Some new results on stochastic comparisons of coherent systems using signatures

TL;DR: It is shown that, in general, similar results may not hold for hazard rate, reversehazard rate, and likelihood ratio orderings for coherent systems with independent and identically distributed components.

Abstract: We consider coherent systems with independent and identically distributed components. While it is clear that the system’s life will be stochastically larger when the components are replaced with stochastically better components, we show that, in general, similar results may not hold for hazard rate, reverse hazard rate, and likelihood ratio orderings. We find sufficient conditions on the signature vector for these results to hold. These results are combined with other well-known results in the literature to get more general results for comparing two systems of the same size with different signature vectors and possibly with different independent and identically distributed component lifetimes. Some numerical examples are also provided to illustrate the theoretical results.

Summary

1 Introduction

• Consider a coherent system consisting of n components with lifetimes X1, . . . ,Xn, which are assumed to be independent and identically distributed (i.i.d.) continuous random variables.
• (1.1) Because of the fundamental property of a systems signature p, namely, that the distribution of the system lifetime T, given i.i.d. components lifetimes with c.d.f.
• These ordering properties are distribution-free in the sense that they do not depend on the common distribution of the components.
• This problem has been pursued further by many other researchers.
• The main results of the paper are given in the next section.

2 Main Results

• Using the above lemma, the authors prove the following result.
• Now using this observation, the second inequality follows from the assumption that uΨ ′(u) Ψ(u) is decreasing and positive for u ∈ (0, v).
• Combining the results of the above theorem with the previous theorem, the authors obtain the following general results which show how the hazard rate, the reversed hazard rate and the likelihood ratio orders between X and Y and signature vectors p and q are preserved by the lifetimes of the corresponding coherent systems.

3 Some numerical examples

• The authors present some coherent systems that satisfy the conditions of Theorem 2.3.
• Below the authors check this condition for various cases of k and r. .
• That is, the condition in Theorem 2.3 (b) is satisfied by such a system.
• The graphs of the functions Δ1(u) and Δ2(u) are given in Figures 2 and 3, respectively.
• The authors now provide a counterexample to illustrate that the condition (iv) in Theorem 2.6 of Navarro et al. (2013) for establishing likelihood ratio ordering is not satisfied, but it satisfies the conditions of their main result on likelihood ratio ordering between systems.

4 Conclusions

• The authors consider the problem of stochastically comparing the lifetimes TX(p) and TY(q) of two coherent systems with signature vectors p and q of the same size and with iid component lifetimes distributed according to X and Y , respectively.
• The results established in this paper generalize some of the known results in the literature.
• The authors first consider the case when a coherent system operates under two different sets of independent and identically distributed component lifetimes.
• The authors find simple sufficient conditions on the distribution of the signature vector under which the two systems are stochastically ordered.
• It will be interesting to examine whether their conjectures are true.

Figures

Table 2. Coherent systems of size 5

Table 1. Coherent systems of sizes 3 and 4

Figure 6: Plot of Δ5(u), for all u ∈ (0, 1)

Figure 1: Plot of the hazard rate functions.

Table 2. Coherent systems of size 5

Table 3. Coherent systems with sizes 4 and 5

Full text

Portland State University Portland State University
PDXScholar PDXScholar
Mathematics and Statistics Faculty
Publications and Presentations
Fariborz Maseeh Department of Mathematics
and Statistics
3-2020
Some New Results on Stochastic Comparisons of Some New Results on Stochastic Comparisons of
Coherent Systems Using Signatures Coherent Systems Using Signatures
Ebrahim Amini-Seresht
Bu-Ali Sina University
Baha-Eldin Khaledi
Florida International University
Subhash C. Kochar
Portland State University
, kochar@pdx.edu
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Citation Details Citation Details
Amini-Seresht, E., Khaledi, B. E., & Kochar, S. (2020). Some new results on stochastic comparisons of
coherent systems using signatures. Journal of Applied Probability, 57(1), 156-173.
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Some New Results on Stochastic Comparisons of
Coherent Systems using Signatures
Ebrahim Amini-Seresht
Department of Statistics
Bu-Ali Sina University, Hamedan, Iran
E-mail: e.amini64@ya hoo.com
Baha-Eldin Khaledi
Department of Statistics
Razi University, Kermanshah, Iran
E-mail: bkhaledi@hotmail.com
Subhash Kochar
Fariborz Maseeh Department of Mathematics and Statistics
Portland State University, Portrland, USA
E-mail:kochar@pdx.edu
June 21, 2019
Abstract
We consider coherent systems with independent and identically distributed components.
While it is clear that the system’s life will be stochastically larger when the components
are replaced with stochastically better components, we show that, in general, similar results
may not hold for hazard rate, reverse hazard rate and likelihood ratio orderings. We find
sufficient conditions on the signature vector for these results to hold. These results are
combined with the other well known results in the literature to get more general results
for comparing two systems of the same size with different signature vectors and possibly
with different independent and identically distributed component lifetimes. Some numerical
examples are also provided to illustrate the theoretical results.
Key words: Likelihood ratio order; Hazard rate order; Reversed hazard rate order; Coherent
system.
1

