and only, if i (t) − i
(3.14)
(3.15)
i∈P
for all t 0. In the next example, (3.15) is not true and hence the series systems are not lr-ordered even if the components are independent. Example 3.6. Let X1:2 = min(X1 , X2 ) be the lifetime of a series system with two independent components with lifetimes X1 and X2 with exponential and positive-truncated normal distributions, respectively. Then, the density 2 2 functions are given by f1 (t)=e−t/1 /1 and f2 (t)=ce−(t−2 ) /22 for t 0, respectively. Therefore, Glaser’s functions are given by 1 (t) = h1 (t) = 1/1 and 2 (t) = (t − 2 )/22 for t 0, respectively. Hence from (3.15), X1 lr X1:2 holds if, and only if, t 2 − 22 /1
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Fig. 1. Glaser’s functions 1 (continuous constant line), 2 (continuous non-constant line), 2:2 (dashed line below) and 1:2 (dashed line above) of X1 , X2 , X2:2 = max(X1 , X2 ) and X1:2 = min(X1 , X2 ), respectively, when X1 has an exponential distribution of mean 1, X2 has a positive-truncated normal distribution with mean 2 and variance 1 and X1 and X2 are independent (see Example 3.6).
for all t 0. Then it holds if, and only if, 2 − 22 /1 0. Note that this last expression is not necessarily true. For example, it is not true when 1 = 1, 2 = 2 and 2 = 1. Therefore, the series systems X1 = X1:1 and X1:2 are not lr-ordered. Fig. 1 shows the functions 1 (constant line), 2 (continuous non-constant line), 2:2 (dashed line below) and 1:2 (dashed line above) of X1 , X2 , X2:2 = max(X1 , X2 ) and X1:2 = min(X1 , X2 ), respectively, when 1 = 1, 2 = 2 and 2 = 1. Note that X1:2 and X2:2 are lr-ordered. However note that X1 and X2:2 are not lr-ordered. Also note that X2 and X2:2 are not lr-ordered. Moreover 1 (Glaser’s function of the best component) and 2:2 are asymptotically (when t → ∞) equivalent. We shall see (in Section 5) that this is a general property. Similar results can be obtained when the joint distribution of the lifetimes of the components is not necessarily exchangeable for the general parallel systems ZP = maxi∈P Xi and ZQ = maxi∈Q Xi when P ⊂ Q by using that the density function gP of ZP can be written as Di F (tP ), gP (t) = i∈P
/ P . For example, we have the following result. where tP = (z1 , z2 , . . . , zn ) with zi = t if i ∈ P and zi = ∞ if i ∈ Proposition 3.7. If F is differentiable and P ⊂ Q, then ZP lr ZQ if, and only if, Di F (tP ) i∈P is decreasing in t Q i∈Q Di F (t )
(3.16)
for all t 0. In particular, if the components are independent with distribution functions Fi , i=1, 2, . . . , n, then (3.16) is equivalent to i∈Q−P
i∈Q hi (t)
Fi (t)
i∈P
hi (t)
is increasing in t
(3.17)
for all t 0, where hi (t)=fi (t)/Fi (t) is the reversed hazard function of Xi . Note that (3.17) holds when the components satisfy the proportional reversed hazard rate model (PRHR), i.e. when hi (t)=i h1 (t) for all t 0 and i =2, 3, . . . , n and, in particular, when they are identically distributed. However, (3.17), and hence Xk−1:k−1 lr Xk:k , are not necessarily
J. Navarro / Journal of Statistical Planning and Inference 138 (2008) 1242 – 1257
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true even if the components are independent (see Fig. 1). Hu et al. (2001, Theorem 2.1) proved that Xk−1:n−1 lr Xk:n for all k = 2, 3, . . . , n when the components are independent and X1 lr X2 lr · · · lr Xn . 4. Properties of finite generalized mixtures Firstly we study properties of the two members generalized mixture density function defined by fp (t) = pf 1 (t) + (1 − p)f0 (t)
(4.1)
for t 0, where f1 and f0 are two density functions and p is a real number. Xp shall denote any random variable having density fp . Thus, we say that Xp is a generalized mixture of X1 and X0 . Without loss of generality we can assume that p 0. However, note that 1 − p can be negative (when p > 1). In this case, we shall say that fp is a negative mixture. If 0 p 1, then fp is a usual or positive mixture and the RHS of (4.1) always defines a density function. In general, it is easy to see that the RHS of (4.1) defines a density function for 0 p pmax where pmax = 1/(1 − ) and = min(1, inf(f1 /f0 )) (we adopt here the convention that 0/0 = ∞). Moreover, the case = 1 leads to the trivial case f1 =a.s. f0 . Obviously, if = 0 then pmax = 1 and the negative mixture cannot be considered. The generalized mixtures were defined in Everitt and Hand (1981), Wu and Lee (1998) and Wu (2001). In practice, they can be used to define families of distributions and to represent the distributions of order statistics and coherent systems as we shall see in the last section (see also Baggs and Nagaraja, 1996; Navarro and Shaked, 2006; Navarro et al., 2007). The negative mixtures are also used to obtain the distribution of the MLEs of exponential parameters under step-stress models (see Balakrishnan and Xie, 2007a, b; Balakrishnan et al., 2007) or progressively censored order statistics (see Balakrishnan and Cramer, 2007). They also appeared as marginal distributions of multivariate models in Kotz et al. (2000, p. 356). Moreover, note that, if 1 < p pmax , then f1 (t) = p −1 fp (t) + p −1 (p − 1)f0 (t),
(4.2)
that is, f1 is a positive (usual) mixture of fp and f0 . If p is Glaser’s function associated to the mixture density fp given by (4.1) and 1 and 0 are Glaser’s functions of the members of the mixture f1 and f0 , then, (t) = W1 (t)1 (t) + (1 − W1 (t))0 (t),
(4.3)
where W1 (t) = pf 1 (t)/fp (t). Thus, if fp is a positive mixture (i.e. 0 p 1), then 0 W1 (t) 1 and min(1 (t), 0 (t))(t) max(1 (t), 0 (t))
(4.4)
for all t 0, that is, Glaser’s function of a positive mixture is between the corresponding functions of the members of the mixture. In particular, if X0 lr X1 , then, from Lemma 2.1, X0 lr Xp lr X1 for 0 p 1. Moreover, if fp is a negative mixture (i.e. 1 < p pmax ) then, from (4.2) and (4.4), we have min(p (t), 0 (t))1 (t) max(p (t), 0 (t))
(4.5)
for all t 0, that is, Glaser’s function of a negative mixture is in the opposite region of 0 with respect to the regions determined by 1 . In particular, if X0 lr X1 , then, from Lemma 2.1, X1 lr Xp for 1 p pmax . In the following proposition we show that Glaser’s functions of two-member generalized mixtures are ordered. Proposition 4.1. If (4.1) holds for 0 p pmax and 0 (t) 1 (t)( ) for a t 0, then j (t)0 jp p
(0).
Proof. From the definitions, we have p (t) = −
fp (t) fp (t)
=−
pf 1 (t) + (1 − p)f0 (t) , pf 1 (t) + (1 − p)f0 (t)
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and by differentiating, we obtain j f0 (t)f1 (t) (1 (t) − 0 (t)), (t) = jp p fp2 (t) and hence, the stated result.
In particular, if the members of the mixture are ordered in the likelihood ratio order X0 lr X1 , then Xp lr Xp for 0 p p pmax . This result extends the result given in Navarro and Hernandez (2004, Proposition 8). Some of the properties given before can be extended to finite generalized mixture density functions defined by n
f (t) =
pi fi (t)
(4.6)
i=1
for t 0, where fi are density functions and pi are real numbers for i = 1, 2, . . . , n. X and Xi shall denote random variables having density functions f and fi , respectively. Thus, we say that X is generalized mixture of X1 , X2 , . . . , Xn . Without loss of generality we can assume that pi = 0 for all i. However, note that some pi can be negative. In this case, we shall say that f is a negative mixture. If 0 pi 1 for all i, then f is a usual or positive mixture and the RHS of (4.1) always defines a density function. In general, it is easy to see that the RHS of (4.1) does not define necessarily a density function. However, we shall suppose that f in (4.6) is a density function. Firstly, note that f can be written as f (t) = pf + (t) + (1 − p)f− (t), (4.7) where p = i:pi >0 pi , f+ (t) = i:pi >0 p −1 pi fi (t) and f− (t) = i:pi <0 (1 − p)−1 pi fi (t). If p > 1, then f is a two members negative mixture of the density functions f+ and f− obtained as positive (usual) mixtures of the density functions with positive and negative coefficients, respectively. If is Glaser’s function associated to the mixture density f given by (4.6) and i are Glaser’s functions of the members of the mixture fi , i = 1, 2, . . . , n, then, (t) =
n
Wi (t)i (t),
(4.8)
i=1
where Wi (t) = pi fi (t)/f (t). Thus, if f is a positive mixture (i.e. 0 pi 1), then 0 Wi (t) 1 and min
i=1,2,...,n
i (t)(t)
max
i=1,2,...,n
i (t)
(4.9)
for all t 0, that is, Glaser’s function of a positive mixture is between the corresponding functions of the members of the mixture. In particular, if Xi lr Xj ( lr ) for all j, then, from Lemma 2.1, Xi lr X( lr ). If f is a negative mixture (i.e. p > 1 in (4.7)), then, from (4.2) and (4.7), we have min((t), − (t))+ (t) max((t), − (t))
(4.10)
for all t 0. A well known property of positive mixtures is that the mixture of DFR (i.e. decreasing failure (hazard) rate) distributions is also DFR. A similar result holds for the IMRL (increasing mean residual life) class. However, the property is not true for IFR (increasing failure rate) and DMRL (decreasing mean residual life) classes. Now, we obtain similar properties for Glaser’s function extending the result given in Navarro and Hernandez (2004, Proposition 10). Proposition 4.2. If (4.6) holds and 1 , 2 , . . . , n are differentiable in t, then (t) =
n
Wi (t)i (t) −
i=1
where Wi (t) = pi fi (t)/f (t).
i
Wi (t)Wj (t)(i (t) − j (t))2 ,
(4.11)
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Proof. From (4.8), we have (t) =
n
Wi (t)i (t) +
i=1
n
Wi (t)i (t),
(4.12)
i=1
where ⎛ ⎞ n Wj (t)(j (t) − i (t))⎠ . Wi (t) = Wi (t)((t) − i (t)) = Wi (t) ⎝ j =1
The proof is completed by replacing the preceding expression in (4.12).
