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Conditional residual lifetimes of coherent systems
Statistics and Probability Letters xx (xxxx) xxx–xxx
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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Conditional residual lifetimes of coherent systems Q1
A. Parvardeh a , N. Balakrishnan b,c,∗ a
Department of Statistics, University of Isfahan, Isfahan, 81744, Iran
b
Department of Mathematics and Statistics, McMaster University, Hamilton, Canada
c
King Abdulaziz University, Jeddah, Saudi Arabia
article
abstract
info
Article history: Received 16 July 2013 Received in revised form 19 August 2013 Accepted 19 August 2013 Available online xxxx MSC: primary 90B25 secondary 60K10
In this paper, we derive mixture representations for the reliability function of the conditional residual lifetime of a coherent system with n independent and identically distributed (i.i.d.) components under the condition that at least j and at most k − 1 (j < k) components have failed by time t. Based on these mixture representations, we then discuss stochastic comparisons of the conditional residual lifetimes of two coherent systems with independent and identical components. © 2013 Elsevier B.V. All rights reserved.
Keywords: Residual lifetime Signature Hazard rate order Likelihood ratio order Reversed hazard rate order
1. Introduction
1
Recently, considerable attention has been given to stochastic and aging properties of coherent systems. A coherent system is a reliability system such that the structure function of the system is monotone in its components and each component of the system is relevant (a component is irrelevant if it does not matter whether or not it is working for the functioning of the system); see Barlow and Proschan (1975). The concept of signature of a coherent system introduced by Samaniego (1985) has become quite useful in studying properties of coherent systems. For a coherent system with n components whose lifetimes X1 , . . . , Xn are independent and identically distributed (i.i.d.) random variables with continuous distribution function F , the lifetime of the system can be expressed as T = τ (X1 , . . . , Xn ), where τ is a structural function; see Barlow and Proschan (1975, p. 12). Then, the signature of the system is defined as a probability vector s = (s1 , . . . , sn ) with si = P (T = Xi:n ),
i = 1, . . . , n,
where Xi:n is the ith order statistic among X1 , . . . , Xn . The signature vector is distribution-free (meaning that it does not depend on the common continuous lifetime |A | distribution F of the components) and si = P (T = Xi:n ) = n!i , where Ai is the set of all permutations of the component lifetimes for which the ith ordered component failure causes the failure of the system and |Ai | denotes the cardinality of the set Ai . Samaniego (1985) showed that the distribution function of T can be expressed as FT (t ) =
n
si Fi:n (t );
i =1
∗
Corresponding author at: Department of Mathematics and Statistics, McMaster University, Hamilton, Canada. E-mail address: [email protected] (N. Balakrishnan).
0167-7152/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.spl.2013.08.010
2 3 4 5 6 7 8 9 10 11 12 13 14 15
16
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx
that is, the distribution function of T can be represented as a mixture of the distributions Fi:n of the ordered component lifetimes Xi:n , i = 1, . . . , n. This representation has been extended to the case of exchangeable components by Navarro Q2 et al. (2005; 2007; 2008b). Recently several authors have discussed recently the reliability properties of the conditional residual lifetime and the conditional inactivity time of coherent systems with specific signature. Interested readers may refer to Balakrishnan et al. (in press), Kelkin-nama and Asadi (in press), Goliforushani et al. (2012), Li and Zhao (2006, 2008), Li and Zhang (2008), Tavangar and Asadi (2010), Poursaeed and Nematollahi (2008, 2010), Navarro et al. (2005, 2008a, in press), Khaledi and Q3 Shaked (2007), Kochar et al. (1999), Zhang (2010a,b), and the references contained therein. In the present work, we consider a coherent system with the signature s = (0, .., 0, sk , sk+1 , . . . , sn ) with sk > 0, for k = 2, . . . , n
(1)
that is, at the (k − 1)th failure, the system continues to work with probability one. We assume that at time t, at least j (j < k) components have failed but the kth failure has not occurred yet. So, the system is still alive and we then consider the residual lifetime of the system, that is, the variable
(T − t |Xj:n < t < Xk:n ) for j = 1, . . . , k − 1.
(2)
19
The rest of this paper is organized as follows. In Section 2, we obtain a mixture representation for the residual lifetime of the system in (2), and then study some of its aging properties. We also use this mixture representation to compare two coherent systems with the same components and ordered signatures. Next, in Section 3, we compare the residual lifetimes of two coherent systems with n components having their lifetimes to be i.i.d. continuous random variables with distribution functions F and G, respectively.
20
2. Residual lifetimes of the system
15 16 17 18
21 22 23 24 25 26 27
28
29
Let T be the lifetime of a coherent system with n i.i.d. components, and let X1 , . . . , Xn be the lifetimes of the components with a common continuous distribution F . We further assume that the signature of the system has the form (1), that is, s = (0, .., 0, sk , sk+1 , . . . , sn ) with sk > 0, for k = 2, . . . , n with sk > 0. Now, we assume that at least j components (j < k) have failed by time t, but the kth failure has not occurred yet so that the system is still alive. We then study the reliability and stochastic properties of the residual lifetime of the system, that is,
(T − t |Xj:n < t < Xk:n ) for j = 1, . . . , k − 1. Theorem 1. Let T be the lifetime of a coherent system with n i.i.d. components whose lifetimes are denoted by X1 , . . . , Xn . Then, P (T − t > x|Xj:n < t < Xk:n ) =
n
si P (Xi:n − t > x|Xj:n < t < Xk:n ).
(3)
i=k
30
31
Proof. Using the law of total probability, for all x ≥ 0 and t > 0, we have P (T − t > x|Xj:n < t < Xk:n ) =
n
P (T − t > x, T = Xi:n |Xj:n < t < Xk:n )
i =k
=
32
n
P (Xi:n − t > x, T = Xi:n |Xj:n < t < Xk:n ).
i =k 33
34
Now, due to fact that the order statistics are independent of their ranks (see Theorem 1 of Kochar et al., 1999), we obtain P (T − t > x|Xj:n < t < Xk:n ) =
n
P (T = Xi:n )P (Xi:n − t > x|Xj:n < t < Xk:n ),
i=k 35
36 37
38 39
which completes the proof of the theorem.
Remark 2. Note that the distribution of the conditional residual lifetime of the coherent system can be represented as a mixture of distributions of the conditional residual lifetimes of (n − i + 1)-out-of-n systems. Now, let us concentrate, for 1 ≤ j < k ≤ i ≤ n, on the random variable
(Xi:n − t |Xj:n < t < Xk:n ),
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx
3
that is, the residual lifetime of a (n − i + 1)-out-of-n system when at least j and at most k − 1 (j < k) components have failed by time t. Note that P (Xi:n − t > x|Xj:n < t < Xk:n ) =
=
3
P (Xi:n − t > x, Xl:n < t < Xl+1:n )
l =j 4
P (Xj:n < t < Xk:n ) k−1
P (Xi:n − t > x|Xl:n < t < Xl+1:n )
l=j
=
2
P (Xi:n − t > x, Xj:n < t < Xk:n ) P (Xj:n < t < Xk:n ) k−1
=
1
k−1
P (Xl:n < t < Xl+1:n )
5
P (Xj:n < t < Xk:n )
P (Xi:n − t > x|Xl:n < t < Xl+1:n )Kln,j,k (t ),
6
l=j
where
7
(F (t )) (1 − F (t )) P (Xl:n < t < Xl+1:n ) = k−1 l , n P (Xj:n < t < Xk:n ) m (1 − F (t ))n−m ( F ( t )) m n
Kln,j,k (t ) =
l
n−l
8
m=j
or
9
(φ(t )) l , n m (φ( t )) m n
Kln,j,k (t ) =
l
k−1
1 ≤ j ≤ l < k ≤ n, with φ(t ) =
F (t ) F (t )
.
10
m =j
Remark 3. From Goliforushani et al. (2012), it is known that
11
d
(Xi:n − t |Xl:n < t < Xl+1:n ) = Xit−l:n−l , where
Xit−l:n−l
12
denotes the (i − l)th order statistic among n − l i.i.d. random variables with the left-truncated distribution
13
F t (x) = [F (t + x)/F (t )].
14
Remark 4. It is also further known from Goliforushani et al. (2012) that
15
d
d
(Xi:n − t |Xl:n < t < Xl+1:n ) = Xit−l:n−l =(Xi−l:n−l − t |X1:n−l > t ),
16
and for i > l ≥ m
17
(Xi:n |Xl:n < t < Xl+1:n ) ≤lr (Xi:n |Xm:n < t < Xm+1:n ).
