International Conference on Mathematical and Statistical Modeling
in Honor of Enrique Castillo. June 28-30, 2006
Reliability properties of mixtures, order statistics
and systems.
Jorge Navarro∗,∗∗
Departamento de Estadı́stica e Investigación Operativa,
Universidad de Murcia.
Pedro J. Hernández
Departamento de Métodos Cuantitativos e Informáticos,
Universidad Politécnica de Cartagena.
Abstract
The paper presents new properties for the failure rate functions of finite mixtures.
Some of these properties are extended to negative finite mixtures, that is, to mixtures with some negative weights. Specifically, we study the shape of the failure rate
of the mixture, its asymptotic behaviour and some ordering properties. We apply
our general results to mixtures of Exponential and Pareto distributions. Moreover,
the properties for negative mixtures are used to obtain properties for the failure
rate functions of order statistics and coherent systems with possibly dependent
components. The properties are illustrated with some examples.
Key Words: Mixtures, failure rate, order statistics, coherent systems.
1
Introduction
In reliability theory, the lifetime of a biological or engineering component
or system is represented by a positive random variable. The mixtures of
distributions are very common models in reliability since they represent
populations with different kinds of components. The properties of mixtures
are well known. The aging process in the units can be represented by the
shape of the failure rate.
Special attention has been paid to the determination of the shape of the
failure rate of the mixture from the shape of the members of the mixture
(see e.g. Barlow and Proschan (1975), Block and Joe (1997), Shaked and
Spizzichino (2001), Block et al. (2003a), Block et al. (2003b), Mi (2004) and
∗
Correspondence to: Jorge Navarro. Facultad de Matemáticas. Universidad de
Murcia. 30100 Murcia, Spain.
∗∗
Partially Supported by Ministerio de Ciencia y Tecnologı́a under grant BFM200302947 and Fundación Seneca under grant 00698/PI/04.
2
J. Navarro and P. J. Hernández
Wondmagegnehu (2004)). In particular, the mixtures can be used to obtain models with bathtub shaped failure rates (see Navarro and Hernandez
(2004)). Moreover, the shape of the failure rate is related to the shape of
the mean residual life (see Mi (1995), Savits (2003), Bekker and Mi (2003)
and Navarro and Hernandez (2004)).
The generalized mixtures are mixtures with possibly negative weights.
Properties and characterizations for these kind of mixtures were given in
Everitt and Hand (1981) 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 (see Bartholomew (1969), Botta et al.
(1987), Harris et al. (1992), Baggs and Nagaraja (1996), Block et al. (2003a)
and Navarro et al. (2007)). For example, Baggs and Nagaraja (1996) studied properties of generalized mixtures of three exponential distributions
applying these properties to obtain results for series and parallel systems
with two components and different joint distributions. Block et al. (2003a)
and Navarro et al. (2007) showed that any coherent system with possibly
dependent components (and hence, any order statistics) is a generalized
mixture of series (or parallel) systems.
In this paper, we study properties for the failure rate functions of generalized mixtures comparing the results obtained to the well known properties
of usual mixtures. We also obtain some new results for positive usual mixtures. Our results can be applied to families of distributions defined as
generalized mixtures and to coherent systems. Specifically, in Section 2,
we study general properties of generalized mixtures extending some results
given in Block et al. (2003a). In Sections 3 & 4, we apply these general
results to mixtures of Exponential and Pareto distributions, respectively.
Finally, in Section 5, we show some applications to order statistics and
coherent systems.
2
2.1
General properties
Definitions
Let X be a random variable with distribution function F (t), density f (t)
and reliability (survival) function R(t) = 1 − F (t). The failure rate of X is
defined by r(t) = f (t)/R(t) for t such that R(t) > 0. It is well known that
3
Reliability properties of mixtures
r uniquely determines R through the following inversion formula
¶
µ Z t
r(x)dx
R(t) = exp −
(2.1)
−∞
We say that a distribution function F is a generalized mixture of the
distribution functions F (·; 1), F (·; 2), ... if
F (t) =
∞
X
wi F (t; i)
(2.2)
i=1
for
P∞all t, where w1 , w2 , ... are real numbers such that
i=1 |wi | < ∞.