As an immediate consequence, we have the following property. Proposition 4.3. If (4.6) holds for 0 pi 1 and i (t) 0 for a t 0 and i = 1, 2, . . . , n, then (t) 0. Moreover, for negative mixtures, we have the following result. Proposition 4.4. If (4.6) holds for p1 1 and pi 0, i = 2, 3, . . . , n, 1 (t) 0 and i (t) 0, i = 2, 3, . . . , n, for a t 0, then (t)0. Proof. The proof for n=2 is obtained from (4.11). Then, the proof for n > 2 is obtained from Proposition 4.3, expression (4.7) and the case n = 2. In particular, if the preceding propositions can be applied for all t 0, we obtain that the positive mixtures of DLR distributions are also DLR and that the negative mixtures of an ILR distribution with a positive weight and DLR distributions with negative weights are ILR. Now, we are going to study properties on the limiting behaviour of function of a finite generalized mixture. A well known property of positive mixtures is that the hazard rate of the mixture has (under some regularity conditions) the same (in different senses) limiting behaviour when t → ∞ as that of the hazard rate of the strongest member of the mixture in the hazard rate order (see Block and Joe, 1997; Mi, 1999; Finkelstein, 2001; Finkelstein and Esaulova, 2006a, b; Navarro and Shaked, 2006). To obtain similar properties on the limiting behaviour of , we need some lemmas which have also interest by themselves. Lemma 4.5. If 1 (t) > 0
for all t t1
(4.13)
and lim inf t→∞
0 (t) > 1, 1 (t)
(4.14)
then lim
t→∞
f0 (t) = 0. f1 (t)
(4.15)
Proof. From (4.14), there exist ε > 0 and t0 > 0 such that 0 (t) >1 + ε 1 (t)
(4.16)
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for t t0 . Moreover, from (4.13) and (4.16), we obtain 0 (t) − 1 (t) > ε1 (t)
(4.17)
for t max(t0 , t1 ). Hence, if a max(t0 , t1 ) and fi (a) > 0 for i = 0, 1, we have t f0 (t) c0 = exp − (0 (x) − 1 (x)) dx f1 (t) c1 a t c0 exp − ε1 (x) dx c1 a =
c01−ε ε f (t) c1 1
for t a, where the equalities are obtained from Lemma 2.1 and the inequality from (4.17). Therefore, (4.15) holds since limt→∞ f1 (t) = 0 and ε > 0. Lemma 4.6. If (4.6), 1 (t) > 0
for all t t1
(4.18)
and lim inf t→∞
i (t) > 1 for i = 2, 3, . . . , n, 1 (t)
(4.19)
hold, then lim
t→∞
f (t) = p1 > 0. f1 (t)
(4.20)
The proof is immediate from Lemma 4.5. Now, we can obtain the main result on the asymptotic behaviour of . Theorem 4.7. If (4.6), (4.18), (4.19) and fi (t) = i t→∞ f (t) 1 lim
where − ∞i ∞
for i = 2, 3, . . . , n
(4.21)
hold, then (t) = 1. t→∞ 1 (t) lim
(4.22)
Proof. Firstly note that, from L’Hôpital’s rule, Lemma 4.5 and (4.21), we have f (t) fi (t) = lim i = i = 0 t→∞ f1 (t) t→∞ f (t) 1
(4.23)
lim
for i = 2, 3, . . . , n. Hence,
n fi (t) (t) f (t) f1 (t) f1 (t) pi = = p1 + 1 (t) f1 (t) f (t) f (t) f1 (t)
(4.24)
i=2
and therefore, from Lemma 4.6 and (4.23), we obtain (4.22).
The next lemma shows that condition (4.21) can be replaced by the stronger condition (4.25). However, in practice, sometimes, (4.25) is easier to check than (4.21) (see Example 5.6).
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Lemma 4.8. If (4.18), (4.19) and lim sup t→∞
i (t) < ∞ for i = 2, 3, . . . , n 1 (t)
(4.25)
hold, then (4.21) holds. Proof. From the definitions, we have f0 (t) f0 (t) 0 (t) = lim t→∞ f (t) t→∞ f1 (t) 1 (t) 1 lim
and, from Lemma 4.5 and (4.25), (4.21) holds.
Block and Joe (1997) and Navarro and Shaked (2006) obtained similar results for the hazard rate function. Block and Joe (1997) also obtain some asymptotic results for infinite countable and continuous mixtures. Analogous results for the mean residual life were obtained in Navarro and Hernandez (2007). The following example show how to apply Theorem 4.7. Example 4.9. Let us suppose that (4.1) holds for 0 < p pmax and for Gamma densities, i.e. fi (t) =
biai ai −1 exp(−bi t) t (ai )
for t 0, ai , bi > 0 and i = 0, 1. Then, i (t) = (1 + bi t − ai )/t. Hence Xi is ILR (DLR) if ai 1 ( ) and lim (t) = bi t→∞ i
> 0.
Therefore Xi is IFR (DFR) if ai 1 ( ) and limt→∞ hi (t) = bi > 0. Note that there is not explicit expression for the hazard function hi (t). We assume b1 b0 (the tail leading member of the mixture in the lr-order has a positive weight). Then let us to consider three cases. Firstly, if a1 > a0 , then = 0, and pmax = 1 (the negative mixture cannot be considered). In this case, 1 < 0 and hence the positive mixtures are lr (fr and mrl) ordered in p (i.e. X0 lr Xp lr X1 ). Secondly, if a1 = a0 and b1 < b0 , then = (b1 /b0 )a1 > 0 and pmax = b0a1 /(b0a1 − b1a1 ) > 1 (the negative mixtures can be considered). Thirdly, if a1 < a0 and b1 b0 , then
a (a0 ) b11 b0 − b1 a0 −a1 a0 −a1 = min 1, >0 e (a1 ) b0a0 a0 − a1 and pmax > 1 (the negative mixtures can be considered). If a1 < a0 and b1 = b0 , then = 0 and pmax = 1 (the negative mixtures cannot be considered). Thus, by applying Theorem 4.7, if b1 < b0 , we obtain that lim hp (t) = lim p (t) = b1 > 0.
t→∞
t→∞
(4.26)
If b1 = b0 , only positive mixtures can be considered and, as they are ordered from (4.4), we also obtain (4.26). In a similar way, if the mean residual life function exists, then limt→∞ mp (t) = 1/b1 . Moreover, if a1 1 and a0 1, then p and hp are decreasing for p 1. Analogously, if b1 < b0 , a1 = a0 = 1, then p and hp are increasing for p > 1. Fig. 2 shows p when a1 = 1, a0 = 2, b1 = 2, b0 = 3 for p = k/5 and k = 0, 1, . . . , 9. In this case, pmax = 9/(9 − 2e) = 2.525651964 . . . and hence, the negative mixtures can be considered for p pmax . The functions 0 and 1 are plotted with dashed lines. Note that 0 is strictly increasing, 1 is constant (i.e. X1 has an exponential distribution),
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J. Navarro / Journal of Statistical Planning and Inference 138 (2008) 1242 – 1257
Fig. 2. Glaser’s functions p in Gamma generalized mixtures when a1 = 1, a0 = 2, b1 = 2, b0 = 3 for p = 0.2 ∗ k and k = 0, 1, . . . , 9 (from the bottom to the top at t = 0). The dashed lines represent Glaser’s functions of the members of the mixture (1 = 2 and 0 ).
but p is not monotone for p = 0, 1. Also note that p (t) are ordered in p at each t and that limt→∞ p (t) = b1 = 2. Therefore, from Lemma 2.2, limt→∞ hp (t) = 2 and limt→∞ mp (t) = 21 . 5. Properties of order statistics and systems Baggs and Nagaraja (1996) showed that F 1:2 (t) + F 2:2 (t) = F 1 (t) + F 2 (t),
(5.1)
where F i (t)=Pr(Xi > t), i=1, 2, are the component (marginal) reliability functions. They considered different bivariate distributions which have exponential marginal distributions and exponential series system distributions. Hence from (5.1), the distribution of the two-component parallel system is a generalized mixture of three exponential distributions. They studied these kind of generalized mixtures and they applied the results to systems. Note that marginal distributions can be seen as one-component series distributions. Hence (5.1) shows that any two-component parallel system is a generalized mixture of series systems. If T is the lifetime of a coherent system with component lifetimes given by X1 , X2 , . . . , Xn , the result given by Baggs and Nagaraja (1996) can be extended to T by using the inclusion–exclusion method given in Agrawal and Barlow (1984) (see also Navarro et al., 2007) showing that any coherent system is a generalized mixture of series systems, i.e.
m m F T (t) = Pr (YPi > t) = F Pi (t) − F Pi ∪Pj (t) + · · · + (−1)m+1 F P1 ∪···∪Pm (t), (5.2) i=1
i=1
i
where F P (t) = Pr(YP > t) is the reliability function of the series system YP = mini∈P Xi with components in P and P1 , . . . , Pm are the minimal path sets of the coherent system. The definition of the minimal path sets of a coherent system can be seen in Barlow and Proschan (1975). A set P of components is a path set if the systems works when all the components in P work. A minimal path set is a path set which does not contain proper path sets. In particular, if the components are independent, then F P (t) = i∈P F i (t). Navarro et al. (2007) gave a similar result for parallel systems by using the minimal cut sets. Moreover, when (X1 , . . . , Xn ) has an exchangeable distribution, the distribution of the series (parallel) system only depends on the number of components. In this case, Navarro et al. (2007) defined the minimal signature of a coherent system T as the vector (1 , . . . , n ) obtained from (5.2) such that F T (t) = ni=1 i F 1:i (t). They also defined the maximal signature in a similar way by using the parallel systems. Hence the results for generalized mixtures can be used to obtain properties for coherent systems from properties of series (or parallel) systems. For example, from Theorem 4.7 and Lemma 4.8, we have the following result.