18
Theorem 5. If the distribution F is absolutely continuous, then for j < k ≤ i ≤ m, we have (Xi:n − t |Xj:n < t < Xk:n ) ≤lr (Xm:n − t |Xj:n < t < Xk:n ).
19 20
Proof. Let fj,i k (x|t ) and fj,mk (x|t ) be the densities of (Xi:n − t |Xj:n < t < Xk:n ) and (Xm:n − t |Xj:n < t < Xk:n ), and let us denote hj,k (t ) = P (Xj:n < t < Xk:n )
−1
21
.
22
Then we have
Q4
P (Xi:n − t > x|Xj:n < t < Xk:n ) =
P (Xi:n > t + x, Xj:n < t < Xk:n ) P (Xj:n < t < Xk:n )
= hj,k (t )P (Xi:n > t + x, Xj:n < t < Xk:n ) k−1 = hj,k (t ) P (Xi:n > t + x, Xl:n < t < Xl:n )
23
24
25
26
l=j
= hj,k (t )
k−1 n−l l=j m=n−i+1
n! l!(n − l − m)!m!
F (t )l [F (t + x) − F (t )]n−l−m (1 − F (t + x))m
27
4
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx
1
= h j ,k ( t )
2
= hj,k (t )
k −1
n!
l!(n − l)! l =j k−1 n
l
l=j
3
4
F (t )l
m
m=n−i+1
n−l−m m n −l n−l F (t + x) − F (t ) 1 − F (t + x) l m 1 − F (t ) 1 − F (t ) l=j m=n−i+1 k −1 n −l n−l−m m n−l n = h j ,k ( t ) F (t )l (1 − F (t ))n−l 1 − F t (x) F t (x) = hj,k (t )
k−1 n
= h j ,k ( t )
F (t )l (1 − F (t ))n−l
l
l =j
5
(n − l)! [F (t + x) − F (t )]n−l−m (1 − F (t + x))m ( n − l − m )! m ! m=n−i+1 n−l n−l [F (t + x) − F (t )]n−l−m (1 − F (t + x))m n −l
F (t )l
k −1 n
l
l =j
F (t )l (1 − F (t ))n−l
= h j ,k ( t )
n l
l =j
7
8
hj,k (t )
k−1 n
l
F (t )l (1 − F (t ))n−l
(n − l)! un−i (1 − u)n−l−(n−i+1) du (n − i)!(n − l − (n − i + 1))!
F t (x)
(n − l)! un−i (1 − u)i−l−1 du. (n − i)!(i − l − 1)!
0
= h j ,k ( t )
F (t )l (1 − F (t ))n−l
k−1 n
l
l =j
10
F t (x)
From the above expression, we find fj,i k (x|t ) to be equal to
l =j
9
0
k−1
6
f ( t + x)
F (t ) (1 − F (t )) l
n−l
= hj,k (t )f (t + x) (1 − F (t + x))n−i
12
13
14
15
16
=
n!
(n − i)!(i − 1)!
=i
n i
(n − l)!
1 − F (t ) (n − i)!(i − l − 1)!
(n − l)! 1 − F (t ) (n − i)!(i − l − 1)!
k−1 n
l
hj,k (t )f (t + x) (1 − F (t + x))n−i
hj,k (t )f (t + x) (1 − F (t + x))
n−i
18
k −1
(1 − F t (x))i−l−1
1 − F ( t + x)
(i − 1)!
l!(i − l − 1)! l =j
k−1 i−1 l
l =j
n−i
F (t + x) − F (t )
1 − F (t )
1 − F (t )
F (t )l (F (t + x) − F (t ))i−l−1
F (t )l (F (t + x) − F (t ))i−l−1 .
In a similar manner, we find fj,mk (x|t ) = m
n m
hj,k (t )f (t + x) (1 − F (t + x))n−m
k−1 m−1 l =j
l
F (t )l (F (t + x) − F (t ))m−l−1 .
Using these two expressions, we readily find
fj,i k (x|t ) fj,mk (x|t )
=
i m
n
hj,k (t )
in m
hj,k (t )
(1 − F (t + x))n−i
k−1
i−1 l =j
(1 − F (t + x))n−m
=
i m
n
hj,k (t )
in m
hj,k (t )
H (x|t ),
l
k−1
F (t )l (F (t + x) − F (t ))k−l−1
m−1 l
F (t )l (F (t + x) − F (t ))m−l−1
say.
Now, we need to prove that H (x|t ) is a decreasing function of x. Note that
(1 − F (t + x))n−i 19
n−i
(n − l)! F (t )l (F (t + x) − F (t ))i−l−1 (n − i)!(i − l − 1)!
l =j
17
F t (x)
f (t + x)
l=j
11
m
m=n−i+1
k−1
i−1 l=j
H (x|t ) =
(1 − F (t + x))n−m
l
k−1
F (t )l (F (t + x) − F (t ))i−l−1
m−1 l=j
l
F (t )l (F (t + x) − F (t ))m−l−1
i−l−1
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx k−1
i−1 l
l =j
= (1 − F (t + x))m−i
F (t )l (F (t + x) − F (t ))i−l−1 1
k−1
m−1 l
l=j
F (t )l (F (t + x) − F (t ))m−l−1 k−1
i−1
l
l =j k−1
i−1
1 − F (t + x)
=
F (t ) F (t +x)−F (t )
(F (t + x) − F (t ))i−1 l=j l −1 (F (t + x) − F (t ))m−1 k m−1
= (1 − F (t + x))m−i
5
m−i
l
l=j
F (t + x) − F (t )
F (t ) F (t +x)−F (t )
l 2
F (t ) F (t +x)−F (t )
l
l 3
k−1
m−1 l
l =j
F (t ) F (t +x)−F (t )
l
= B(t , x)Φ (A(t , x)),
4
where
5
k−1
i −1
Φ ( u) =
l
l =j
ul
,
k−1
m−1 l
l =j
A(t , x) = B(t , x) =
6
ul
F (t ) F (t + x) − F (t )
,
1 − F (t + x)
7
m−i
F (t + x) − F (t )
.
8
We now have
9
k −1
d du
Φ ( u) =
l 1 =j
i−1 l1
l1 ul1 −1
k −1 l 2 =j
m−1 l2
ul 2 −
k −1 l 1 =j
l =j
l
ul 1
k −1 l 2 =j
m−1 l2
l 2 ul 2 − 1
.
2
k−1
m−1
i −1 l1
10
ul
The numerator of the above fraction is seen to be equal to
k−1 k−1 i−1 m−1 l1 =j l2 =j
l1
l2
u
l1 +l2 −1
(l1 − l2 ) =
11
l1 k−1 i−1 m−1 l1 =j l2 =j
+
l1
l1
l 1 =j l 2 =j
l1
l2
l1
l2
l1 k−1 i−1 m−1 l 1 =j l 2 =j
+
l1
l2
l2
l1
ul1 +l2 −1 (l1 − l2 )
ul1 +l2 −1 (l1 − l2 )
l1 k−1 i−1 m−1 l 1 =j l 2 =j
ul1 +l2 −1 (l1 − l2 )
ul1 +l2 −1 (l1 − l2 )
l2 k−1 i−1 m−1 l 2 =j l 1 =j
=
l2
l1 k−1 i−1 m−1
+
ul1 +l2 −1 (l1 − l2 )
k−1 k−1 i−1 m−1 l1 =j l2 =l1
=
l2
ul1 +l2 −1 (l2 − l1 )
12
13
14
15
16
17
6
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx
=
1
l1 k−1
u
l 1 +l 2 −1
(l1 − l2 )
i−1
−
2
3
i−1
l1
m−1 l2
−
i−1
l2
m−1 l1
=
5
9
10
12 13
14
15
16
17 18 19 20
,
(i − 1)!(m − 1)! 1 l1 !l2 ! (i − 1 − l1 )!(m − 1 − l2 )! 1 − . (i − 1 − l2 )!(m − 1 − l1 )!
(i − 1 − l2 )(i − 1 − l2 − 1) · · · (i − l1 ) ≤ (m − 1 − l2 )(m − 1 − l2 − 1) · · · (m − l1 ), or
(i − 1 − l2 )! (m − 1 − l2 )! ≤ , (i − 1 − l1 )! (m − 1 − l1 )! or equivalently 1
11
l2
Now, since l2 ≤ l1 and i ≤ m, we have
7 8
m−1 l1
l2
where, for i < m, we have
4
6
i−1
m−1
l1
l1 =j l2 =j
(i − 1 − l1 )!(m − 1 − l2 )!