1 Introduction
Consider a coherent system consisting of n components with lifetimes X
1
,...,X
n
,whichare
assumed to be independent and identically distributed (i.i.d.) continuous random variables. It
follows from the coherent property of the system that the lifetime of the system T
X
corresponds
to exactly one of the order statistics, X
i:n
, i =1,...,n. [Samaniego (1985)] introduced the
concept of “signature” of a system which depends on the design of the system. Let
p
i
= P [T
X
= X
i:n
],i=1,...,n and

n
i=1
p
i
=1,
be the probability that the system fails upon the occurrence of the ith component failure. The
vector p =(p
1
,...,p
n
) is called the signature of the system. The survival function of the lifetime
of the underlying coherent system can be expressed as
P (T
X
(p) >t)=
n

i=1
p
i
P (X
i:n
>t). (1.1)
Because of the fundamental property of a systems signature p, namely, that the distribution
of the system lifetime T, given i.i.d. components lifetimes with c.d.f. F, can be expressed as a
function of p and F alone, we use the notation T
X
(p to denote the life time of the system.
This representation was used by to stochastically compare two systems with each with same
i.i.d.components. These ordering properties are distribution-free in the sense that they do not
depend on the common distribution of the components. As mentioned in Kochar et al. (1999)
and also explained in Navarro et al. (2008), the above results continue to hold when the vector
(X
1
,...,X
n
) has an exchangeable distribution. Moreover, they obtain distribution-free ordering
properties used to compare systems having different numbers of exchangeable components. For
further references on this topic, see Samaniego (2007) and Navarro et al. (2010)
First we review some notions of stochastic orderings. Throughout this paper increasing and
decreasing stand for non-decreasing and non-increasing, respectively.
Let X and Y be two non-negative continuous random variables with density functions f and
g, distribution functions F and G, survival functions
F =1− F and G, hazard rate functions
h
X
= f/F and h
Y
, and reversed hazard rate functions
˜
h
X
= f/F and
˜
h
Y
, respectively.
(a) X is said to be larger than Y in the llikelihood ratio order (denoted by X ≥
lr
Y )if
f(t)
g(t)
is
increasing in t.
(b) X is said to be larger than Y in the hazard rate order (denoted by X ≥
hr
Y )if
F (t)
G(t)
is
increasing in t, or, equivalently, h
X
(t) ≤ h
Y
(t) for all t.
(c) X is said to be larger than Y in the reversed hazard rate order (denoted by X ≥
rh
Y )if
F (t)
G(t)
is increasing in t, or, equivalently,
˜
h
X
(t) ≥
˜
h
Y
(t) for all t.
2

(d) X is said to be larger than Y in the usual stochastic order (denoted by X ≥
st
Y )if
F (t) ≥ G(t) for all t.
The above stochastic orders can be defined on the same lines to compare two discrete random
variables with sample space {1,...,n}. For an n-dimensional probability vector p =(p
1
,...,p
n
),
we denote by h
p
(j)=
p
j

n
i=j
p
i
, the hazard rate of p and
˜
h
(j)
=
p
j

j
i=1
p
i
, as the reverse hazard rate
of p. For two discrete distributions p and q on the integers {1,...,n},wewrite
(a) p ≥
st
q if and only if