J. Navarro / Journal of Statistical Planning and Inference 138 (2008) 1242 – 1257
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Proposition 5.1. If T is a coherent system with reliability F T , Glaser’s function T and minimal path sets P1 , . . . , Pm such that P1 (t)0
for all t > t1 ,
(5.3)
lim inf
P (t) > 1, P1 (t)
(5.4)
lim sup
P (t) < ∞, P1 (t)
(5.5)
t→∞
and t→∞
where P = lim
t→∞
i∈A Pi
for all A ⊆ {1, . . . , m} (A = {1}), then
T (t) = 1. P1 (t)
(5.6)
Remark 5.2. If (5.3)–(5.5) hold, then the path set P1 can be called the leading path set. Hence the tail behaviour of Glaser’s function of the system is equivalent to that of the series system obtained from its leading path set. Note that, from Theorem 4.7, (5.5) can be replaced by the weaker condition lim f (t)/fP 1 (t) = P , t→∞ P where −∞P ∞ and P =
(5.7)
i∈A Pi
for all A ⊆ {1, . . . , m}.
Remark 5.3. In Example 3.6, the parallel system X2:2 has minimal path sets P1 = {1} and P2 = {2} and Glaser’s functions of the components satisfy t − 2 2 (t) = ∞. = lim 1 t→∞ 1 (t) t→∞ 22 lim
Hence (5.5) does not hold. However, f2 (t) =0 t→∞ f (t) 1 lim
and hence (5.7) holds for P = P2 . This example shows that condition (5.7) is strictly weaker than condition (5.5). It is easy to see that (5.3), (5.4) and (5.7) hold for P2 and P = {1, 2}. Therefore, limt→∞ 2:2 (t) = P1 (t) = 1/1 (see Fig. 1). Remark 5.4. If the components are independent, then h (t) P (t) = hi (t) − i∈P i i∈P hi (t) i∈P
and if they have exponential distributions with hazard rate function hi (t) = i , then P (t) = i∈P i and (5.3)–(5.5) hold for P1 such that i∈P1 i < i∈Pj i for j = 2, 3, . . . , m. If the components are independent and they satisfy the PHR model then, from (3.13), the series systems are lr-ordered and hence (5.4) and (5.5) can be replaced by 1 < lim inf t→∞
Pj (t) P1 (t)
lim sup t→∞
Pj (t) P1 (t)
<∞
for j = 2, 3, . . . , m.
Example 5.5. Consider a 2-out-of-3 system with lifetime X2:3 and minimal path sets P1 = {1, 2}, P2 = {2, 3} and P3 = {1, 3}. Thus, from (5.2), the system reliability is given by F 2:3 (t) = F {1,2} (t) + F {2,3} (t) + F {1,3} (t) − 2F 1:3 (t).
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Fig. 3. Glaser’s functions of all the coherent system with three independent exponential components with 1 = 2 = 3 = 1. The dashed lines represent the asymptotic behaviours (see Example 5.5).
If the components have exchangeable joint distribution, then F 2:3 (t)=3F 1:2 (t)−2F 1:3 (t). Hence its minimal signature is (0, 3, −2). If we suppose that the components are independent and that they have exponential distributions with hi (t) = i for t 0 and 1 < 2 < 3 , then the system distribution is a negative mixture of four exponential distributions. Moreover, from the preceding proposition we have limt→∞ 2:3 (t) = 1 + 2 . Note that, if 1 = 2 = 3 , then it is a negative mixture of two exponential distributions and hence it is ILR. Fig. 3 shows Glaser’s functions of all the coherent systems with three independent exponential components with 1 = 2 = 3 = 1. The definitions of these systems can be seen in Shaked and Suarez-Llorens (2003) and their minimal signatures are (0, 0, 1), (0, 2, −1), (0, 3, −2), (1, 1, −1) and (3, −3, 1) (see Navarro and Shaked, 2006 or Navarro et al., 2007). Note that they are lr-ordered. Also note that asymptotically they are equivalent to exponential distributions. Example 5.6. If the lifetimes (X1 , X2 , . . . , Xn ) of the components in a system have the Farlie–Gumbel–Morgenstern exchangeable distribution with standard exponential marginals given in Example 3.3, then it is not hard to verify that the reliability function of the series system YP = mini∈P Xi for P ⊂ {1, 2, . . . , n} is F P (t) = exp(−|P |t) for t 0, where |P | represents the number of elements in P. Hence it has an exponential distribution and P (t) = |P |. Analogously, when P = {1, 2, . . . , n}, the reliability function is given by F P (t) = F 1:n (t) = exp(−nt)(1 + (1 − e−t )n ) for t 0. A straightforward calculation gives 1:n (t) = n − e−t (1 − e−t )n−2
1 + n(1 − 2e−t ) 1 + (1 − 2e−t )(1 − e−t )n−1
for t 0. Hence limt→∞ 1:n (t) = n. Therefore, the series systems are asymptotically lr-ordered and (5.3)–(5.5) hold for any coherent system. Note that, in this example, (5.5) is easier to check than (5.7). For example, if we consider the k-out-of-n system whose lifetime is represented by the order statistics Xn−k+1:n , its minimal path sets are all the sets with k elements. Therefore, from (5.2), its reliability function is given by F n−k+1:n (t) =
n j =k
(−1)j −k
n j −1 j
k−1
exp(−jt)
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for t 0. Hence, from Proposition 5.1, limt→∞ n−k+1:n (t) = k. Moreover, from Lemma 2.2, limt→∞ hn−k+1:n (t) = k and limt→∞ mn−k+1:n (t) = 1/k. Acknowledgements I want to thank Professor Moshe Shaked and the anonymous referees for several helpful comments during the preparation of this paper. References Agrawal, A., Barlow, R.E., 1984. A survey of network reliability and domination theory. Oper. Res. 12, 611–631. Baggs, G.E., Nagaraja, H.N., 1996. Reliability properties of order statistics from bivariate exponential distributions. Commun. Statist. Stochastic Models 12, 611–631. Balakrishnan, N., Cramer, E., 2007. Progressive censoring from heterogeneous distributions with applications to robustness. Ann. Inst. Statist. Math., in press, doi: 10.1007/s10463-006-0070-8. Balakrishnan, N., Xie, Q., 2007a. Exact inference for a simple step-stress model with type-I hybrid censored data from the exponential distribution. J. Statist. Plann. Inference, in press, doi: 10.1016/j.jspi.2007.03.011. Balakrishnan, N., Xie, Q., 2007b. Exact inference for a simple step-stress model with type-II hybrid censored data from the exponential distribution. J. Statist. Plann. Inference 137, 2543–2563. Balakrishnan, N., Xie, Q., Kundu, D., 2007. Exact inference for a simple step-stress model from the exponential distribution under time constraint. Ann. Inst. Statist. Math., to appear. Barlow, R.E., Proschan, F., 1975. Statistical Theory of Reliability and Life Testing: Reliability Models. Holt, Rinehart & Winston, New York. Block, H.W., Joe, H., 1997. Tail behavior of the failure rate functions of mixtures. Lifetime Data Anal. 3, 269–288. Everitt, B.S., Hand, D.J., 1981. Finite Mixture Distributions. Chapman & Hall, New York. Finkelstein, M.S., 2001. The failure rate and the mean residual lifetime of mixtures. System and Bayesian Reliability, Ser. Qual. Reliab. Eng. Stat., vol. 5. World Scientific Publishing, River Edge, NJ, pp. 165–183. Finkelstein, M.S., Esaulova, V., 2006a. Asymptotic behavior of a general class of mixture failure rates. Adv. in Appl. Probab. 38 (1), 244–262. Finkelstein, M.S., Esaulova, V., 2006b. On mixture failure rates ordering. Comm. Statist. Theory Methods 35 (11), 1943–1955. Glaser, R.E., 1980. Bathtub and related failure rate characterizations. J. Amer. Statist. Assoc. 75 (371), 667–672. Gupta, R.C., Warren, R., 2001. Determination of change points of non-monotonic failure rates. Comm. Statist. Theory Methods 30, 1903–1920. Hu, T., Zhu, Z., Wei, Y., 2001. Likelihood ratio and mean residual life orders for order statistics of heterogeneous random variables. Probab. Eng. Inform. Sci. 15, 259–272. Kotz, S., Balakrishnan, N., Johnson, N.L., 2000. Continuous Multivariate Distributions. Wiley, New York. Lillo, R.E., Nanda, A.K., Shaked, M., 2001. Preservation of some likelihood ratio stochastic orders by order statistics. Statist. Probab. Lett. 51 (2), 111–119. Mi, J., 1999. Age-mooth properties of mixture models. Statist. Probab. Lett. 43, 225–236. Mi, J., 2004. A general approach to the shape of failure rate and MRL functions. Naval Res. Logist. 51, 543–556. Navarro, J., Eryilmaz, S., 2007. Mean residual lifetimes of Consecutive k-out-of-n systems. J. Appl. Probab. 44, 82–98. Navarro, J., Hernandez, P.J., 2004. How to obtain bathtub-shaped failure rate models from normal mixtures. Probab. Eng. Inform. Sci. 18 (4), 511–531. Navarro, J., Hernandez, P.J., 2007. Mean residual life functions of finite mixtures, order statistics and coherent systems. Metrika, in press, doi: 10.1007/s00184-007-0133-8. Navarro, J., Shaked, M., 2006. Hazard rate ordering of order statistics and systems. J. Appl. Probab. 43, 391–408. Navarro, J., Ruiz, J.M., Sandoval, C.J., 2007. Modelling coherent systems under dependence. Comm. Statist. Theory Methods 36 (1), 175–191. Shaked, M., Shanthikumar, J.G., 2007. Stochastic Orders. Springer Series in Statistics. Springer, New York. Shaked, M., Spizzichino, F., 2001. Mixtures and monotonicity of failure rate functions. Advances in Reliability. North-Holland, Amsterdam. pp. 185–198. Shaked, M., Suarez-Llorens, A., 2003. On the comparison of reliability experiments based on the convolution order. J. Amer. Statist. Assoc. 98, 693–702. Shanthikumar, J.G., Yao, D.D., 1986a. The preservation of likelihood ratio ordering under convolutions. Stochastic Process. Appl. 23, 259–267. Shanthikumar, J.G., Yao, D.D., 1986b. The effect of increasing service rates in a closed queueing network. J. Appl. Probab. 23, 474–483. Shanthikumar, J.G., Yao, D.D., 1988. Second-order properties of the throughput of a closed queueing network. Math. Oper. Res. 13 (3), 524–534. Wu, J.W., 2001. Characterizations of generalized mixtures of geometric and exponential distributions based on upper record values. Statist. Papers 42, 123–133. Wu, J.W., Lee, W.C., 1998. Characterization of generalized mixtures of exponential distribution based on conditional expectation of order statistics. J. Japan Statist. Soc. 28, 39–44.