≤
1
(i − 1 − l2 )!(m − 1 − l1 )!
.
Thus, we see Φ to be a decreasing function. On the other hand, A(t , x) = Φ (A(t , x)) is also a decreasing function. Next, we find
F (t ) F (t +x)−F (t )
−f (x + t )(F (t + x) − F (t )) − f (x + t )(1 − F (x + t )) ∂ B(t , x) = (m − i) ∂x (F (t + x) − F (t ))2 m−i−1 f (x + t )F (t ) − f (x + t ) 1 − F (t + x) = ( m − i) (F (t + x) − F (t ))2 F (t + x) − F (t ) m−i−1 −f (x + t )(1 − F (t )) 1 − F (t + x) = ( m − i) ≤ 0. (F (t + x) − F (t ))2 F (t + x) − F (t )
is also a decreasing function, and so
1 − F ( t + x)
m−i−1
F (t + x) − F (t )
Thus, B(t , x) is a decreasing function and so H (x|t ) = B(t , x)Φ (A(t , x)) is also a decreasing function. This completes the proof of the theorem. We now present a result regarding the comparison of two coherent systems with the same components and ordered signatures.
23
Theorem 6. Let s1 = (0, .., 0, s1,k , s1,k+1 , . . . , s1,n ) and s2 = (0, .., 0, s2,k , s2,k+1 , . . . , s2,n ) be the signatures of two coherent systems T1 = φ1 (X1 , .., Xn ) and T2 = φ2 (X1 , .., Xn ) whose lifetimes X1 , . . . , Xn are i.i.d. with a common continuous distribution function F .
24
(a) If s1 ≤st s2 , then T1 − t |Xj:n < t < Xk:n ≤st T2 − t |Xj:n < t < Xk:n .
21 22
25 26 27 28
(b) If s1 ≤rh s2 , then T1 − t |Xj:n < t < Xk:n ≤rh T2 − t |Xj:n < t < Xk:n . (c) If s1 ≤lr s2 , then T1 − t |Xj:n < t < Xk:n ≤lr T2 − t |Xj:n < t < Xk:n . Proof. The proof of this theorem follows from the representation in (3) and the mixture preservation results in Shaked and Shanthikumar (2007).
31
Now, we show that when the lifetimes of the components of the system are IFR (increasing failure rate), then the survival function of the conditional residual lifetime of the coherent system, i.e., P (T − t > x|Xj:n < t < Xk:n ), is a decreasing function of time t. For establishing this result, we first need to show the result for (n − i + 1)-out-of-n systems.
32
Theorem 7. If X ′ s are IFR, then for all x > 0, P (Xi:n − t > x|Xj:n < t < Xk:n ) is a decreasing function of t.
33
Proof. Note that
29 30
34
P (Xi:n − t > x|Xj:n < t < Xk:n ) =
k−1 l =j
P (Xi:n − t > x|Xl:n < t < Xl+1:n )Kln,j,k (t ),
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx
7
and we know that Rl,i (t , x) = P (Xi:n − t > x|Xl:n < t < Xl+1:n ) is decreasing in t for all x ≥ 0 (Goliforushani et al., 2012). We then have
∂ P (Xi:n − t > x|Xj:n < t < Xk:n ) = ∂t
k−1
l=j
∂ ∂ n Rl,i (t , x) Kln,j,k (t ) + Rl,i (t , x) K (t ) . ∂t ∂ t l ,j ,k l =j k−1
k−1
k−1
R l ,i ( t , x )
d n Kl,j,k (t ) =
l =j
Um (t ) − Ul (t )
m=j
dt
l=j
k−1
Rl,i (t , x) Ul′ (t )
k −1
k−1
m=j
2
2
The first term n on the right-hand side is negative. Hence, we just need to prove that the second term is negative. On taking Um (t ) = m t m , we have
1
3
4 5
′ Um (t )
.
6
Um (t )
m=j
The numerator of the above expression can be written as k−1
Rl,i (t , x) Ul′ (t )
l=j
=
k−1
Um (t ) − Ul (t )
Ul′ (t )Um (t )Rl,i (t , x) −
k−1 k−1
k−1 k−1
8
′ Ul (t )Um (t )Rl,i (t , x)
9
l=j m=j
Ul′ (t )Um (t )Rl,i (t , x) −
k−1 k−1
l=j m=j
=
′ Um (t )
k−1 k−1
l=j m=j
=
k−1 m=j
m=j k−1 k−1
7
Um (t )Ul′ (t )Rm,i (t , x)
10
l=j m=j
Ul′ (t )Um (t ) Rl,i (t , x) − Rm,i (t , x)
11
l=j m=j
=
k−1 l
Ul′ (t )Um (t ) Rl,i (t , x) − Rm,i (t , x) +
l=j m=j
=
k−1 l
k −1 k−1
Ul′ (t )Um (t ) Rl,i (t , x) − Rm,i (t , x)
12
l=j m=l
′ Ul′ (t )Um (t ) − Um (t )Ul (t )
Rl,i (t , x) − Rm,i (t , x)
13
l=j m=j
= φ ′ (t )
k−1 l
( l − m)
n n
l=j m=j
m
l
φ(t )l+m−1 Rl,i (t , x) − Rm,i (t , x)
≤ 0,
14
15
where the last inequality follows from the fact that, for m ≤ l < i, we have
(Xi:n − t |Xl:n < t < Xl+1:n ) ≤lr (Xi:n − t |Xm:n < t < Xm+1:n ), and so Rl,i (t , x) ≤ Rm,i (t , x). This completes the proof of the theorem.
16 17
We can now present the main result.
18 19
Theorem 8. If X s are IFR, then for all x > 0, P (T − t > x|Xj:n < t < Xk:n ) is a decreasing function of t > 0.
20
Proof. From Theorem 7, we know that for all x > 0, P (Xi:n − t > x|Xj:n < t < Xk:n ) is a decreasing function of t > 0, and so
21
′
P (T − t > x|Xj:n < t < Xk:n ) =
n
si P (Xi:n − t > x|Xj:n < t < Xk:n )
22
i=k
is a decreasing function of t > 0.
3. Stochastic comparisons between two coherent systems In this section, we consider two coherent systems with n i.i.d. components with their lifetimes denoted by X1 , . . . , Xn and Y1 , . . . , Yn , having continuous distribution functions F and G, respectively. Suppose TX and TY are the lifetimes of two coherent systems with the same structure, respectively. In this section, we wish to show that if X1 ≤hr Y1 , then the system with lifetime TX is less reliable than the system with lifetime TY . For establishing this result, we first need the following lemma that states the same result for the (n − i + 1)-out-of-n systems.
23
24
25 26 27 28 29
8
1 2
3 4 5
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx
Lemma 9. Let X ≤hr Y and j < k ≤ i. Then,
(Xi:n − t |Xj:n < t < Xk:n ) ≤st (Yi:n − t |Yj:n < t < Yk:n ). Proof. We know from Goliforushani et al. (2012) that X ≤hr Y if and only if
(Xi:n − t |Xl:n < t < Xl+1:n ) ≤st (Yi:n − t |Yl:n < t < Yl+1:n ) for all t > 0. Let us assume that
6
RXl,i (t , x) = P (Xi:n − t > x|Xl:n < t < Xl+1:n ),
7
RYl,i (t , x) = P (Yi:n − t > x|Yl:n < t < Yl+1:n ).