P∞
i=1 wi
= 1 and
Without loss of generality, we can assume F (·, i) 6= F (·, j) for i 6= j.
We can also suppose that, if wi = 0, then wj = 0 for j > i and, in this
case, we say that F is a finite generalized mixture. When w1 , w2 , ... are
positive real numbers we say that F is a positive mixture. In this case,
the right part of expression (2.2) is always a distribution function. If for
some indexes i, wi < 0, then we say that F is a negative mixture. In
this case, some conditions are needed in order to guarantee that the right
part of expression (2.2) is a genuine distribution function. However, for
our purposes, we can assume these conditions hold and F is a distribution
function (see Section 5).
If all F (·; i) are absolutely continuous with densities f (·; i), i = 1, 2, ...,
then the generalized
P∞ mixture F is absolutely continuous with derivative
′
f (t) = F (t) = i=1 wi f (t; i). In this case, (2.2) defines a genuine distribution function iff f (t) ≥ 0 for all t.
First, we note that any generalized mixture given by (2.2) can be seen
as a generalized mixture with two components, since
F (t) = pF1 (t) + (1 − p)F0 (t)
where
p=
X
wi > 1,
wi >0
F1 (t) =
P
wi >0 wi F (t; i)
P
wi >0 wi
,
(2.3)
4
J. Navarro and P. J. Hernández
and
F0 (t) =
P
wi <0 wi F (t; i)
P
wi <0 wi
.
Note that F1 and F0 are distribution functions obtained from positive
mixtures. However, note that they can have different properties from those
of the components. For example, F (·; i) can be exponential distributions
and F1 and F0 can be different distributions.
The preceding property can be used to give a general definition of generalized mixtures.
Definition 2.1. We say that a distribution function F is a generalized
mixture if
Z
Z
F (t) = p F1 (t; θ)dG1 (θ) + (1 − p) F0 (t; θ)dG0 (θ)
(2.4)
for all t, where p > 0, G1 and G0 are two distribution functions and F1 (·; θ)
and F0 (·; θ) are distribution functions for all θ. If p > 1, we say that F is
a negative mixture.
2.2
Generalized mixtures with two components
First, we consider generalized mixtures defined by (2.3). However, note that
some properties can be applied to generalized mixtures defined by (2.4).
The following proposition characterizes the generalized mixtures defined
by (2.3).
Proposition 2.1. If f1 and f0 are two density functions and p ≥ 0, then
f = pf1 + (1 − p)f0
is a density function for 0 ≤ p ≤ pmax , where pmax = 1/(1 − α) and α =
min(1, inf(f1 /f0 )) (we adopt here the convention that 0/0 = ∞). Moreover,
the case α = 1 leads to the trivial case f1 =a.s. f0 .
The proof is easy. Obviously, if α = 0 then pmax = 1 and the negative
mixture cannot be considered. Moreover, note that if 1 < p ≤ pmax , then
we have
F1 (t) = p−1 F (t) + p−1 (p − 1)F0 (t),
(2.5)
Reliability properties of mixtures
5
that is, F1 (the positive member) is a positive mixture of F (the negative
mixture) and F0 (the negative member).
Thus, we can use here all the properties for positive mixtures. For
example, it is well known that the failure rate of a positive mixture is
between the failure rates of the weaker and the stronger components of
the mixture in the failure rate order (see
R e.g. Block and Joe (1997)). For
example, if F is a mixture, F (t) = F (t; θ)dG(θ) and the failure rates
satisfy r(t; θ0 ) ≤ r(t; θ) (≥) for a t and all θ, then the mixture failure rate
satisfies
Z
R(t; θ)
R
r(t; θ)dG(θ)
r(t) =
R(t; θ)dG(θ)
Z
R(t; θ)
≥ r(t; θ0 ) R
dG(θ) = r(t; θ0 )(≤).