Likelihood ratio ordering of order statistics, mixtures and systems夡 Jorge Navarro∗ Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain Received 20 February 2007; received in revised form 24 April 2007; accepted 27 April 2007 Available online 7 May 2007
Abstract Let X=(X1 , X2 , . . . , Xn ) be an exchangeable random vector, and denote X1:i =min{X1 , X2 , . . . , Xi } and Xi:i =max{X1 , X2 , . . . , Xi }, 1 i n. These order statistics represent the lifetimes of the series and the parallel systems, respectively, with component lifetimes Xi . In this paper we obtain conditions under which X1:i (or Xi:i ) decreases (increases) in i in the likelihood ratio (lr) order. An even more general result involving general (that is, not necessary exchangeable) random vectors is also derived for general series (or parallel) systems. We show that the series (parallel) systems are not necessarily lr-ordered even if the components are independent. The likelihood ratio order can be characterized in terms of Glaser’s function, defined by (t) = −f (t)/f (t) where f is the density function. This function is also a very useful tool to study the shape of hazard (or failure) rate and the mean residual life functions (see Glaser, R.E., 1980. Bathtub and related failure rate characterizations. J. Amer. Statist. Assoc. 75 (371), 667–672). It is also useful to study the likelihood ratio ordering and the increasing (or decreasing) likelihood ratio ILR (DLR) class. In this paper we also study properties of Glaser’s function of mixtures. Specifically, we study ordering properties, monotonicity and the limiting behaviour. We show that, under some conditions, the limiting behaviour is similar to that of the strongest member (in the likelihood ratio order) of the mixture. We also consider the case of finite negative mixtures (i.e. mixtures which have some negative coefficients) which is applied to study Glaser’s function of general coherent systems and order statistics and, in particular, the likelihood ratio ordering of coherent systems. The results are illustrated through a series of examples. © 2007 Elsevier B.V. All rights reserved. MSC: 60E15; 60K10 Keywords: Likelihood ratio order; ILR; DLR; Mixture; Hazard rate; Order statistics; k-Out-of-n system; Coherent system
1. Introduction The failure (hazard) rate and the mean residual life functions are very important in reliability and survival theories because they describe the aging process. It is well known that both of them uniquely determine the distribution function, i.e. they have all the information about the model. However, in several models (including mixture models), it is not easy to determine the shape of these functions.
夡
Partially supported by Ministerio de Ciencia y Tecnología under Grant MTM2006-12834 and Fundación Seneca under Grant 00698/PI/04.
∗ Fax: +34 968364 182.
E-mail address: [email protected]. 0378-3758/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2007.04.022
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For that reason, Glaser (1980) introduced the function (t) = −f (t)/f (t), where f is the density function, as a useful tool to determine the shape of the hazard rate. For example, he proved that if is increasing, then the hazard rate h is increasing. Thus, one can use function to prove that the hazard rate of normal and gamma models are increasing. Note that the hazard rate functions of these models do not have explicit expressions. A discrete version of Glaser’s function (called ‘equilibrium rate’) was defined in Shanthikumar and Yao (1986a, b, 1988). Glaser’s results were extended by Gupta and Warren (2001) and Navarro and Hernandez (2004). Moreover, can also be used to determine the shape of the mean residual life function and to study the likelihood ratio order and the increasing and decreasing likelihood ratio (ILR and DLR) classes (see Mi, 2004; Navarro and Hernandez, 2004). The mixtures of distributions are common models in reliability since they represent populations with different kinds of units. They can also be used to represent the distributions of coherent system lifetimes. Even if the shapes of the hazard rate functions of the members in the mixture are known, it is not easy to determine the shape of the hazard rate of the mixture. The same holds for the mean residual life. There exist several papers studying properties for the hazard rate and mean residual life functions of mixtures and systems (see, e.g., Block and Joe, 1997; Shaked and Spizzichino, 2001; Finkelstein, 2001; Navarro and Hernandez, 2004, 2007; Finkelstein and Esaulova, 2006a, b; Navarro and Shaked, 2006; Navarro and Eryilmaz, 2007 and the references given there). The purpose of this paper is to study properties for Glaser’s function of order statistics, systems and mixtures and apply them to study likelihood ratio ordering properties. We can also derive some properties for the hazard rate and the mean residual life functions. We also obtain properties for negative mixtures (i.e. mixtures with some negative coefficients) which can be applied to obtain properties of coherent systems, and in particular, for k-out-of-n systems (order statistics), by using that their distributions are negative mixtures of the distributions of series systems (see Navarro and Shaked, 2006; Navarro et al., 2007). The negative mixtures are also used to obtain the distribution of the MLEs of exponential parameters under step-stress models (see Balakrishnan and Xie, 2007a, b and Balakrishnan et al., 2007) or progressively censored order statistics (see Balakrishnan and Cramer, 2007). The paper is organized as follows. In Section 2, we introduce the basic definitions and known properties used in the paper. The main results are given in Section 3 which is devoted to study the likelihood ratio ordering of series and parallel systems. In Section 4, we study Glaser’s functions of generalized mixtures (i.e. mixtures which can have some negative weights). Finally, in Section 5, we apply the results on series systems and negative mixtures to obtain properties for order statistics and general coherent systems. Throughout the paper, ‘increasing’ stands for ‘non-decreasing’ and ‘decreasing’ stands for ‘non-increasing’. 2. Definitions and basic properties In this section we give basic definitions and known properties that we need to obtain the results included in this paper. We shall suppose that all the random variables considered in the paper have support (0, ∞) and differentiable density functions with continuous derivative in (0, ∞). However, the results in the paper can be easily extended to random variables with common support (, ∞) where −∞ < ∞. If X is a random variable with density function f satisfying these assumptions, Glaser’s function of X is defined by (t) = −f (t)/f (t) for t such that f (t) > 0. The random variable X belongs to increasing (decreasing) likelihood ratio ILR (DLR) class if f is log-concave (logconvex). A random variable X is less than or equal in the likelihood ratio order (denoted by X lr Y ) than another random variable Y if fX (t)/fY (t) decreases in t (see Shaked and Shanthikumar, 2007). Then, it is easy to obtain the following basic properties of . The proof of the first property is obtained by a straightforward integration of . Lemma 2.1. Under the preceding assumptions: (1) (2) (3) (4) (5)
t If t0 > 0 and f (t0 ) > 0, then f (t) = c exp(− t0 (x) dx) for t > 0. X is ILR (DLR) if, and only if, is increasing (decreasing) for t > 0. X lr Y if, and only t if, X Y . ∞ If f (t0 ) > 0, then t0 (x) dx is finite for all t > 0 and t0 (x) dx = ∞. If g(t) is a continuous function for t ∈ (0, ∞) satisfying the properties given in the preceding item, then g is Glaser’s function of a random variable X with support (0, ∞).