8 9
10
Then, we have P (Xi:n − t > x|Xj:n < t < Xk:n ) − P (Yi:n − t > x|Yj:n < t < Yk:n )
=
k−1
n ,X
RXl,i (t , x)Kl,j,k (t ) −
l=j k−1
11
=
l =j
RXl,i (t , x)
k−1
n l
m
(φX (t ))
m=j
RXl,i (t , x)
=
m
m
(φY (t ))l
(φY (t ))m
n
m=j
m=j
m=j
l
l =j
k−1 k−1 n n
+
m
l
l=j
(φX (t ))l (φY (t ))m RXl,i (t , x) − RYl,i (t , x)
k−1 k−1 n n m=j
19
n
l
k−1 k−1 n n (φX (t ))l (φY (t ))m RXl,i (t , x) − (φX (t ))m (φY (t ))l RYl,i (t , x)
16
17
k−1
n
The numerator of the right hand side can be rewritten as
m=j
18
l =j
RYl,i (t , x)
k−1 k−1 k −1 n n Y n l m l m (φ ( t )) (φ ( t )) R ( t , x ) (φ ( t )) (φ ( t )) X Y Y X l ,i l m l m l =k l =k m=k m=k . − k −1 k −1 k −1 k −1 n n n n (φY (t ))m (φY (t ))m (φX (t ))m (φX (t ))m m m m m
=
m=j
15
m
m=j
14
k−1
(φX (t ))l −
k−1
m=j
13
n ,Y
RYl,i (t , x)Kl,j,k (t )
l =j
n
12
k−1
m
l
l=j
(φX (t ))l (φY (t ))m − (φX (t ))m (φY (t ))l RYl,i (t , x).
(4)
The assumption that X ≤hr Y implies HiX,l,n (t , x) − HiY,l,n (t , x) ≤ 0, and so the first term in the above expression is negative. Next, after some algebraic manipulations, we can show that the second term can be expressed as k−1 m n n (φX (t ))l (φY (t ))m − (φX (t ))m (φY (t ))l RYl,i (t , x)
m
m=j l=j
+
20
l
k−1 k−1 n n
m
m=j l=m
l
(φX (t ))l (φY (t ))m − (φX (t ))m (φY (t ))l RYl,i (t , x)
j −1
21
=
m n n m=j l=j
22 23 24
25 26 27 28
m
l
(φX (t ))m (φY (t ))m (φX (t ))l−m − (φY (t ))l−m RYl,i (t , x) − RYm,i (t , x) .
Now, since (Yi:n − t |Yl:n < t < Yl+1:n ) ≥st (Yi:n − t |Ym:n < t < Ym+1:n ) for l ≤ m, we have RYl,i (t , x) − RYm,i (t , x) ≥ 0. Moreover,
(φX (t ))l−m − (φY (t ))l−m ≤ 0. Hence, the second term in (4) is also negative, which completes the proof of the lemma.
Now, we are ready to present the following main result. Theorem 10. Let TX (TY ) be the lifetime of a coherent system with n components whose lifetimes Xi (Yi ), i = 1, . . . , n, are independent and identically distributed (i.i.d.) random variables with an absolutely continuous distribution function F (G). Suppose two systems have the same structure and a common signature of the form in (1). If X1 ≤hr Y1 , then (TX − t |Xj:n < t < Xk:n ) ≤st (TY − t |Yj:n < t < Yk:n ).
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx
9
Proof. If X1 ≤hr Y1 , then from Lemma 9, we readily have
1
(Xi:n − t |Xj:n < t < Xk:n ) ≤st (Yi:n − t |Yj:n < t < Yk:n ).
2
F
G
On the other hand, the signature vector is distribution-free, and so s = s , where s (0, . . . , sGk , . . . , sGn ) are the signature vectors of TX and TY . Therefore, we have
F
= (0, . . . , , . . . , ) and s = sFk
sFn
G
3 4
n
P (TX − t > x|Xj:n < t < Xk:n ) =
sFk P (Xi:n − t > x|Xj:n < t < Xk:n )
5
sFk P (Yi:n − t > x|Yj:n < t < Yk:n )
6
sGk P (Yi:n − t > x|Yj:n < t < Yk:n )
7
i=k
≤
n i=k
=
n i=k
= P (TY − t > x|Yj:n < t < Yk:n ), which completes the proof of the theorem.
8
9
Remark 11. If we now consider the mean residual life MjT,k (t ) = E (T − t |Yj:n < t < Yk:n ), we have
10
n
MjT,k (t ) = E (T − t |Yj:n < t < Yk:n ) =
si E (Xi:n − t |Yj:n < t < Yk:n ),
11
i=k
and from Theorem 8 it is clear that if the lifetimes of the components of the system are IFR, then MjT,k (t ) is a decreasing
12
function of time (in t). Further, under the assumption of Theorem 10, we have Mj,Xk (t ) ≤ Mj,Yk (t ).
13
T
T
Uncited references
14
Q5
Belzunce et al., 2011, Goliforushani and Asadi, 2011, Navarro and Rychlik, 2007 and Zhang and Yang, 2010.
15
Acknowledgments
16
The authors express their sincere thanks to an anonymous reviewer and Professor Hira Koul (Editor-in-Chief) for their useful comments and suggestions on an earlier version of this manuscript which led to this improved version.
17 18
References Balakrishnan, N., Barmalzan, G., Haidari, A., 2013. Stochastic orderings and ageing properties of residual life-lengths of live components in (n − k + 1)out-of-n systems. J. Appl. Probab. (in press). Barlow, R.E., Proschan, F., 1975. Statistical Theory of Reliability and Life Testing: Reliability Models. Holt, Rinehart and Winston, New York. Belzunce, F., Gurler, S., Ruiz, J.M., 2011. Revisiting multivariate likelihood ratio ordering results for order statistics. Probab. Eng. Inform. Sci. 25, 355–368. Goliforushani, S., Asadi, M., 2011. Stochastic ordering among inactivity times of coherent systems. Sankhya¯ 73, 241–262. Goliforushani, S., Asadi, M., Balakrishnan, N., 2012. On the residual and inactivity times of the components of used coherent systems. J. Appl. Probab. 49, 385–404. Kelkin-nama, M., Asadi, M., 2013. Stochastic properties of components in a used coherent system. Methodol. Comput. Appl. Probab. (in press). Khaledi, B., Shaked, M., 2007. Ordering conditional lifetimes of coherent systems. J. Statist. Plann. Inference 137, 1173–1184. Kochar, S.C., Mukerjee, H., Samaniego, F.J., 1999. The signature of a coherent system and its application to comparisons among systems. Naval Res. Logist. 46, 507–523. Li, X., Zhang, Z., 2008. Some stochastic comparisons of conditional coherent systems. Appl. Stoch. Models Bus. Ind. 24, 541–549. Li, X., Zhao, P., 2006. Some ageing properties of the residual life of k-out-of-n systems. IEEE Trans. Reliab. 55, 535–541. Li, X., Zhao, P., 2008. Stochastic comparison on general inactivity time and general residual life of k-out-of-n systems. Comm. Statist. Simulation Comput. 37, 1005–1019. Navarro, J., Balakrishnan, N., Samaniego, F.J., 2008a. Mixture representations of residual lifetimes of used systems. J. Appl. Probab. 45, 1097–1112. Navarro, J., Ruiz, J.M., Sandoval, C.J., 2005. A note on comparisons among coherent systems with dependent components using signatures. Statist. Probab. Lett. 72, 179–185. Navarro, J., Rychlik, T., 2007. Reliability and expectation bounds for coherent systems with exchangeable components. J. Multivariate Anal. 98, 102–113. Navarro, J., Samaniego, F.J., Balakrishnan, N., 2013. Representations for the joint distribution of lifetimes of two coherent systems with shared components. Adv. Appl. Probab. (in press). Navarro, J., Samaniego, F.J., Balakrishnan, N., Bhattacharya, D., 2008b. On the application and extension of system signatures in engineering reliability. Naval Res. Logist. 55, 313–327. Poursaeed, M.H., Nematollahi, A., 2008. On the mean past and the mean residual life under double monitoring. Comm. Statist. Theory Methods 37, 1119–1133. Poursaeed, M.H., Nematollahi, A., 2010. Some aspects of the mean past lifetime of a parallel system under double regularly checking. Statistics 44, 505–515. Samaniego, F.J., 1985. On the closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. 34, 69–72. Shaked, M., Shanthikumar, J.G., 2007. Stochastic Orders. Springer, New York. Tavangar, M., Asadi, M., 2010. A study on the mean past lifetime of the components of (n − k + 1)-out-of-n system at the system level. Metrika 72, 59–73. Zhang, Z., 2010a. Ordering conditional general coherent systems with exchangeable components. J. Statist. Plann. Inference 140, 454–460. Zhang, Z., 2010b. Mixture representations of inactivity times of conditional coherent systems and their applications. J. Appl. Probab. 47, 876–885. Zhang, Z., Yang, Y., 2010. Ordered properties of the residual lifetime and inactivity time of (n − k + 1)-out-of-n systems under double monitoring. Statist. Probab. Lett. 80, 711–717.