R(t; θ)dG(θ)
In particular, if (2.3) holds for 0 ≤ p ≤ 1, then
min(r1 (t), r0 (t)) ≤ r(t) ≤ max(r1 (t), r0 (t))
(2.6)
Hence, from(2.5), we have the following immediate property.
Proposition 2.2. If F is distribution function and (2.3) holds for p > 1,
then
min(r(t), r0 (t)) ≤ r1 (t) ≤ max(r(t), r0 (t)),
where r, r1 and r0 are the failure rates of F, F1 and F0 , respectively.
Remark 2.1. This property can be extended to mixtures defined by (2.4)
when r0 (t; θ) ≤ r1 (t; θ′ ) (≥) for all θ, θ′ . Note that hence, at any t, if r0 (t) ≤
r1 (t) (≥), then r1 (t) ≤ r(t) (≥). In particular, if the components are
ordered with respect to the failure rate order F0 ≤f r F1 (≥) (i.e. r0 (t) ≥
r1 (t) (≤) for all t), then F1 ≤f r F (≥). Moreover, if rp (t) is the failure
rate of a generalized mixture defined by (2.3), it is immediate to show that
∂
∂p rp (t) has the same sign as r1 (t) − r0 (t) and hence,
r0 (t) ≤ r1 (t) ( ≥ ) ⇒ rp (t) ≤ rp′ (t) ( ≥ )
for 0 ≤ p < p′ ≤ pmax . In particular, if F0 ≤f r F1 (≥), then Fp ≤f r Fp′
(≥) for 0 ≤ p < p′ ≤ pmax . A similar property holds for the stochastic,
likelihood ratio and mean residual life orders.
6
J. Navarro and P. J. Hernández
Another well known property of positive mixtures is that the failure
rate of the mixture has (under some regularity conditions) the same limiting
behaviour as that of the failure rate of the strongest member of the mixture
in the failure rate order (see Block et al. (1993), Block and Joe (1997) and
Block et al. (2003a)). Hence, from (2.5), if p > 1 and F0 ≤f r F1 , then the
mixture failure rate has the same limiting behaviour as that of the failure
rate of the strongest component (F1 ). However, if p > 1 and F0 ≥f r F1 , then
this result is, in general, not true. Moreover, to have a valid generalized
mixture, the failure rates of both components must have the same limiting
behaviour.
Next, we obtain some properties on the limiting behaviour of failure
rates of generalized mixtures. First, we obtain a new result for positive
mixtures which extends Theorem 2.3 in Block et al. (2003a).
Proposition 2.3. If (2.3) holds for 0 < p < 1,
lim r1 (t) = λ1 ∈ [0, ∞]
(2.7)
r1 (t) ≤ r0 (t) for all t ≥ t′ ,
(2.8)
lim r(t) = ξ ∈ [0, ∞],
(2.9)
t→∞
and
t→∞
then ξ = λ1 .
Remark 2.2. Note that, (2.8) holds when lim inf t→∞ r0 (t) > λ1 and, in
particular, when limt→∞ r0 (t) = ∞ (the condition of Theorem 2.3 in Block
et al. (2003a)). Also note that, from (2.6), if λ1 = ∞ in (2.7), then (2.9)
holds with ξ = ∞.
We have obtained a similar result for negative mixtures.
Proposition 2.4. If F is a distribution function, (2.3) holds for p > 1 and
(2.7) holds, then the following properties hold.
1. If (2.8) and (2.9) hold, then ξ = λ1 .
2. If λ1 = ∞ and r0 (t) ≤ r1 (t) for all t ≥ t′ , then limt→∞ r(t) = ∞.
3. lim supt→∞ (r0 (t) − r1 (t)) ≥ 0.
Reliability properties of mixtures
7
Figure 1: Generalized mixtures of two exponential distributions with µ1 = 1,
µ0 = 1/2 and p = 0.2k for k = 0, ..., 10 (from the top to the bottom). The dashed
lines represent the failure rates of the members of the mixture (r1 = 1 < r0 = 2).