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The hazard rate h and the mean residual life m functions of a random variable X are defined by h(t) = f (t)/F (t) and ∞ 1 F (x) dx, (2.1) m(t) = E(X − t | X t) = F (t) t respectively, for t such that F (t) > 0, where F (t) = P (X t) is the reliability (or survival) function. When we consider m, we shall assume that it exists. The following lemma gives some relationships between functions , h and m. It is based on L’Hôpital’s rule and on properties given in Glaser (1980) and Navarro and Hernandez (2004). More properties are given in Gupta and Warren (2001) and Mi (2004). Lemma 2.2. Under the preceding assumptions: (1) (2) (3) (4)
If is increasing (decreasing), then h is increasing (decreasing) and m is decreasing (increasing). If limt→∞ (t) = where 0 ∞, then limt→∞ h(t) = . If limt→∞ h(t) = where 0 ∞, then limt→∞ m(t) = 1/. If limt→∞ (t) = where 0 ∞, then lim −
t→∞
log f (t) = . t
3. Likelihood ratio ordering of series and parallel systems Let (X1 , X2 , . . . , Xn ) be a random vector representing the lifetimes of n components in a system with absolutely continuous joint reliability (survival) function F (x1 , x2 , . . . , xn )=Pr(X1 x1 , X2 x2 , . . . , Xn xn ). Let X1:n X2:n · · · Xn:n be the corresponding order statistics which also represent the lifetimes of the k-out-of-n systems. In particular, X1:i = min(X1 , X2 , . . . , Xi ) represents the series system and Xi:i = min(X1 , X2 , . . . , Xi ) the parallel system for i = 1, 2, . . . , n. If ti = (t, t, . . . , t , 0, 0, . . . , 0) i times
n−i times
the reliability function of X1:i is F 1:i (t) = F (ti ). In general, the lifetime of the series system with components in P will be represented by YP = mini∈P Xi and its reliability function by F P (t). It is well known that the series systems are ordered in the usual stochastic order, i.e. X1:i st X1:i+1 . Hu et al. (2001) proved that they are lr-ordered when the components are independent and X1 lr X2 lr . . . lr Xn . Navarro and Shaked (2006) showed that the series systems X1:i and X1:i+1 are not necessarily ordered in the hazard rate (hr) order when the components are dependent. Therefore, as the likelihood ratio (lr) order is stronger than the hr-order, in general, the series systems are not likelihood ratio ordered. The purpose of this section is to obtain some conditions to have the lr-ordering of series systems. As a consequence, we will have that they are also hr and mrl ordered. Similar results are obtained for the parallel systems. Also, these results can be applied to obtain tail ordering properties of coherent systems and order statistics (see Section 5). We will say that F is exchangeable if F (x1 , x2 , . . . , xn ) = F (x(1) , x(2) , . . . , x(n) ) for any permutation . Clearly, if F is exchangeable and P has cardinality i, then YP =st X1:i , where =st denotes equality in law. We shall use the following notation: Di F (x1 , x2 , . . . , xn ) =
j F (x1 , x2 , . . . , xn ) jxi
and Di,j F (x1 , x2 , . . . , xn ) =
j j F (x1 , x2 , . . . , xn ). jxj jxi
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The main result of this section is given in the following proposition: Proposition 3.1. If F is exchangeable, twice differentiable and satisfies D1,1 F (tk ) D1 F (tk )
D1,1 F (tk+1 )
(3.1)
D1 F (tk+1 )
and (k − 1)
D1,2 F (tk ) D1 F (tk )
k
D1,2 F (tk+1 )
(3.2)
D1 F (tk+1 )
for all t 0, then X1:k lr X1:k+1 . Proof. The reliability function of X1:k is F 1:k (t) = F (tk ). Hence its density function is given by f1:k (t) = −
k
Di F (tk ).
(3.3)
i=1
Therefore, Glaser’s function 1:k of X1:k is given by k k (t) f1:k i=1 j =1 Di,j F (tk ) 1:k (t) = − =− . k f1:k (t) i=1 Di F (tk ) Using now that F is exchangeable, we have f1:k (t) = −kD 1 F (tk )
(3.4)
and 1:k (t) = −
(k − 1)D1,2 F (tk ) + D1,1 F (tk ) D1 F (tk )
.
Hence X1:k lr X1:k+1 if, and only if, (k − 1)D1,2 F (tk ) + D1,1 F (tk ) D1 F (tk )
kD 1,2 F (tk+1 ) + D1,1 F (tk+1 ) D1 F (tk+1 )
.
Finally, it is easy to check that this condition holds when (3.1) and (3.2) hold.
Condition (3.1) can be interpreted as follows. First, note that Glaser’s function of the random variable (X1 | X2 > t, . . . , Xk > t) at time t is (X1 |X2 >t,...,Xk >t) (t) = −
D1,1 F (tk ) D1 F (tk )
.
Hence (3.1) can also be written as (X1 |X2 >t,...,Xk >t) (t)(X1 |X2 >t,...,Xk+1 >t) (t).
(3.5)
This condition is similar to that used in Navarro and Shaked (2006) to obtain the hazard rate ordering of series systems. It is not clear the interpretation of (3.2). In particular, when k = 1, (3.2) can be written as D1,2 F (t2 )0 since D1 F 0. Moreover, in this case, we have D1,2 F (t2 ) = D1,2 F 1,2 (t, t) = f1,2 (t, t) 0,
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where F 1,2 (x1 , x2 )=Pr(X1 x1 , X2 x2 ) and f1,2 (x1 , x2 )=D1,2 F 1,2 (x1 , x2 ) are the reliability and density functions, respectively, of the random vector (X1 , X2 ). Hence, (3.2) always holds for k = 1. We do not know if (3.2) always holds for any k. If the components are independent and identically distributed (i.i.d.), then it is easy to see that (3.1) (or equivalently (3.5)) and (3.2) always holds and hence X1:k lr X1:k+1 . This result can also be obtained from Lillo et al. (2001) or Hu et al. (2001, Theorem 2.2) (see also Shaked and Shanthikumar, 2007, Theorem 1.C.31). The next proposition gives a sufficient condition to have (3.2). Proposition 3.2. If F is exchangeable, twice differentiable and satisfies D1,2 F (tk )D1,2 F (tk+1 )
(3.6)
for all t 0, then (3.2) holds for all t 0. Proof. Firstly, note that the joint reliability function can be written as F (x1 , . . . , xn ) = F 3,...,n (x3 , . . . , xn ) Pr(X1 x1 , X2 x2 | X3 x3 , . . . , Xn xn ),
(3.7)
where F 3,...,n (x3 , . . . , xn ) = Pr(X3 x3 , . . . , Xn xn ) is the reliability function of the random vector (X3 , . . . , Xn ). Therefore, D1,2 F (x1 , . . . , xn ) = F 3,...,n (x3 , . . . , xn )g(x1 , x2 | x3 , . . . , xn ) 0,
(3.8)
where g(x1 , x2 | x3 , . . . , xn ) is the density function of the random vector (X1 , X2 | X3 x3 , . . . , Xn xn ). Analogously, we can obtain that D1,k+1 F (x1 , . . . , xn ) 0
(3.9)
for k = 2, 3, . . . , n − 1. Hence D1 F (x1 , . . . , xn ) is increasing in xk+1 and 0 − (k − 1)
1 D1 F (tk )
−k
1 D1 F (tk+1 )
holds for all t 0 and for k = 2, 3, . . . , n − 1. Finally, if (3.6) holds, from (3.8) and (3.10), we obtain (3.2).
(3.10)
The next example shows that (3.6) is not necessary for (3.2) to hold. Example 3.3. Let (X1 , X2 , . . . , Xn ) have the following Farlie–Gumbel–Morgenstern exchangeable distribution with standard exponential marginals; that is, suppose that the reliability (survival) function of (X1 , X2 , . . . , Xn ) is given by
n n F (x1 , x2 , . . . , xn ) = e− i=1 xi 1 + (1 − e−xi ) , (x1 , x2 , . . . , xn ) (0, 0, . . . , 0), i=1
where ||1. It is not hard to verify that (3.1) and (3.2) hold for k = 1, 2, . . . , n − 2 since D1 F (tk ) = −F (tk ), and D1,1 F (tk ) = D1,2 F (tk ) = −D1 F (tk ) = F (tk ), for k = 1, 2, . . . , n − 1 when n 3. Therefore, X1:k lr X1:k+1 for k = 1, 2, . . . , n − 2 and n 3. However, note that (3.6) does not hold. Moreover, a tedious calculation (which can be obtained from the author) proves that (3.1) and (3.2) hold when 0 and therefore, X1:1 lr X1:2 · · · lr X1:n . The next proposition gives a necessary and sufficient condition to have the lr-ordering of series systems.
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Proposition 3.4. If F is exchangeable and differentiable, then X1:k lr X1:k+1 if, and only if, D1 F (tk ) D1 F (tk+1 )
is increasing in t
(3.11)
for all t 0. Proof. From the definition we have that X1:k lr X1:k+1 if, and only if, f1:k (t)/f1:k+1 (t) is increasing in t. Therefore, the stated result is obtained from (3.4). Similar results can be obtained when the joint distribution of the lifetimes of the components is not necessarily exchangeable for the general series systems YP = mini∈P Xi and YQ = mini∈Q Xi when P ⊂ Q by using that the density function fP of YP can be written as fP (t) = − Di F (tP ), i∈P
where tP = (z1 , z2 , . . . , zn ) with zi = t if i ∈ P and zi = 0 if i ∈ / P . For example, we have the following result: Proposition 3.5. If F is differentiable and P ⊂ Q, then YP lr YQ if, and only if, Di F (tP ) is increasing in t i∈P i∈Q Di F (tQ )
(3.12)
for all t 0. In particular, if the components are independent with reliability functions F i , i =1, 2, . . . , n, then (3.12) is equivalent to
i∈Q hi (t)
F i (t)
i∈Q−P
i∈P
hi (t)
is decreasing in t,
(3.13)
where hi (t) = fi (t)/F i (t) is the hazard function of Xi . Note that (3.13) holds when the components satisfy the proportional hazard rate model (PHR), i.e. when hi (t) = i h1 (t) for all t 0 and i = 2, 3, . . . , n and, in particular, when they are identically distributed. However, (3.13) is not necessarily true. For example, if Q = P ∪ {i} and the components in P are exponential, i.e. hj (t) = j for t 0 and j ∈ P , then (3.13) is equivalent to fi (t) F i (t) + is decreasing in t j ∈P j for all t 0. Differentiating, (3.14) holds if, and only, if i (t) − i
(3.14)
(3.15)
i∈P
for all t 0. In the next example, (3.15) is not true and hence the series systems are not lr-ordered even if the components are independent. Example 3.6. Let X1:2 = min(X1 , X2 ) be the lifetime of a series system with two independent components with lifetimes X1 and X2 with exponential and positive-truncated normal distributions, respectively. Then, the density 2 2 functions are given by f1 (t)=e−t/1 /1 and f2 (t)=ce−(t−2 ) /22 for t 0, respectively. Therefore, Glaser’s functions are given by 1 (t) = h1 (t) = 1/1 and 2 (t) = (t − 2 )/22 for t 0, respectively. Hence from (3.15), X1 lr X1:2 holds if, and only if, t 2 − 22 /1
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Fig. 1. Glaser’s functions 1 (continuous constant line), 2 (continuous non-constant line), 2:2 (dashed line below) and 1:2 (dashed line above) of X1 , X2 , X2:2 = max(X1 , X2 ) and X1:2 = min(X1 , X2 ), respectively, when X1 has an exponential distribution of mean 1, X2 has a positive-truncated normal distribution with mean 2 and variance 1 and X1 and X2 are independent (see Example 3.6).