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Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Conditional residual lifetimes of coherent systems Q1
A. Parvardeh a , N. Balakrishnan b,c,∗ a
Department of Statistics, University of Isfahan, Isfahan, 81744, Iran
b
Department of Mathematics and Statistics, McMaster University, Hamilton, Canada
c
King Abdulaziz University, Jeddah, Saudi Arabia
article
abstract
info
Article history: Received 16 July 2013 Received in revised form 19 August 2013 Accepted 19 August 2013 Available online xxxx MSC: primary 90B25 secondary 60K10
In this paper, we derive mixture representations for the reliability function of the conditional residual lifetime of a coherent system with n independent and identically distributed (i.i.d.) components under the condition that at least j and at most k − 1 (j < k) components have failed by time t. Based on these mixture representations, we then discuss stochastic comparisons of the conditional residual lifetimes of two coherent systems with independent and identical components. © 2013 Elsevier B.V. All rights reserved.
Keywords: Residual lifetime Signature Hazard rate order Likelihood ratio order Reversed hazard rate order
1. Introduction
1
Recently, considerable attention has been given to stochastic and aging properties of coherent systems. A coherent system is a reliability system such that the structure function of the system is monotone in its components and each component of the system is relevant (a component is irrelevant if it does not matter whether or not it is working for the functioning of the system); see Barlow and Proschan (1975). The concept of signature of a coherent system introduced by Samaniego (1985) has become quite useful in studying properties of coherent systems. For a coherent system with n components whose lifetimes X1 , . . . , Xn are independent and identically distributed (i.i.d.) random variables with continuous distribution function F , the lifetime of the system can be expressed as T = τ (X1 , . . . , Xn ), where τ is a structural function; see Barlow and Proschan (1975, p. 12). Then, the signature of the system is defined as a probability vector s = (s1 , . . . , sn ) with si = P (T = Xi:n ),
i = 1, . . . , n,
where Xi:n is the ith order statistic among X1 , . . . , Xn . The signature vector is distribution-free (meaning that it does not depend on the common continuous lifetime |A | distribution F of the components) and si = P (T = Xi:n ) = n!i , where Ai is the set of all permutations of the component lifetimes for which the ith ordered component failure causes the failure of the system and |Ai | denotes the cardinality of the set Ai . Samaniego (1985) showed that the distribution function of T can be expressed as FT (t ) =
n
si Fi:n (t );
i =1
∗
Corresponding author at: Department of Mathematics and Statistics, McMaster University, Hamilton, Canada. E-mail address: [email protected] (N. Balakrishnan).
0167-7152/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.spl.2013.08.010
2 3 4 5 6 7 8 9 10 11 12 13 14 15
16
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx
that is, the distribution function of T can be represented as a mixture of the distributions Fi:n of the ordered component lifetimes Xi:n , i = 1, . . . , n. This representation has been extended to the case of exchangeable components by Navarro Q2 et al. (2005; 2007; 2008b). Recently several authors have discussed recently the reliability properties of the conditional residual lifetime and the conditional inactivity time of coherent systems with specific signature. Interested readers may refer to Balakrishnan et al. (in press), Kelkin-nama and Asadi (in press), Goliforushani et al. (2012), Li and Zhao (2006, 2008), Li and Zhang (2008), Tavangar and Asadi (2010), Poursaeed and Nematollahi (2008, 2010), Navarro et al. (2005, 2008a, in press), Khaledi and Q3 Shaked (2007), Kochar et al. (1999), Zhang (2010a,b), and the references contained therein. In the present work, we consider a coherent system with the signature s = (0, .., 0, sk , sk+1 , . . . , sn ) with sk > 0, for k = 2, . . . , n
(1)
that is, at the (k − 1)th failure, the system continues to work with probability one. We assume that at time t, at least j (j < k) components have failed but the kth failure has not occurred yet. So, the system is still alive and we then consider the residual lifetime of the system, that is, the variable
(T − t |Xj:n < t < Xk:n ) for j = 1, . . . , k − 1.
(2)
19
The rest of this paper is organized as follows. In Section 2, we obtain a mixture representation for the residual lifetime of the system in (2), and then study some of its aging properties. We also use this mixture representation to compare two coherent systems with the same components and ordered signatures. Next, in Section 3, we compare the residual lifetimes of two coherent systems with n components having their lifetimes to be i.i.d. continuous random variables with distribution functions F and G, respectively.
20
2. Residual lifetimes of the system
15 16 17 18
21 22 23 24 25 26 27
28
29
Let T be the lifetime of a coherent system with n i.i.d. components, and let X1 , . . . , Xn be the lifetimes of the components with a common continuous distribution F . We further assume that the signature of the system has the form (1), that is, s = (0, .., 0, sk , sk+1 , . . . , sn ) with sk > 0, for k = 2, . . . , n with sk > 0. Now, we assume that at least j components (j < k) have failed by time t, but the kth failure has not occurred yet so that the system is still alive. We then study the reliability and stochastic properties of the residual lifetime of the system, that is,
(T − t |Xj:n < t < Xk:n ) for j = 1, . . . , k − 1. Theorem 1. Let T be the lifetime of a coherent system with n i.i.d. components whose lifetimes are denoted by X1 , . . . , Xn . Then, P (T − t > x|Xj:n < t < Xk:n ) =
n
si P (Xi:n − t > x|Xj:n < t < Xk:n ).
(3)
i=k
30
31
Proof. Using the law of total probability, for all x ≥ 0 and t > 0, we have P (T − t > x|Xj:n < t < Xk:n ) =
n
P (T − t > x, T = Xi:n |Xj:n < t < Xk:n )
i =k
=
32
n
P (Xi:n − t > x, T = Xi:n |Xj:n < t < Xk:n ).
i =k 33
34
Now, due to fact that the order statistics are independent of their ranks (see Theorem 1 of Kochar et al., 1999), we obtain P (T − t > x|Xj:n < t < Xk:n ) =
n
P (T = Xi:n )P (Xi:n − t > x|Xj:n < t < Xk:n ),
i=k 35
36 37
38 39
which completes the proof of the theorem.
Remark 2. Note that the distribution of the conditional residual lifetime of the coherent system can be represented as a mixture of distributions of the conditional residual lifetimes of (n − i + 1)-out-of-n systems. Now, let us concentrate, for 1 ≤ j < k ≤ i ≤ n, on the random variable
(Xi:n − t |Xj:n < t < Xk:n ),
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx
3
that is, the residual lifetime of a (n − i + 1)-out-of-n system when at least j and at most k − 1 (j < k) components have failed by time t. Note that P (Xi:n − t > x|Xj:n < t < Xk:n ) =
=
3
P (Xi:n − t > x, Xl:n < t < Xl+1:n )
l =j 4
P (Xj:n < t < Xk:n ) k−1
P (Xi:n − t > x|Xl:n < t < Xl+1:n )
l=j
=
2
P (Xi:n − t > x, Xj:n < t < Xk:n ) P (Xj:n < t < Xk:n ) k−1
=
1
k−1
P (Xl:n < t < Xl+1:n )
5
P (Xj:n < t < Xk:n )
P (Xi:n − t > x|Xl:n < t < Xl+1:n )Kln,j,k (t ),
6
l=j
where
7
(F (t )) (1 − F (t )) P (Xl:n < t < Xl+1:n ) = k−1 l , n P (Xj:n < t < Xk:n ) m (1 − F (t ))n−m ( F ( t )) m n
Kln,j,k (t ) =
l
n−l
8
m=j
or
9
(φ(t )) l , n m (φ( t )) m n
Kln,j,k (t ) =
l
k−1
1 ≤ j ≤ l < k ≤ n, with φ(t ) =
F (t ) F (t )
.
10
m =j
Remark 3. From Goliforushani et al. (2012), it is known that
11
d
(Xi:n − t |Xl:n < t < Xl+1:n ) = Xit−l:n−l , where
Xit−l:n−l
12
denotes the (i − l)th order statistic among n − l i.i.d. random variables with the left-truncated distribution
13
F t (x) = [F (t + x)/F (t )].
14
Remark 4. It is also further known from Goliforushani et al. (2012) that
15
d
d
(Xi:n − t |Xl:n < t < Xl+1:n ) = Xit−l:n−l =(Xi−l:n−l − t |X1:n−l > t ),
16
and for i > l ≥ m
17
(Xi:n |Xl:n < t < Xl+1:n ) ≤lr (Xi:n |Xm:n < t < Xm+1:n ).