Remark 2.3. Note that condition (2.8) holds when lim inf t→∞ r0 (t) > λ1
holds and, in particular, when limt→∞ r0 (t) = ∞ and λ1 < ∞ hold. In a
similar way, the condition in item 2 about r0 holds when lim supt→∞ r0 (t) <
∞ and, in particular, when limt→∞ r0 (t) < ∞. Note that item 3 gives a
necessary condition to have a negative mixture. Moreover, if λ1 ∈ [0, ∞)
and limt→∞ r0 (t) = λ0 ∈ [0, ∞), then λ1 ≤ λ0 . For example, if we consider
exponential negative mixtures defined by (2.3) with p > 1, then r1 (t) =
1/µ1 ≤ r0 (t) = 1/µ0 .
Another well known property (see e.g. Barlow and Proschan (1975)) of
positive mixtures is that the mixture of two DFR (i.e. non-increasing failure rate) distributions, is also DFR. However, this property is, in general,
not true for negative mixtures. For example, if the components have exponential distributions with µ0 < µ1 and 1 < p < pmax = µ1 /(µ1 − µ0 ), then
r(t) is strictly increasing for t > 0 (see Figure 1). For negative mixtures we
have the following result.
8
J. Navarro and P. J. Hernández
Proposition 2.5. If F is a distribution function, (2.3) holds for p > 1,
r1′ (t) ≥ 0 and r0′ (t) ≤ 0 then r′ (t) ≥ 0, where r, r1 and r0 are the failure
rates of F , F1 and F0 , respectively.
Remark 2.4. In particular, if (2.3) holds for p > 1, F1 is IFR (i.e. nondecreasing failure rate) and F0 is DFR, then F is IFR. Moreover, if for a
fixed t, r1 (t) 6= r0 (t) and Ri (t) > 0 hold for i = 0, 1, then p(t) > 1 and
r′ (t) > 0. For example, the negative mixtures of two exponential distributions are strictly IFR.
Corollary 2.1. If F is a distribution function, (2.3) holds for p > 1, F1
is IFR and F0 is DFR, then r(t) ≤ r1 (t) ≤ r0 (t) for all t and
0 < lim r1 (t) = lim r(t) ≤ lim r0 (t) < ∞.
t→∞
2.3
t→∞
t→∞
(2.10)
Finite generalized mixtures
First, we study the asymptotic behaviour of the failure rate of finite generalized mixtures. For this purpose, we assume that the reliability functions Ri ¿0 for all t and i. Hence, the failure rate function ri are defined for t → ∞. The following proposition extends Lemma 2.1 in Block
et al. (2003a), Proposition 8 in Navarro and HernanNavarro and Hernandez
(2004) and Lemma 2.5 in Li (2005) on positive mixtures. Moreover, note
that it can also be applied to finite negative mixtures.
Proposition 2.6. If F is a distribution function, (2.2) holds for F1 , ..., Fn
and
lim inf ri (t)/r1 (t) = ξi ∈ (1, ∞] for i = 2, 3, ..., n,
(2.11)
t→∞
then the following properties hold.
1. limt→∞ Ri (t)/R1η (t) = 0 for any i 6= 1 and η such that 1 ≤ η < ξi .
2. limt→∞ R(t)/R1 (t) = w1 > 0.
3. If lim supt→∞ ri (t)/r1 (t) < ∞, then limt→∞ r(t)/r1 (t) = 1.
4. If limt→∞ fi (t)/f1 (t) = λi for i = 2, 3, ..., n, then λi = 0 for i =
2, 3, ..., n and limt→∞ r(t)/r1 (t) = 1.
9
Reliability properties of mixtures
Here, ri , fi and Ri are the failure rate, the density and the reliability
functions of Fi , respectively.
The following example shows that we need some conditions (as that
included in the preceding proposition) to obtain limt→∞ r(t)/ri (t) = 1 for
some index i in negative mixtures.
Example 2.1. Let us to consider the reliability functions R1 (t) = exp(−λt)
and R2 (t) = 2 exp(−λt) − exp(−2λt) for t ≥ 0. It is easy to prove that
R2 is a genuine reliability function and that, from the preceding proposition, limt→∞ r2 (t) = λ. Hence the exponential reliability function R(t) =
exp(−2λt) can be written as the following negative mixture R(t) = 2R1 (t)−
R2 (t). However, limt→∞ r(t) = r(t) = 2λ whereas limt→∞ r2 (t) = r1 (t) =
λ.