n
i=j
p
i
≥

n
i=j
q
i
for j =1,...,n− 1.
(b) p ≥
hr
q if and only if h
p
(i) ≤ h
q
(i)fori =1,...,n.
(c) p ≥
rhr
q if and only if
˜
h
p
(i) ≥
˜
h
q
(i)fori =1,...,n.
(d) p ≥
lr
q if and only if
p
i
q
i
is increasing in i for i =1,...,n.
It is well known that
X ≥
lr
Y =⇒ X ≥
hr[rh]
Y =⇒ X ≥
st
Y, (1.2)
but neither the reversed hazard rate nor hazard rate orders imply each other. one may refer to
Shaked (2007) and Muller and Stoyan (2002) for more details.
Let T
Y
(q) be the lifetime of another coherent system with signature vector q =(q
1
,...,q
n
)
and with component lifetimes Y
1
,...,Y
n
. It is of interest to compare two systems T
X
(p)and
T
Y
(q) according to various stochastic orders. Kochar et al. (1999) proved that if random
lifetimes X
1
,...,X
n
are i.i.d., then
p ≥
∗
q =⇒ T
X
(p) ≥
∗
T
X
(q), (1.3)
where ∗ order stands for lr, hr, rh and st orders. In fact, they pointed out that the above results
hold when X
1
,...,X
n
are exchangeable random variables and the corresponding consecutive or-
der statistics are ordered according to ∗ ordering, that is if X
i+1:n
≥
∗
X
i:n
i =1, 2,...,n −
1. This problem has been pursued further by many other researchers. For instance, the
reader may refer to [Belzunce et al.(2001a)], [Khaledi and Shaked (2007)], [Nanda et al. (1998)],
[Navarro et al. (2008)], [Zhang (2010)] and [Zhang and Meeker (2013)], among others.
A k-out-of-n system consisting of n components with lifetimes X
1
,...,X
n
, functions if and
only if at least k out of the n components function. That is, the lifetime of the system corresponds
to the (n − k +1)th order statistic, X
(n−k+1:n)
. Therefore, stochastically comparing two k-out-
of-n systems is equivalent to comparing the corresponding order statistics. Let X
1
,...,X
n
and
Y
1
,...,Y
n
be two sets of i.i.d. random variables. Then, it is known that
X
1
≥
∗
Y
1
=⇒ X
i:n
≥
∗
Y
i:n
, (1.4)
where ∗ order stands for lr, hr, rh and st orders. Note that X
i:n
and Y
i:n
are the lifetimes of
two coherent systems with common signature vector p = q =(0, 0,...,0, 1, 0,...,0) ( 1 being
3

at the ith position), but with different lifetime distributions. It is of interest to generalize the
results of (1.4) to compare coherent systems with more general signature vectors.
It follows from the fact that
X
1
≥
st
Y
1
=⇒ X
i:n
≥
st
Y
i:n
, for i =1,...,n (1.5)
and the equation (1.1) that when X
1
,...,X
n
(Y
1
,...,Y
n
) are independent and identically dis-
tributed, then
X
1
≥
st
Y
1
⇒ T
X
(p) ≥
st
T
Y
(p) (1.6)
However, such a result may not hold for other stochastic orders like likelihood ratio, hazard rate
and reverse hazard rate orders as shown in the following counter example.
T
X

p

T
Y
p
5.5
6.0
6.5
7.0
7.5
8
.0
0
.0
1
0
.0
2
0
.0
3
0
.0
4
Figure 1: Plot of the hazard rate functions.
Example 1.1. Consider a coherent system of order 4 with lifetime
T
X
(p)=max(X
1
, min(X
2
,X
3
,X
4
))
with signature vector p =(0,
1
2
,
1
4
,
1
4
),whereX
i
, i =1,...,4 are independent exponential random
variables with common hazard rate 0.7. Let Y
i
, i =1,...,4 be another set of independent
exponential random variables with common hazard rate 1. In Figure 1 we plot the hazard rate
functions of T
X
(p) and T
Y
(p) and it can be seen that the hazard rate functions cross at t =5.5
even though X
1
≥
hr
Y
1
. Therefore, a result similar to (1.6) does not hold, in general, for hazard
rate ordering.
In the second section, we find sufficient conditions on the signature vector p under which
X
1
≥
∗
Y
1
⇒ T
X
(p) ≥
∗
T
Y
(p) (1.7)
holds for hazard rate, reverse hazard and likelihood ratio orderings. This problem has also been
studied by Navarro et al. (2013), but as will be seen later in the paper, their results seem to be
4

Frequently asked questions

Q1. What are the contributions in "Some new results on stochastic comparisons of coherent systems using signatures" ?