18
Theorem 5. If the distribution F is absolutely continuous, then for j < k ≤ i ≤ m, we have (Xi:n − t |Xj:n < t < Xk:n ) ≤lr (Xm:n − t |Xj:n < t < Xk:n ).
19 20
Proof. Let fj,i k (x|t ) and fj,mk (x|t ) be the densities of (Xi:n − t |Xj:n < t < Xk:n ) and (Xm:n − t |Xj:n < t < Xk:n ), and let us denote hj,k (t ) = P (Xj:n < t < Xk:n )
−1
21
.
22
Then we have
Q4
P (Xi:n − t > x|Xj:n < t < Xk:n ) =
P (Xi:n > t + x, Xj:n < t < Xk:n ) P (Xj:n < t < Xk:n )
= hj,k (t )P (Xi:n > t + x, Xj:n < t < Xk:n ) k−1 = hj,k (t ) P (Xi:n > t + x, Xl:n < t < Xl:n )
23
24
25
26
l=j
= hj,k (t )
k−1 n−l l=j m=n−i+1
n! l!(n − l − m)!m!
F (t )l [F (t + x) − F (t )]n−l−m (1 − F (t + x))m
27
4
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx
1
= h j ,k ( t )
2
= hj,k (t )
k −1
n!
l!(n − l)! l =j k−1 n
l
l=j
3
4
F (t )l
m
m=n−i+1
n−l−m m n −l n−l F (t + x) − F (t ) 1 − F (t + x) l m 1 − F (t ) 1 − F (t ) l=j m=n−i+1 k −1 n −l n−l−m m n−l n = h j ,k ( t ) F (t )l (1 − F (t ))n−l 1 − F t (x) F t (x) = hj,k (t )
k−1 n
= h j ,k ( t )
F (t )l (1 − F (t ))n−l
l
l =j
5
(n − l)! [F (t + x) − F (t )]n−l−m (1 − F (t + x))m ( n − l − m )! m ! m=n−i+1 n−l n−l [F (t + x) − F (t )]n−l−m (1 − F (t + x))m n −l
F (t )l
k −1 n
l
l =j
F (t )l (1 − F (t ))n−l
= h j ,k ( t )
n l
l =j
7
8
hj,k (t )
k−1 n
l
F (t )l (1 − F (t ))n−l
(n − l)! un−i (1 − u)n−l−(n−i+1) du (n − i)!(n − l − (n − i + 1))!
F t (x)
(n − l)! un−i (1 − u)i−l−1 du. (n − i)!(i − l − 1)!
0
= h j ,k ( t )
F (t )l (1 − F (t ))n−l
k−1 n
l
l =j
10
F t (x)
From the above expression, we find fj,i k (x|t ) to be equal to
l =j
9
0
k−1
6
f ( t + x)
F (t ) (1 − F (t )) l
n−l
= hj,k (t )f (t + x) (1 − F (t + x))n−i
12
13
14
15
16
=
n!
(n − i)!(i − 1)!
=i
n i
(n − l)!
1 − F (t ) (n − i)!(i − l − 1)!
(n − l)! 1 − F (t ) (n − i)!(i − l − 1)!
k−1 n
l
hj,k (t )f (t + x) (1 − F (t + x))n−i
hj,k (t )f (t + x) (1 − F (t + x))
n−i
18
k −1
(1 − F t (x))i−l−1
1 − F ( t + x)
(i − 1)!
l!(i − l − 1)! l =j
k−1 i−1 l
l =j
n−i
F (t + x) − F (t )
1 − F (t )
1 − F (t )
F (t )l (F (t + x) − F (t ))i−l−1
F (t )l (F (t + x) − F (t ))i−l−1 .
In a similar manner, we find fj,mk (x|t ) = m
n m
hj,k (t )f (t + x) (1 − F (t + x))n−m
k−1 m−1 l =j
l
F (t )l (F (t + x) − F (t ))m−l−1 .
Using these two expressions, we readily find
fj,i k (x|t ) fj,mk (x|t )
=
i m
n
hj,k (t )
in m
hj,k (t )
(1 − F (t + x))n−i
k−1
i−1 l =j
(1 − F (t + x))n−m
=
i m
n
hj,k (t )
in m
hj,k (t )
H (x|t ),
l
k−1
F (t )l (F (t + x) − F (t ))k−l−1
m−1 l
F (t )l (F (t + x) − F (t ))m−l−1
say.
Now, we need to prove that H (x|t ) is a decreasing function of x. Note that
(1 − F (t + x))n−i 19
n−i
(n − l)! F (t )l (F (t + x) − F (t ))i−l−1 (n − i)!(i − l − 1)!
l =j
17
F t (x)
f (t + x)
l=j
11
m
m=n−i+1
k−1
i−1 l=j
H (x|t ) =
(1 − F (t + x))n−m
l
k−1
F (t )l (F (t + x) − F (t ))i−l−1
m−1 l=j
l
F (t )l (F (t + x) − F (t ))m−l−1
i−l−1
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx k−1
i−1 l
l =j
= (1 − F (t + x))m−i
F (t )l (F (t + x) − F (t ))i−l−1 1
k−1
m−1 l
l=j
F (t )l (F (t + x) − F (t ))m−l−1 k−1
i−1
l
l =j k−1
i−1
1 − F (t + x)
=
F (t ) F (t +x)−F (t )
(F (t + x) − F (t ))i−1 l=j l −1 (F (t + x) − F (t ))m−1 k m−1
= (1 − F (t + x))m−i
5
m−i
l
l=j
F (t + x) − F (t )
F (t ) F (t +x)−F (t )
l 2
F (t ) F (t +x)−F (t )
l
l 3
k−1
m−1 l
l =j
F (t ) F (t +x)−F (t )
l
= B(t , x)Φ (A(t , x)),
4
where
5
k−1
i −1
Φ ( u) =
l
l =j
ul
,
k−1
m−1 l
l =j
A(t , x) = B(t , x) =
6
ul
F (t ) F (t + x) − F (t )
,
1 − F (t + x)
7
m−i
F (t + x) − F (t )
.
8
We now have
9
k −1
d du
Φ ( u) =
l 1 =j
i−1 l1
l1 ul1 −1
k −1 l 2 =j
m−1 l2
ul 2 −
k −1 l 1 =j
l =j
l
ul 1
k −1 l 2 =j
m−1 l2
l 2 ul 2 − 1
.
2
k−1
m−1
i −1 l1
10
ul
The numerator of the above fraction is seen to be equal to
k−1 k−1 i−1 m−1 l1 =j l2 =j
l1
l2
u
l1 +l2 −1
(l1 − l2 ) =
11
l1 k−1 i−1 m−1 l1 =j l2 =j
+
l1
l1
l 1 =j l 2 =j
l1
l2
l1
l2
l1 k−1 i−1 m−1 l 1 =j l 2 =j
+
l1
l2
l2
l1
ul1 +l2 −1 (l1 − l2 )
ul1 +l2 −1 (l1 − l2 )
l1 k−1 i−1 m−1 l 1 =j l 2 =j
ul1 +l2 −1 (l1 − l2 )
ul1 +l2 −1 (l1 − l2 )
l2 k−1 i−1 m−1 l 2 =j l 1 =j
=
l2
l1 k−1 i−1 m−1
+
ul1 +l2 −1 (l1 − l2 )
k−1 k−1 i−1 m−1 l1 =j l2 =l1
=
l2
ul1 +l2 −1 (l2 − l1 )
12
13
14
15
16
17
6
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx
=
1
l1 k−1
u
l 1 +l 2 −1
(l1 − l2 )
i−1
−
2
3
i−1
l1
m−1 l2
−
i−1
l2
m−1 l1
=
5
9
10
12 13
14
15
16
17 18 19 20
,
(i − 1)!(m − 1)! 1 l1 !l2 ! (i − 1 − l1 )!(m − 1 − l2 )! 1 − . (i − 1 − l2 )!(m − 1 − l1 )!
(i − 1 − l2 )(i − 1 − l2 − 1) · · · (i − l1 ) ≤ (m − 1 − l2 )(m − 1 − l2 − 1) · · · (m − l1 ), or
(i − 1 − l2 )! (m − 1 − l2 )! ≤ , (i − 1 − l1 )! (m − 1 − l1 )! or equivalently 1
11
l2
Now, since l2 ≤ l1 and i ≤ m, we have
7 8
m−1 l1
l2
where, for i < m, we have
4
6
i−1
m−1
l1
l1 =j l2 =j
(i − 1 − l1 )!(m − 1 − l2 )!