Next, we study the asymptotic monotonicity of the mixture failure rate
extending Theorem 2.4 in Block et al. (2003a).
Proposition 2.7. If F is a distribution function, (2.2) holds for F1 , ..., Fn
and the failure rates are differentiable, then
r′ (t) =
n
X
i=1
wi
X
Ri (t) ′
Ri (t) Rj (t)
ri (t) −
wi w j
(rj (t) − ri (t))2
R(t)
R(t) R(t)
(2.12)
i<j
where ri and Ri are the failure rate and the reliability functions of Fi ,
respectively. Moreover, if the failure rate functions satisfy (2.11) and the
following conditions,
lim sup ri′ (t)/r1′ (t) < ∞ for i = 2, 3, ..., n,
(2.13)
t→∞
and
lim Ri (t)Rj (t)R1−2 (t)(rj (t) − ri (t))2 /r1′ (t) = 0 for i 6= j,
t→∞
(2.14)
then limt→∞ r′ (t)/r1′ (t) = 1. In particular, if r1 is ultimately strictly increasing (decreasing), then r is ultimately strictly increasing (decreasing).
Remark 2.5. In particular, if F1 is IFR, Fi is DFR and wi < 0 for
i = 2, 3, ..., n, then, from (2.12), F is IFR. Note that, from Proposition
2.6, (2.14) holds when
lim sup(rj (t) − ri (t))2 /r1′ (t) < ∞
t→∞
(2.15)
10
J. Navarro and P. J. Hernández
holds for i 6= j, and, in particular, when r2 , r3 , ..., rn and 1/r1′ are bounded
as t → ∞. Also note that, from (2.12) and Proposition 2.6, (2.15) can be
replaced by
lim sup Riγ (t)Rjγ (t)(rj (t) − ri (t))2 /r1′ (t) < ∞,
t→∞
where γ = η − 1 and 1 < η < min(ξi , ξj ). Moreover, from (2.12), if r1 is
ultimately strictly increasing (decreasing) and (2.14) holds for i and j such
that wi wj > 0, then r is ultimately strictly increasing (decreasing).
Remark 2.6. We can also use (2.12) to study the initial behaviour of r(t)
for finite generalized mixtures obtaining the same result as that of Theorem
2.5 in Block et al. (2003a).
Example 2.2. If ri (t) = ai t + bi for t ≥ 0, 0 < a1 < ai , bi ≥ 0 and
i = 1, 2, ..., n, then (2.11)-(2.14) hold and hence, r is ultimately strictly
increasing and it approaches to r1 as t → ∞. Moreover, the initial behavior
can be determined from (2.12) through
r′ (0+) =
n
X
i=1
3
wi ai −
X
wi wj (bj − bi )2 .
i<j
Generalized mixtures of exponential distributions
The family of distributions obtained as generalized mixtures of exponential
distributions is called the class of generalized hyperexponential (GH) distributions (see Bartholomew (1969), Botta et al. (1987) and Harris et al.
(1992)). This family of distributions can be used to study the distribution
of coherent systems with different kind of joint multivariate exponential
distributions (see Baggs and Nagaraja (1996) and Navarro et al. (2007)).
First, we study the easiest case, a mixture with two components. The
following proposition covers all the possible cases.
Proposition 3.1. If (2.3) holds for Fi (t) = 1 − exp(−t/µi ), i = 0, 1 and
µ0 < µ1 , then the following properties hold.
1. If 0 < p < 1, then r(t) is strictly decreasing to 1/µ1 .
Reliability properties of mixtures
11
2. If 1 < p < pmax = µ1 /(µ1 − µ0 ), then r(t) is strictly increasing to
1/µ1 .