The authors consider coherent systems with independent and identically distributed components. While it is clear that the system ’ s life will be stochastically larger when the components are replaced with stochastically better components, the authors show that, in general, similar results may not hold for hazard rate, reverse hazard rate and likelihood ratio orderings. 

Q2. What have the authors stated for future works in "Some new results on stochastic comparisons of coherent systems using signatures" ?

The authors plan to pursue this problem in the near future. 

Q3. What is the hazard rate of a coherent system?

Consider a coherent system of order 4 with lifetimeTX(p) = max(X1,min(X2,X3,X4))with signature vector p = (0, 12 , 1 4 , 1 4 ), where Xi, i = 1, . . . , 4 are independent exponential random variables with common hazard rate 0.7. 

Q4. What is the order of random lifetimes?

Kochar et al. (1999) proved that if random lifetimes X1, . . . ,Xn are i.i.d., thenp ≥∗ q =⇒ TX(p) ≥∗ TX(q), (1.3)where ∗ order stands for lr, hr, rh and st orders. 

Q5. what is the hazard rate function of tx?

(2.2)and the hazard rate function of TX(p) ishTX(p)(t) =∑n i=1 ipi (n i ) F i−1(t)F n−i+1(t)∑ni=1 pi ∑i−1 j=1 (n j ) F j(t)F n−j (t) hX(t)=∑n−1 i=0 (n− i)pi+1 ( n i ) F i(t)Fn−i (t)∑n−1i=0 (∑n j=i+1 pj ) (n i ) F i(t)F n−i (t) hX(t)= Ψ1( F (t)F (t)) hX(t) (2.3)whereΨ1(x) =∑n−1 i=0 (n− i) (n i ) pi+1xi∑n−1 i=0 (∑n j=i+1 pj ) (n i ) xi(2.4)The reverse hazard rate function of TX(p) ish̃TX(p)(t) =∑n i=1 ipi (n i ) F i(t)Fn−i (t)∑ni=1 (∑i j=1 pj ) ( n i ) F i(t)F n−i (t) h̃X(t)= Ψ2(x)( F (t)F (t)) h̃X(t), (2.5)whereΨ2(x) =∑n i=1 ipi (n i ) x1∑ni=1 (∑i j=1 pj ) (n i ) xi(t)(2.6)First the authors prove the following lemma which gives simple sufficient conditions for the functionsΨ1 and Ψ2 to be monotone. 

Q6. What is the likelihood ratio order of a coherent system?

Consider a coherent system of order 4 with signature vector p = (0, 23 , 1 3 , 0) , and lifetime T = max(min(X1,X2),min(X3,X4)), where X1,X2,X3,X4 are independent and identically distributed. 

Q7. What is the hazard rate function of X1?

In Figure 1 the authors plot the hazard rate functions of TX(p) and TY (p) and it can be seen that the hazard rate functions cross at t = 5.5 even though X1 ≥hr Y1. 

Q8. what is the survival function of xi?

It is known that the survival function of Xi:n can be written as (see [Gupta (2002, p. 839)])P (Xi:n > t) = i−1∑ j=0 (−1)i−j−1 ( n j )( n− j − 1 n− i ) F n−j (t),from which its density function isfi:n(t) = hX(t) i−1∑ j=0 (−1)i−j−1 ( n j )( n− j − 1 n− i ) (n − j)F n−j(t),where hX(t) is the hazard rate function of X. 

Q9. What is the survival function of TX(p)?

From (1.1) the survival function of TX(p) can be written asF TX(p)(t) = n∑ i=1 pi i−1∑ j=0( nj) F j(t)F n−j (t)= n−1∑ j=0⎛ ⎝ n∑i=j+1pi⎞ ⎠(nj) F j(t)F n−j (t), (2.1)by changing the order of summation.