≤
1
(i − 1 − l2 )!(m − 1 − l1 )!
.
Thus, we see Φ to be a decreasing function. On the other hand, A(t , x) = Φ (A(t , x)) is also a decreasing function. Next, we find
F (t ) F (t +x)−F (t )
−f (x + t )(F (t + x) − F (t )) − f (x + t )(1 − F (x + t )) ∂ B(t , x) = (m − i) ∂x (F (t + x) − F (t ))2 m−i−1 f (x + t )F (t ) − f (x + t ) 1 − F (t + x) = ( m − i) (F (t + x) − F (t ))2 F (t + x) − F (t ) m−i−1 −f (x + t )(1 − F (t )) 1 − F (t + x) = ( m − i) ≤ 0. (F (t + x) − F (t ))2 F (t + x) − F (t )
is also a decreasing function, and so
1 − F ( t + x)
m−i−1
F (t + x) − F (t )
Thus, B(t , x) is a decreasing function and so H (x|t ) = B(t , x)Φ (A(t , x)) is also a decreasing function. This completes the proof of the theorem. We now present a result regarding the comparison of two coherent systems with the same components and ordered signatures.
23
Theorem 6. Let s1 = (0, .., 0, s1,k , s1,k+1 , . . . , s1,n ) and s2 = (0, .., 0, s2,k , s2,k+1 , . . . , s2,n ) be the signatures of two coherent systems T1 = φ1 (X1 , .., Xn ) and T2 = φ2 (X1 , .., Xn ) whose lifetimes X1 , . . . , Xn are i.i.d. with a common continuous distribution function F .
24
(a) If s1 ≤st s2 , then T1 − t |Xj:n < t < Xk:n ≤st T2 − t |Xj:n < t < Xk:n .
21 22
25 26 27 28
(b) If s1 ≤rh s2 , then T1 − t |Xj:n < t < Xk:n ≤rh T2 − t |Xj:n < t < Xk:n . (c) If s1 ≤lr s2 , then T1 − t |Xj:n < t < Xk:n ≤lr T2 − t |Xj:n < t < Xk:n . Proof. The proof of this theorem follows from the representation in (3) and the mixture preservation results in Shaked and Shanthikumar (2007).
31
Now, we show that when the lifetimes of the components of the system are IFR (increasing failure rate), then the survival function of the conditional residual lifetime of the coherent system, i.e., P (T − t > x|Xj:n < t < Xk:n ), is a decreasing function of time t. For establishing this result, we first need to show the result for (n − i + 1)-out-of-n systems.
32
Theorem 7. If X ′ s are IFR, then for all x > 0, P (Xi:n − t > x|Xj:n < t < Xk:n ) is a decreasing function of t.
33
Proof. Note that
29 30
34
P (Xi:n − t > x|Xj:n < t < Xk:n ) =
k−1 l =j
P (Xi:n − t > x|Xl:n < t < Xl+1:n )Kln,j,k (t ),
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx
7
and we know that Rl,i (t , x) = P (Xi:n − t > x|Xl:n < t < Xl+1:n ) is decreasing in t for all x ≥ 0 (Goliforushani et al., 2012). We then have
∂ P (Xi:n − t > x|Xj:n < t < Xk:n ) = ∂t
k−1
l=j
∂ ∂ n Rl,i (t , x) Kln,j,k (t ) + Rl,i (t , x) K (t ) . ∂t ∂ t l ,j ,k l =j k−1
k−1
k−1
R l ,i ( t , x )
d n Kl,j,k (t ) =
l =j
Um (t ) − Ul (t )
m=j
dt
l=j
k−1
Rl,i (t , x) Ul′ (t )
k −1
k−1
m=j
2
2
The first term n on the right-hand side is negative. Hence, we just need to prove that the second term is negative. On taking Um (t ) = m t m , we have
1
3
4 5
′ Um (t )
.
6
Um (t )
m=j
The numerator of the above expression can be written as k−1
Rl,i (t , x) Ul′ (t )
l=j
=
k−1
Um (t ) − Ul (t )
Ul′ (t )Um (t )Rl,i (t , x) −
k−1 k−1
k−1 k−1
8
′ Ul (t )Um (t )Rl,i (t , x)
9
l=j m=j
Ul′ (t )Um (t )Rl,i (t , x) −
k−1 k−1
l=j m=j
=
′ Um (t )
k−1 k−1
l=j m=j
=
k−1 m=j
m=j k−1 k−1
7
Um (t )Ul′ (t )Rm,i (t , x)
10
l=j m=j
Ul′ (t )Um (t ) Rl,i (t , x) − Rm,i (t , x)
11
l=j m=j
=
k−1 l
Ul′ (t )Um (t ) Rl,i (t , x) − Rm,i (t , x) +
l=j m=j
=
k−1 l
k −1 k−1
Ul′ (t )Um (t ) Rl,i (t , x) − Rm,i (t , x)
12
l=j m=l
′ Ul′ (t )Um (t ) − Um (t )Ul (t )
Rl,i (t , x) − Rm,i (t , x)
13
l=j m=j
= φ ′ (t )
k−1 l
( l − m)
n n
l=j m=j
m
l
φ(t )l+m−1 Rl,i (t , x) − Rm,i (t , x)
≤ 0,
14
15
where the last inequality follows from the fact that, for m ≤ l < i, we have
(Xi:n − t |Xl:n < t < Xl+1:n ) ≤lr (Xi:n − t |Xm:n < t < Xm+1:n ), and so Rl,i (t , x) ≤ Rm,i (t , x). This completes the proof of the theorem.
16 17
We can now present the main result.
18 19
Theorem 8. If X s are IFR, then for all x > 0, P (T − t > x|Xj:n < t < Xk:n ) is a decreasing function of t > 0.
20
Proof. From Theorem 7, we know that for all x > 0, P (Xi:n − t > x|Xj:n < t < Xk:n ) is a decreasing function of t > 0, and so
21
′
P (T − t > x|Xj:n < t < Xk:n ) =
n
si P (Xi:n − t > x|Xj:n < t < Xk:n )
22
i=k
is a decreasing function of t > 0.
3. Stochastic comparisons between two coherent systems In this section, we consider two coherent systems with n i.i.d. components with their lifetimes denoted by X1 , . . . , Xn and Y1 , . . . , Yn , having continuous distribution functions F and G, respectively. Suppose TX and TY are the lifetimes of two coherent systems with the same structure, respectively. In this section, we wish to show that if X1 ≤hr Y1 , then the system with lifetime TX is less reliable than the system with lifetime TY . For establishing this result, we first need the following lemma that states the same result for the (n − i + 1)-out-of-n systems.
23
24
25 26 27 28 29
8
1 2
3 4 5
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx
Lemma 9. Let X ≤hr Y and j < k ≤ i. Then,
(Xi:n − t |Xj:n < t < Xk:n ) ≤st (Yi:n − t |Yj:n < t < Yk:n ). Proof. We know from Goliforushani et al. (2012) that X ≤hr Y if and only if
(Xi:n − t |Xl:n < t < Xl+1:n ) ≤st (Yi:n − t |Yl:n < t < Yl+1:n ) for all t > 0. Let us assume that
6
RXl,i (t , x) = P (Xi:n − t > x|Xl:n < t < Xl+1:n ),
7
RYl,i (t , x) = P (Yi:n − t > x|Yl:n < t < Yl+1:n ).
8 9
10
Then, we have P (Xi:n − t > x|Xj:n < t < Xk:n ) − P (Yi:n − t > x|Yj:n < t < Yk:n )
=
k−1
n ,X
RXl,i (t , x)Kl,j,k (t ) −
l=j k−1
11
=
l =j
RXl,i (t , x)
k−1
n l
m
(φX (t ))
m=j
RXl,i (t , x)
=
m
m
(φY (t ))l
(φY (t ))m
n
m=j
m=j
m=j
l
l =j
k−1 k−1 n n
+
m
l
l=j
(φX (t ))l (φY (t ))m RXl,i (t , x) − RYl,i (t , x)
k−1 k−1 n n m=j
19
n
l
k−1 k−1 n n (φX (t ))l (φY (t ))m RXl,i (t , x) − (φX (t ))m (φY (t ))l RYl,i (t , x)
16
17
k−1
n
The numerator of the right hand side can be rewritten as
m=j
18
l =j
RYl,i (t , x)
k−1 k−1 k −1 n n Y n l m l m (φ ( t )) (φ ( t )) R ( t , x ) (φ ( t )) (φ ( t )) X Y Y X l ,i l m l m l =k l =k m=k m=k . − k −1 k −1 k −1 k −1 n n n n (φY (t ))m (φY (t ))m (φX (t ))m (φX (t ))m m m m m
=
m=j
15
m
m=j
14
k−1
(φX (t ))l −
k−1
m=j
13
n ,Y
RYl,i (t , x)Kl,j,k (t )
l =j
n
12
k−1
m
l
l=j
(φX (t ))l (φY (t ))m − (φX (t ))m (φY (t ))l RYl,i (t , x).