Remark 3.1. Obviously, the cases p = 0, 1 are trivial. The property for 0 <
p < 1 was given in Barlow and Proschan (1975). If µ0 < µ1 , then f1 /f0
decreases, α = inf(f1 /f0 )) = µ0 /µ1 < 1, pmax = 1/(1 − α) = µ1 /(µ1 − µ0 )
and hence, the generalized mixture is valid for 1 < p < pmax . This property
was given by Bartholomew (1969). Note that the case p < 0 (or µ0 > µ1 )
is not possible. Moreover, from (??), r(t) is increasing. This property
was given by Baggs and Nagaraja (1996). They also studied generalized
exponential mixtures with three components. Figure 1 shows the failure
rates for exponential generalized mixtures with µ1 = 1 and µ0 = 1/2. The
dashed lines represent the failure rates r1 and r0 . The positive mixtures are
decreasing and the negative increasing. Note that rp > rp′ for p < p′ .
Next, we study the finite case defined by (2.2), where Fi (t) = 1 −
exp(−t/µi ) for t > 0 and i = 1, 2, ..., n. Without loss of generality we can
assume that
wk+1 , ..., wn < 0 < w1 , ..., wk ,
(3.1)
µi 6= µj for i 6= j,
(3.2)
µ1 > µ2 > ... > µk , µk+1 > µk+2 > ... > µn
(3.3)
and
for a k such that 1 ≤ k < n.
Note that this case cannot be reduced to a generalized mixture of two
exponential distributions since the mixtures of exponential distributions are
not, in general, exponential. We have obtained the following properties.
Proposition 3.2. If (2.2), (3.1), (3.2) and (3.3) hold for Fi (t) = 1 −
exp(−t/µi ), i = 1, 2, ..., n then
1. µ1 > µk+1 and limt→∞ r(t) = 1/µ1 .
2. If k = 1, then r(t) is strictly increasing to 1/µ1 .
3. If k > 1 and µ2 > µk+1 (<), then r(t) is ultimately strictly decreasing
(increasing) to 1/µ1 .
12
J. Navarro and P. J. Hernández
4
Generalized mixtures of Pareto distributions
Now we consider generalized mixtures of two Pareto type II (Lomax) distributions with the same shape parameter, i.e. Fi (t) = 1 − (1 + ai t)−c , t > 0,
ai , c > 0, i = 0, 1. The following proposition covers all the possible cases.
Proposition 4.1. If (2.3) holds for 1 < p < pmax and Fi (t) = 1 − (1 +
ai t)−c , i = 0, 1, then the following properties hold.
1. If a1 > a0 (<), then pmax = ac1 /(ac1 − ac0 ) (p∗max = a0 /(a0 − a1 )).
2. limt→∞ R/R1 = (1 − p/pmax )ac1 /ac0 .
3. limt→∞ R/R0 = (1 − p/pmax ).
4. If a1 > a0 (<), then r0 < r1 < r (>).
5. If p 6= pmax , then limt→∞ r(t)/ri (t) = 1, i = 0, 1.
6. If a1 > a0 and p = pmax , then limt→∞ r(t)/r1 (t) = (c + 1)/c.
7. If p 6= pmax , then limt→∞ r′ (t)/r1′ (t) = 1 and r(t) is ultimately strictly
decreasing to 0.
Remark 4.1. Note that we can have a negative mixture (p > 1) with
r0 < r1 . The case a1 < a0 with 1 < p < p∗max is equivalent to the case
a1 > a0 with p∗max ≤ p < 0. That is, we can consider generalized mixtures
of Pareto distributions with a1 > a0 for pmin = a0 /(a0 − a1 ) ≤ p ≤ pmax =
ac1 /(ac1 − ac0 ). In this case, we can study the initial behavior of r(t) by using
(2.12), obtaining that r′ (0) < 0 iff p1 < p < p2 , where
p
c(a1 − a0 ) + a0 + a1 − (c + 1)2 (a20 + a21 ) + 2a1 a0 (1 − c2 )
p1 =
2c(a1 − a0 )
and
p2 =
c(a1 − a0 ) + a0 + a1 +
p
(c + 1)2 (a20 + a21 ) + 2a1 a0 (1 − c2 )
2c(a1 − a0 )
Figure 2 shows the mixture failure rates for a1 = 2, a0 = 1 and c = 1.