for all t 0. Then it holds if, and only if, 2 − 22 /1 0. Note that this last expression is not necessarily true. For example, it is not true when 1 = 1, 2 = 2 and 2 = 1. Therefore, the series systems X1 = X1:1 and X1:2 are not lr-ordered. Fig. 1 shows the functions 1 (constant line), 2 (continuous non-constant line), 2:2 (dashed line below) and 1:2 (dashed line above) of X1 , X2 , X2:2 = max(X1 , X2 ) and X1:2 = min(X1 , X2 ), respectively, when 1 = 1, 2 = 2 and 2 = 1. Note that X1:2 and X2:2 are lr-ordered. However note that X1 and X2:2 are not lr-ordered. Also note that X2 and X2:2 are not lr-ordered. Moreover 1 (Glaser’s function of the best component) and 2:2 are asymptotically (when t → ∞) equivalent. We shall see (in Section 5) that this is a general property. Similar results can be obtained when the joint distribution of the lifetimes of the components is not necessarily exchangeable for the general parallel systems ZP = maxi∈P Xi and ZQ = maxi∈Q Xi when P ⊂ Q by using that the density function gP of ZP can be written as Di F (tP ), gP (t) = i∈P
/ P . For example, we have the following result. where tP = (z1 , z2 , . . . , zn ) with zi = t if i ∈ P and zi = ∞ if i ∈ Proposition 3.7. If F is differentiable and P ⊂ Q, then ZP lr ZQ if, and only if, Di F (tP ) i∈P is decreasing in t Q i∈Q Di F (t )
(3.16)
for all t 0. In particular, if the components are independent with distribution functions Fi , i=1, 2, . . . , n, then (3.16) is equivalent to i∈Q−P
i∈Q hi (t)
Fi (t)
i∈P
hi (t)
is increasing in t
(3.17)
for all t 0, where hi (t)=fi (t)/Fi (t) is the reversed hazard function of Xi . Note that (3.17) holds when the components satisfy the proportional reversed hazard rate model (PRHR), i.e. when hi (t)=i h1 (t) for all t 0 and i =2, 3, . . . , n and, in particular, when they are identically distributed. However, (3.17), and hence Xk−1:k−1 lr Xk:k , are not necessarily
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true even if the components are independent (see Fig. 1). Hu et al. (2001, Theorem 2.1) proved that Xk−1:n−1 lr Xk:n for all k = 2, 3, . . . , n when the components are independent and X1 lr X2 lr · · · lr Xn . 4. Properties of finite generalized mixtures Firstly we study properties of the two members generalized mixture density function defined by fp (t) = pf 1 (t) + (1 − p)f0 (t)
(4.1)
for t 0, where f1 and f0 are two density functions and p is a real number. Xp shall denote any random variable having density fp . Thus, we say that Xp is a generalized mixture of X1 and X0 . Without loss of generality we can assume that p 0. However, note that 1 − p can be negative (when p > 1). In this case, we shall say that fp is a negative mixture. If 0 p 1, then fp is a usual or positive mixture and the RHS of (4.1) always defines a density function. In general, it is easy to see that the RHS of (4.1) defines a density function for 0 p pmax where pmax = 1/(1 − ) and = min(1, inf(f1 /f0 )) (we adopt here the convention that 0/0 = ∞). Moreover, the case = 1 leads to the trivial case f1 =a.s. f0 . Obviously, if = 0 then pmax = 1 and the negative mixture cannot be considered. The generalized mixtures were defined in Everitt and Hand (1981), Wu and Lee (1998) and Wu (2001). In practice, they can be used to define families of distributions and to represent the distributions of order statistics and coherent systems as we shall see in the last section (see also Baggs and Nagaraja, 1996; Navarro and Shaked, 2006; Navarro et al., 2007). The negative mixtures are also used to obtain the distribution of the MLEs of exponential parameters under step-stress models (see Balakrishnan and Xie, 2007a, b; Balakrishnan et al., 2007) or progressively censored order statistics (see Balakrishnan and Cramer, 2007). They also appeared as marginal distributions of multivariate models in Kotz et al. (2000, p. 356). Moreover, note that, if 1 < p pmax , then f1 (t) = p −1 fp (t) + p −1 (p − 1)f0 (t),
(4.2)
that is, f1 is a positive (usual) mixture of fp and f0 . If p is Glaser’s function associated to the mixture density fp given by (4.1) and 1 and 0 are Glaser’s functions of the members of the mixture f1 and f0 , then, (t) = W1 (t)1 (t) + (1 − W1 (t))0 (t),
(4.3)
where W1 (t) = pf 1 (t)/fp (t). Thus, if fp is a positive mixture (i.e. 0 p 1), then 0 W1 (t) 1 and min(1 (t), 0 (t))(t) max(1 (t), 0 (t))
(4.4)
for all t 0, that is, Glaser’s function of a positive mixture is between the corresponding functions of the members of the mixture. In particular, if X0 lr X1 , then, from Lemma 2.1, X0 lr Xp lr X1 for 0 p 1. Moreover, if fp is a negative mixture (i.e. 1 < p pmax ) then, from (4.2) and (4.4), we have min(p (t), 0 (t))1 (t) max(p (t), 0 (t))
(4.5)
for all t 0, that is, Glaser’s function of a negative mixture is in the opposite region of 0 with respect to the regions determined by 1 . In particular, if X0 lr X1 , then, from Lemma 2.1, X1 lr Xp for 1 p pmax . In the following proposition we show that Glaser’s functions of two-member generalized mixtures are ordered. Proposition 4.1. If (4.1) holds for 0 p pmax and 0 (t) 1 (t)( ) for a t 0, then j (t)0 jp p
(0).
Proof. From the definitions, we have p (t) = −
fp (t) fp (t)
=−
pf 1 (t) + (1 − p)f0 (t) , pf 1 (t) + (1 − p)f0 (t)
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and by differentiating, we obtain j f0 (t)f1 (t) (1 (t) − 0 (t)), (t) = jp p fp2 (t) and hence, the stated result.
In particular, if the members of the mixture are ordered in the likelihood ratio order X0 lr X1 , then Xp lr Xp for 0 p p pmax . This result extends the result given in Navarro and Hernandez (2004, Proposition 8). Some of the properties given before can be extended to finite generalized mixture density functions defined by n
f (t) =
pi fi (t)
(4.6)
i=1
for t 0, where fi are density functions and pi are real numbers for i = 1, 2, . . . , n. X and Xi shall denote random variables having density functions f and fi , respectively. Thus, we say that X is generalized mixture of X1 , X2 , . . . , Xn . Without loss of generality we can assume that pi = 0 for all i. However, note that some pi can be negative. In this case, we shall say that f is a negative mixture. If 0 pi 1 for all i, then f is a usual or positive mixture and the RHS of (4.1) always defines a density function. In general, it is easy to see that the RHS of (4.1) does not define necessarily a density function. However, we shall suppose that f in (4.6) is a density function. Firstly, note that f can be written as f (t) = pf + (t) + (1 − p)f− (t), (4.7) where p = i:pi >0 pi , f+ (t) = i:pi >0 p −1 pi fi (t) and f− (t) = i:pi <0 (1 − p)−1 pi fi (t). If p > 1, then f is a two members negative mixture of the density functions f+ and f− obtained as positive (usual) mixtures of the density functions with positive and negative coefficients, respectively. If is Glaser’s function associated to the mixture density f given by (4.6) and i are Glaser’s functions of the members of the mixture fi , i = 1, 2, . . . , n, then, (t) =
n
Wi (t)i (t),
(4.8)
i=1
where Wi (t) = pi fi (t)/f (t). Thus, if f is a positive mixture (i.e. 0 pi 1), then 0 Wi (t) 1 and min
i=1,2,...,n
i (t)(t)
max
i=1,2,...,n
i (t)
(4.9)
for all t 0, that is, Glaser’s function of a positive mixture is between the corresponding functions of the members of the mixture. In particular, if Xi lr Xj ( lr ) for all j, then, from Lemma 2.1, Xi lr X( lr ). If f is a negative mixture (i.e. p > 1 in (4.7)), then, from (4.2) and (4.7), we have min((t), − (t))+ (t) max((t), − (t))
(4.10)
for all t 0. A well known property of positive mixtures is that the mixture of DFR (i.e. decreasing failure (hazard) rate) distributions is also DFR. A similar result holds for the IMRL (increasing mean residual life) class. However, the property is not true for IFR (increasing failure rate) and DMRL (decreasing mean residual life) classes. Now, we obtain similar properties for Glaser’s function extending the result given in Navarro and Hernandez (2004, Proposition 10). Proposition 4.2. If (4.6) holds and 1 , 2 , . . . , n are differentiable in t, then (t) =
n
Wi (t)i (t) −
i=1
where Wi (t) = pi fi (t)/f (t).
i
Wi (t)Wj (t)(i (t) − j (t))2 ,
(4.11)
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Proof. From (4.8), we have (t) =
n
Wi (t)i (t) +
i=1
n
Wi (t)i (t),
(4.12)
i=1
where ⎛ ⎞ n Wj (t)(j (t) − i (t))⎠ . Wi (t) = Wi (t)((t) − i (t)) = Wi (t) ⎝ j =1
The proof is completed by replacing the preceding expression in (4.12).