(4)
The assumption that X ≤hr Y implies HiX,l,n (t , x) − HiY,l,n (t , x) ≤ 0, and so the first term in the above expression is negative. Next, after some algebraic manipulations, we can show that the second term can be expressed as k−1 m n n (φX (t ))l (φY (t ))m − (φX (t ))m (φY (t ))l RYl,i (t , x)
m
m=j l=j
+
20
l
k−1 k−1 n n
m
m=j l=m
l
(φX (t ))l (φY (t ))m − (φX (t ))m (φY (t ))l RYl,i (t , x)
j −1
21
=
m n n m=j l=j
22 23 24
25 26 27 28
m
l
(φX (t ))m (φY (t ))m (φX (t ))l−m − (φY (t ))l−m RYl,i (t , x) − RYm,i (t , x) .
Now, since (Yi:n − t |Yl:n < t < Yl+1:n ) ≥st (Yi:n − t |Ym:n < t < Ym+1:n ) for l ≤ m, we have RYl,i (t , x) − RYm,i (t , x) ≥ 0. Moreover,
(φX (t ))l−m − (φY (t ))l−m ≤ 0. Hence, the second term in (4) is also negative, which completes the proof of the lemma.
Now, we are ready to present the following main result. Theorem 10. Let TX (TY ) be the lifetime of a coherent system with n components whose lifetimes Xi (Yi ), i = 1, . . . , n, are independent and identically distributed (i.i.d.) random variables with an absolutely continuous distribution function F (G). Suppose two systems have the same structure and a common signature of the form in (1). If X1 ≤hr Y1 , then (TX − t |Xj:n < t < Xk:n ) ≤st (TY − t |Yj:n < t < Yk:n ).
A. Parvardeh, N. Balakrishnan / Statistics and Probability Letters xx (xxxx) xxx–xxx
9
Proof. If X1 ≤hr Y1 , then from Lemma 9, we readily have
1
(Xi:n − t |Xj:n < t < Xk:n ) ≤st (Yi:n − t |Yj:n < t < Yk:n ).
2
F
G
On the other hand, the signature vector is distribution-free, and so s = s , where s (0, . . . , sGk , . . . , sGn ) are the signature vectors of TX and TY . Therefore, we have
F
= (0, . . . , , . . . , ) and s = sFk
sFn
G
3 4
n
P (TX − t > x|Xj:n < t < Xk:n ) =
sFk P (Xi:n − t > x|Xj:n < t < Xk:n )
5
sFk P (Yi:n − t > x|Yj:n < t < Yk:n )
6
sGk P (Yi:n − t > x|Yj:n < t < Yk:n )
7
i=k
≤
n i=k
=
n i=k
= P (TY − t > x|Yj:n < t < Yk:n ), which completes the proof of the theorem.
8
9
Remark 11. If we now consider the mean residual life MjT,k (t ) = E (T − t |Yj:n < t < Yk:n ), we have
10
n
MjT,k (t ) = E (T − t |Yj:n < t < Yk:n ) =
si E (Xi:n − t |Yj:n < t < Yk:n ),
11
i=k
and from Theorem 8 it is clear that if the lifetimes of the components of the system are IFR, then MjT,k (t ) is a decreasing
12
function of time (in t). Further, under the assumption of Theorem 10, we have Mj,Xk (t ) ≤ Mj,Yk (t ).
13
T
T
Uncited references
14
Q5
Belzunce et al., 2011, Goliforushani and Asadi, 2011, Navarro and Rychlik, 2007 and Zhang and Yang, 2010.
15
Acknowledgments
16
The authors express their sincere thanks to an anonymous reviewer and Professor Hira Koul (Editor-in-Chief) for their useful comments and suggestions on an earlier version of this manuscript which led to this improved version.
17 18
References Balakrishnan, N., Barmalzan, G., Haidari, A., 2013. Stochastic orderings and ageing properties of residual life-lengths of live components in (n − k + 1)out-of-n systems. J. Appl. Probab. (in press). Barlow, R.E., Proschan, F., 1975. Statistical Theory of Reliability and Life Testing: Reliability Models. Holt, Rinehart and Winston, New York. Belzunce, F., Gurler, S., Ruiz, J.M., 2011. Revisiting multivariate likelihood ratio ordering results for order statistics. Probab. Eng. Inform. Sci. 25, 355–368. Goliforushani, S., Asadi, M., 2011. Stochastic ordering among inactivity times of coherent systems. Sankhya¯ 73, 241–262. Goliforushani, S., Asadi, M., Balakrishnan, N., 2012. On the residual and inactivity times of the components of used coherent systems. J. Appl. Probab. 49, 385–404. Kelkin-nama, M., Asadi, M., 2013. Stochastic properties of components in a used coherent system. Methodol. Comput. Appl. Probab. (in press). Khaledi, B., Shaked, M., 2007. Ordering conditional lifetimes of coherent systems. J. Statist. Plann. Inference 137, 1173–1184. Kochar, S.C., Mukerjee, H., Samaniego, F.J., 1999. The signature of a coherent system and its application to comparisons among systems. Naval Res. Logist. 46, 507–523. Li, X., Zhang, Z., 2008. Some stochastic comparisons of conditional coherent systems. Appl. Stoch. Models Bus. Ind. 24, 541–549. Li, X., Zhao, P., 2006. Some ageing properties of the residual life of k-out-of-n systems. IEEE Trans. Reliab. 55, 535–541. Li, X., Zhao, P., 2008. Stochastic comparison on general inactivity time and general residual life of k-out-of-n systems. Comm. Statist. Simulation Comput. 37, 1005–1019. Navarro, J., Balakrishnan, N., Samaniego, F.J., 2008a. Mixture representations of residual lifetimes of used systems. J. Appl. Probab. 45, 1097–1112. Navarro, J., Ruiz, J.M., Sandoval, C.J., 2005. A note on comparisons among coherent systems with dependent components using signatures. Statist. Probab. Lett. 72, 179–185. Navarro, J., Rychlik, T., 2007. Reliability and expectation bounds for coherent systems with exchangeable components. J. Multivariate Anal. 98, 102–113. Navarro, J., Samaniego, F.J., Balakrishnan, N., 2013. Representations for the joint distribution of lifetimes of two coherent systems with shared components. Adv. Appl. Probab. (in press). Navarro, J., Samaniego, F.J., Balakrishnan, N., Bhattacharya, D., 2008b. On the application and extension of system signatures in engineering reliability. Naval Res. Logist. 55, 313–327. Poursaeed, M.H., Nematollahi, A., 2008. On the mean past and the mean residual life under double monitoring. Comm. Statist. Theory Methods 37, 1119–1133. Poursaeed, M.H., Nematollahi, A., 2010. Some aspects of the mean past lifetime of a parallel system under double regularly checking. Statistics 44, 505–515. Samaniego, F.J., 1985. On the closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. 34, 69–72. Shaked, M., Shanthikumar, J.G., 2007. Stochastic Orders. Springer, New York. Tavangar, M., Asadi, M., 2010. A study on the mean past lifetime of the components of (n − k + 1)-out-of-n system at the system level. Metrika 72, 59–73. Zhang, Z., 2010a. Ordering conditional general coherent systems with exchangeable components. J. Statist. Plann. Inference 140, 454–460. Zhang, Z., 2010b. Mixture representations of inactivity times of conditional coherent systems and their applications. J. Appl. Probab. 47, 876–885. Zhang, Z., Yang, Y., 2010. Ordered properties of the residual lifetime and inactivity time of (n − k + 1)-out-of-n systems under double monitoring. Statist. Probab. Lett. 80, 711–717.
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Q8 Q9
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