As an immediate consequence, we have the following property. Proposition 4.3. If (4.6) holds for 0 pi 1 and i (t) 0 for a t 0 and i = 1, 2, . . . , n, then (t) 0. Moreover, for negative mixtures, we have the following result. Proposition 4.4. If (4.6) holds for p1 1 and pi 0, i = 2, 3, . . . , n, 1 (t) 0 and i (t) 0, i = 2, 3, . . . , n, for a t 0, then (t)0. Proof. The proof for n=2 is obtained from (4.11). Then, the proof for n > 2 is obtained from Proposition 4.3, expression (4.7) and the case n = 2. In particular, if the preceding propositions can be applied for all t 0, we obtain that the positive mixtures of DLR distributions are also DLR and that the negative mixtures of an ILR distribution with a positive weight and DLR distributions with negative weights are ILR. Now, we are going to study properties on the limiting behaviour of function of a finite generalized mixture. A well known property of positive mixtures is that the hazard rate of the mixture has (under some regularity conditions) the same (in different senses) limiting behaviour when t → ∞ as that of the hazard rate of the strongest member of the mixture in the hazard rate order (see Block and Joe, 1997; Mi, 1999; Finkelstein, 2001; Finkelstein and Esaulova, 2006a, b; Navarro and Shaked, 2006). To obtain similar properties on the limiting behaviour of , we need some lemmas which have also interest by themselves. Lemma 4.5. If 1 (t) > 0
for all t t1
(4.13)
and lim inf t→∞
0 (t) > 1, 1 (t)
(4.14)
then lim
t→∞
f0 (t) = 0. f1 (t)
(4.15)
Proof. From (4.14), there exist ε > 0 and t0 > 0 such that 0 (t) >1 + ε 1 (t)
(4.16)
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for t t0 . Moreover, from (4.13) and (4.16), we obtain 0 (t) − 1 (t) > ε1 (t)
(4.17)
for t max(t0 , t1 ). Hence, if a max(t0 , t1 ) and fi (a) > 0 for i = 0, 1, we have t f0 (t) c0 = exp − (0 (x) − 1 (x)) dx f1 (t) c1 a t c0 exp − ε1 (x) dx c1 a =
c01−ε ε f (t) c1 1
for t a, where the equalities are obtained from Lemma 2.1 and the inequality from (4.17). Therefore, (4.15) holds since limt→∞ f1 (t) = 0 and ε > 0. Lemma 4.6. If (4.6), 1 (t) > 0
for all t t1
(4.18)
and lim inf t→∞
i (t) > 1 for i = 2, 3, . . . , n, 1 (t)
(4.19)
hold, then lim
t→∞
f (t) = p1 > 0. f1 (t)
(4.20)
The proof is immediate from Lemma 4.5. Now, we can obtain the main result on the asymptotic behaviour of . Theorem 4.7. If (4.6), (4.18), (4.19) and fi (t) = i t→∞ f (t) 1 lim
where − ∞i ∞
for i = 2, 3, . . . , n
(4.21)
hold, then (t) = 1. t→∞ 1 (t) lim
(4.22)
Proof. Firstly note that, from L’Hôpital’s rule, Lemma 4.5 and (4.21), we have f (t) fi (t) = lim i = i = 0 t→∞ f1 (t) t→∞ f (t) 1
(4.23)
lim
for i = 2, 3, . . . , n. Hence,
n fi (t) (t) f (t) f1 (t) f1 (t) pi = = p1 + 1 (t) f1 (t) f (t) f (t) f1 (t)
(4.24)
i=2
and therefore, from Lemma 4.6 and (4.23), we obtain (4.22).
The next lemma shows that condition (4.21) can be replaced by the stronger condition (4.25). However, in practice, sometimes, (4.25) is easier to check than (4.21) (see Example 5.6).
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Lemma 4.8. If (4.18), (4.19) and lim sup t→∞
i (t) < ∞ for i = 2, 3, . . . , n 1 (t)
(4.25)
hold, then (4.21) holds. Proof. From the definitions, we have f0 (t) f0 (t) 0 (t) = lim t→∞ f (t) t→∞ f1 (t) 1 (t) 1 lim
and, from Lemma 4.5 and (4.25), (4.21) holds.
Block and Joe (1997) and Navarro and Shaked (2006) obtained similar results for the hazard rate function. Block and Joe (1997) also obtain some asymptotic results for infinite countable and continuous mixtures. Analogous results for the mean residual life were obtained in Navarro and Hernandez (2007). The following example show how to apply Theorem 4.7. Example 4.9. Let us suppose that (4.1) holds for 0 < p pmax and for Gamma densities, i.e. fi (t) =
biai ai −1 exp(−bi t) t (ai )
for t 0, ai , bi > 0 and i = 0, 1. Then, i (t) = (1 + bi t − ai )/t. Hence Xi is ILR (DLR) if ai 1 ( ) and lim (t) = bi t→∞ i
> 0.
Therefore Xi is IFR (DFR) if ai 1 ( ) and limt→∞ hi (t) = bi > 0. Note that there is not explicit expression for the hazard function hi (t). We assume b1 b0 (the tail leading member of the mixture in the lr-order has a positive weight). Then let us to consider three cases. Firstly, if a1 > a0 , then = 0, and pmax = 1 (the negative mixture cannot be considered). In this case, 1 < 0 and hence the positive mixtures are lr (fr and mrl) ordered in p (i.e. X0 lr Xp lr X1 ). Secondly, if a1 = a0 and b1 < b0 , then = (b1 /b0 )a1 > 0 and pmax = b0a1 /(b0a1 − b1a1 ) > 1 (the negative mixtures can be considered). Thirdly, if a1 < a0 and b1 b0 , then
a (a0 ) b11 b0 − b1 a0 −a1 a0 −a1 = min 1, >0 e (a1 ) b0a0 a0 − a1 and pmax > 1 (the negative mixtures can be considered). If a1 < a0 and b1 = b0 , then = 0 and pmax = 1 (the negative mixtures cannot be considered). Thus, by applying Theorem 4.7, if b1 < b0 , we obtain that lim hp (t) = lim p (t) = b1 > 0.
t→∞
t→∞
(4.26)
If b1 = b0 , only positive mixtures can be considered and, as they are ordered from (4.4), we also obtain (4.26). In a similar way, if the mean residual life function exists, then limt→∞ mp (t) = 1/b1 . Moreover, if a1 1 and a0 1, then p and hp are decreasing for p 1. Analogously, if b1 < b0 , a1 = a0 = 1, then p and hp are increasing for p > 1. Fig. 2 shows p when a1 = 1, a0 = 2, b1 = 2, b0 = 3 for p = k/5 and k = 0, 1, . . . , 9. In this case, pmax = 9/(9 − 2e) = 2.525651964 . . . and hence, the negative mixtures can be considered for p pmax . The functions 0 and 1 are plotted with dashed lines. Note that 0 is strictly increasing, 1 is constant (i.e. X1 has an exponential distribution),
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J. Navarro / Journal of Statistical Planning and Inference 138 (2008) 1242 – 1257
Fig. 2. Glaser’s functions p in Gamma generalized mixtures when a1 = 1, a0 = 2, b1 = 2, b0 = 3 for p = 0.2 ∗ k and k = 0, 1, . . . , 9 (from the bottom to the top at t = 0). The dashed lines represent Glaser’s functions of the members of the mixture (1 = 2 and 0 ).
but p is not monotone for p = 0, 1. Also note that p (t) are ordered in p at each t and that limt→∞ p (t) = b1 = 2. Therefore, from Lemma 2.2, limt→∞ hp (t) = 2 and limt→∞ mp (t) = 21 . 5. Properties of order statistics and systems Baggs and Nagaraja (1996) showed that F 1:2 (t) + F 2:2 (t) = F 1 (t) + F 2 (t),
(5.1)
where F i (t)=Pr(Xi > t), i=1, 2, are the component (marginal) reliability functions. They considered different bivariate distributions which have exponential marginal distributions and exponential series system distributions. Hence from (5.1), the distribution of the two-component parallel system is a generalized mixture of three exponential distributions. They studied these kind of generalized mixtures and they applied the results to systems. Note that marginal distributions can be seen as one-component series distributions. Hence (5.1) shows that any two-component parallel system is a generalized mixture of series systems. If T is the lifetime of a coherent system with component lifetimes given by X1 , X2 , . . . , Xn , the result given by Baggs and Nagaraja (1996) can be extended to T by using the inclusion–exclusion method given in Agrawal and Barlow (1984) (see also Navarro et al., 2007) showing that any coherent system is a generalized mixture of series systems, i.e.
m m F T (t) = Pr (YPi > t) = F Pi (t) − F Pi ∪Pj (t) + · · · + (−1)m+1 F P1 ∪···∪Pm (t), (5.2) i=1
i=1
i
where F P (t) = Pr(YP > t) is the reliability function of the series system YP = mini∈P Xi with components in P and P1 , . . . , Pm are the minimal path sets of the coherent system. The definition of the minimal path sets of a coherent system can be seen in Barlow and Proschan (1975). A set P of components is a path set if the systems works when all the components in P work. A minimal path set is a path set which does not contain proper path sets. In particular, if the components are independent, then F P (t) = i∈P F i (t). Navarro et al. (2007) gave a similar result for parallel systems by using the minimal cut sets. Moreover, when (X1 , . . . , Xn ) has an exchangeable distribution, the distribution of the series (parallel) system only depends on the number of components. In this case, Navarro et al. (2007) defined the minimal signature of a coherent system T as the vector (1 , . . . , n ) obtained from (5.2) such that F T (t) = ni=1 i F 1:i (t). They also defined the maximal signature in a similar way by using the parallel systems. Hence the results for generalized mixtures can be used to obtain properties for coherent systems from properties of series (or parallel) systems. For example, from Theorem 4.7 and Lemma 4.8, we have the following result.
J. Navarro / Journal of Statistical Planning and Inference 138 (2008) 1242 – 1257
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Proposition 5.1. If T is a coherent system with reliability F T , Glaser’s function T and minimal path sets P1 , . . . , Pm such that P1 (t)0
for all t > t1 ,
(5.3)
lim inf
P (t) > 1, P1 (t)
(5.4)
lim sup
P (t) < ∞, P1 (t)
(5.5)
t→∞
and t→∞
where P = lim
t→∞
i∈A Pi
for all A ⊆ {1, . . . , m} (A = {1}), then
T (t) = 1. P1 (t)
(5.6)
Remark 5.2. If (5.3)–(5.5) hold, then the path set P1 can be called the leading path set. Hence the tail behaviour of Glaser’s function of the system is equivalent to that of the series system obtained from its leading path set. Note that, from Theorem 4.7, (5.5) can be replaced by the weaker condition lim f (t)/fP 1 (t) = P , t→∞ P where −∞P ∞ and P =
(5.7)
i∈A Pi
for all A ⊆ {1, . . . , m}.
Remark 5.3. In Example 3.6, the parallel system X2:2 has minimal path sets P1 = {1} and P2 = {2} and Glaser’s functions of the components satisfy t − 2 2 (t) = ∞. = lim 1 t→∞ 1 (t) t→∞ 22 lim
Hence (5.5) does not hold. However, f2 (t) =0 t→∞ f (t) 1 lim
and hence (5.7) holds for P = P2 . This example shows that condition (5.7) is strictly weaker than condition (5.5). It is easy to see that (5.3), (5.4) and (5.7) hold for P2 and P = {1, 2}. Therefore, limt→∞ 2:2 (t) = P1 (t) = 1/1 (see Fig. 1). Remark 5.4. If the components are independent, then h (t) P (t) = hi (t) − i∈P i i∈P hi (t) i∈P
and if they have exponential distributions with hazard rate function hi (t) = i , then P (t) = i∈P i and (5.3)–(5.5) hold for P1 such that i∈P1 i < i∈Pj i for j = 2, 3, . . . , m. If the components are independent and they satisfy the PHR model then, from (3.13), the series systems are lr-ordered and hence (5.4) and (5.5) can be replaced by 1 < lim inf t→∞
Pj (t) P1 (t)
lim sup t→∞
Pj (t) P1 (t)
<∞
for j = 2, 3, . . . , m.
Example 5.5. Consider a 2-out-of-3 system with lifetime X2:3 and minimal path sets P1 = {1, 2}, P2 = {2, 3} and P3 = {1, 3}. Thus, from (5.2), the system reliability is given by F 2:3 (t) = F {1,2} (t) + F {2,3} (t) + F {1,3} (t) − 2F 1:3 (t).
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Fig. 3. Glaser’s functions of all the coherent system with three independent exponential components with 1 = 2 = 3 = 1. The dashed lines represent the asymptotic behaviours (see Example 5.5).
If the components have exchangeable joint distribution, then F 2:3 (t)=3F 1:2 (t)−2F 1:3 (t). Hence its minimal signature is (0, 3, −2). If we suppose that the components are independent and that they have exponential distributions with hi (t) = i for t 0 and 1 < 2 < 3 , then the system distribution is a negative mixture of four exponential distributions. Moreover, from the preceding proposition we have limt→∞ 2:3 (t) = 1 + 2 . Note that, if 1 = 2 = 3 , then it is a negative mixture of two exponential distributions and hence it is ILR. Fig. 3 shows Glaser’s functions of all the coherent systems with three independent exponential components with 1 = 2 = 3 = 1. The definitions of these systems can be seen in Shaked and Suarez-Llorens (2003) and their minimal signatures are (0, 0, 1), (0, 2, −1), (0, 3, −2), (1, 1, −1) and (3, −3, 1) (see Navarro and Shaked, 2006 or Navarro et al., 2007). Note that they are lr-ordered. Also note that asymptotically they are equivalent to exponential distributions. Example 5.6. If the lifetimes (X1 , X2 , . . . , Xn ) of the components in a system have the Farlie–Gumbel–Morgenstern exchangeable distribution with standard exponential marginals given in Example 3.3, then it is not hard to verify that the reliability function of the series system YP = mini∈P Xi for P ⊂ {1, 2, . . . , n} is F P (t) = exp(−|P |t) for t 0, where |P | represents the number of elements in P. Hence it has an exponential distribution and P (t) = |P |. Analogously, when P = {1, 2, . . . , n}, the reliability function is given by F P (t) = F 1:n (t) = exp(−nt)(1 + (1 − e−t )n ) for t 0. A straightforward calculation gives 1:n (t) = n − e−t (1 − e−t )n−2
1 + n(1 − 2e−t ) 1 + (1 − 2e−t )(1 − e−t )n−1
for t 0. Hence limt→∞ 1:n (t) = n. Therefore, the series systems are asymptotically lr-ordered and (5.3)–(5.5) hold for any coherent system. Note that, in this example, (5.5) is easier to check than (5.7). For example, if we consider the k-out-of-n system whose lifetime is represented by the order statistics Xn−k+1:n , its minimal path sets are all the sets with k elements. Therefore, from (5.2), its reliability function is given by F n−k+1:n (t) =
n j =k
(−1)j −k
n j −1 j
k−1
exp(−jt)
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for t 0. Hence, from Proposition 5.1, limt→∞ n−k+1:n (t) = k. Moreover, from Lemma 2.2, limt→∞ hn−k+1:n (t) = k and limt→∞ mn−k+1:n (t) = 1/k. Acknowledgements I want to thank Professor Moshe Shaked and the anonymous referees for several helpful comments during the preparation of this paper. References Agrawal, A., Barlow, R.E., 1984. A survey of network reliability and domination theory. Oper. Res. 12, 611–631. Baggs, G.E., Nagaraja, H.N., 1996. Reliability properties of order statistics from bivariate exponential distributions. Commun. Statist. Stochastic Models 12, 611–631. Balakrishnan, N., Cramer, E., 2007. Progressive censoring from heterogeneous distributions with applications to robustness. Ann. Inst. Statist. Math., in press, doi: 10.1007/s10463-006-0070-8. Balakrishnan, N., Xie, Q., 2007a. Exact inference for a simple step-stress model with type-I hybrid censored data from the exponential distribution. J. Statist. Plann. Inference, in press, doi: 10.1016/j.jspi.2007.03.011. Balakrishnan, N., Xie, Q., 2007b. Exact inference for a simple step-stress model with type-II hybrid censored data from the exponential distribution. J. Statist. Plann. Inference 137, 2543–2563. Balakrishnan, N., Xie, Q., Kundu, D., 2007. Exact inference for a simple step-stress model from the exponential distribution under time constraint. Ann. Inst. Statist. Math., to appear. Barlow, R.E., Proschan, F., 1975. Statistical Theory of Reliability and Life Testing: Reliability Models. Holt, Rinehart & Winston, New York. Block, H.W., Joe, H., 1997. Tail behavior of the failure rate functions of mixtures. Lifetime Data Anal. 3, 269–288. Everitt, B.S., Hand, D.J., 1981. Finite Mixture Distributions. Chapman & Hall, New York. Finkelstein, M.S., 2001. The failure rate and the mean residual lifetime of mixtures. System and Bayesian Reliability, Ser. Qual. Reliab. Eng. Stat., vol. 5. World Scientific Publishing, River Edge, NJ, pp. 165–183. Finkelstein, M.S., Esaulova, V., 2006a. Asymptotic behavior of a general class of mixture failure rates. Adv. in Appl. Probab. 38 (1), 244–262. Finkelstein, M.S., Esaulova, V., 2006b. On mixture failure rates ordering. Comm. Statist. Theory Methods 35 (11), 1943–1955. Glaser, R.E., 1980. Bathtub and related failure rate characterizations. J. Amer. Statist. Assoc. 75 (371), 667–672. Gupta, R.C., Warren, R., 2001. Determination of change points of non-monotonic failure rates. Comm. Statist. Theory Methods 30, 1903–1920. Hu, T., Zhu, Z., Wei, Y., 2001. Likelihood ratio and mean residual life orders for order statistics of heterogeneous random variables. Probab. Eng. Inform. Sci. 15, 259–272. Kotz, S., Balakrishnan, N., Johnson, N.L., 2000. Continuous Multivariate Distributions. Wiley, New York. Lillo, R.E., Nanda, A.K., Shaked, M., 2001. Preservation of some likelihood ratio stochastic orders by order statistics. Statist. Probab. Lett. 51 (2), 111–119. Mi, J., 1999. Age-mooth properties of mixture models. Statist. Probab. Lett. 43, 225–236. Mi, J., 2004. A general approach to the shape of failure rate and MRL functions. Naval Res. Logist. 51, 543–556. Navarro, J., Eryilmaz, S., 2007. Mean residual lifetimes of Consecutive k-out-of-n systems. J. Appl. Probab. 44, 82–98. Navarro, J., Hernandez, P.J., 2004. How to obtain bathtub-shaped failure rate models from normal mixtures. Probab. Eng. Inform. Sci. 18 (4), 511–531. Navarro, J., Hernandez, P.J., 2007. Mean residual life functions of finite mixtures, order statistics and coherent systems. Metrika, in press, doi: 10.1007/s00184-007-0133-8. Navarro, J., Shaked, M., 2006. Hazard rate ordering of order statistics and systems. J. Appl. Probab. 43, 391–408. Navarro, J., Ruiz, J.M., Sandoval, C.J., 2007. Modelling coherent systems under dependence. Comm. Statist. Theory Methods 36 (1), 175–191. Shaked, M., Shanthikumar, J.G., 2007. Stochastic Orders. Springer Series in Statistics. Springer, New York. Shaked, M., Spizzichino, F., 2001. Mixtures and monotonicity of failure rate functions. Advances in Reliability. North-Holland, Amsterdam. pp. 185–198. Shaked, M., Suarez-Llorens, A., 2003. On the comparison of reliability experiments based on the convolution order. J. Amer. Statist. Assoc. 98, 693–702. Shanthikumar, J.G., Yao, D.D., 1986a. The preservation of likelihood ratio ordering under convolutions. Stochastic Process. Appl. 23, 259–267. Shanthikumar, J.G., Yao, D.D., 1986b. The effect of increasing service rates in a closed queueing network. J. Appl. Probab. 23, 474–483. Shanthikumar, J.G., Yao, D.D., 1988. Second-order properties of the throughput of a closed queueing network. Math. Oper. Res. 13 (3), 524–534. Wu, J.W., 2001. Characterizations of generalized mixtures of geometric and exponential distributions based on upper record values. Statist. Papers 42, 123–133. Wu, J.W., Lee, W.C., 1998. Characterization of generalized mixtures of exponential distribution based on conditional expectation of order statistics. J. Japan Statist. Soc. 28, 39–44.