The dashed lines represent the components failure rates (r1 > r0 ). In
particular, in this case, pmin = −1, pmax = 2, p1 = −0.2360679775 and
p2 = 4.2360679775. Moreover, if p = 2, we have limt→∞ r(t)/ri (t) = 2 for
i = 0, 1, that is, the asymptotic behaviour is not equivalent to that of the
members of the mixture. Also note that r(t) is decreasing for p > p1 .
Reliability properties of mixtures
13
Figure 2: Generalized mixtures of two Pareto type II distributions with a1 = 2,
a0 = 1, c = 1 and p = 0.2k, for k = −5, −4, ..., 10 (from the bottom to the
top). The dashed lines represent the failure rates of the members of the mixture
(r1 > r0 ).
5
Applications to coherent systems
Let us suppose that the random vector (X1 , ..., Xn ) represents the lifetimes
of n (possibly) dependent components in a system. Here, the dependence
is represented by the joint distribution function F (x1 , ..., xn ) = Pr(X1 ≤
x1 , ..., Xn ≤ xn ) or the joint reliability (survival) function R(x1 , ..., xn ) =
Pr(X1 > x1 , ..., Xn > xn ). In this section, we suppose that R is continuous.
If we consider the (increasing) ordered random vector (X(1,n) , ..., X(n,n) ),
then the lifetime of the k-out-of-n system is represented by the order statistics X(n−k+1,n) . In particular, the series system is represented by the minimum X(1,n) and the parallel by the maximum X(n,n) and its reliability
functions are given by R(1,n) (t) = R(t, ..., t) and R(n,n) (t) = 1 − F (t, ..., t),
respectively.
14
J. Navarro and P. J. Hernández
Baggs and Nagaraja (1996) showed that
R(1,2) (t) + R(2,2) (t) = R1 (t) + R2 (t),
(5.1)
where Ri (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 (or parallel)
distribution. Hence, from (5.1), the parallel (or series) distribution is a
generalized mixture of three exponential distributions. They studied this
kind of generalized mixtures and applied the results to systems. Note that
marginal distributions can be seen as one-component series distributions.
Hence, (5.1) shows that any two-components parallel system is a generalized mixture of series systems. Expression (3.1) in Block et al. (2003a)
extended this result showing that any coherent system is a generalized mixture of series systems, i.e.
X
RT (t) =
(−1)1+|A| RA (t)
A
where RA (t) = Pr(Xi > t for all i ∈ ∪j∈A Pj ), P1 , ..., Pm are the minimal
path sets of the coherent system T , A ⊆ {1, ..., m} and | A | is the number
of elements in A. The definition of the minimal path sets of a coherent
system can be seen in Barlow and Proschan (1975). Note that RA (t) is
the reliability of the series system with components ∪j∈A Pj . Obviously,
in
Q
particular, if the components are independent, then RA (t) = i∈A Ri (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 series (parallel) reliability only depends on the number of
components. Thus, they defined the minimal signature
of a coherent system
P
T as the vector (α1 , ..., αn ) such that RT (t) = ni=1 αi R(1,i) (t). They also
defined the maximal signature by using 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 Proposition 2.6, we have the following result.
Proposition 5.1. If T is a coherent system with reliability RT , failure rate
rT and minimal path sets P1 , .., Pm such that
lim inf rA (t)/r{1} (t) = ξA ∈ (1, ∞],
t→∞
for all A ⊆ {1, ..., m} (A 6= {1}), then the following properties hold.
(5.2)
Reliability properties of mixtures
15
η
1. limt→∞ RA (t)/R{1}
(t) = 0 for any η with 1 ≤ η < ξA .
2. limt→∞ RT (t)/R{1} (t) = 1.
3. If lim supt→∞ rA (t)/r{1} (t) < ∞, holds for all A ⊆ {1, ..., m}, then
limt→∞ rT (t)/r{1} (t) = 1.
Here, rA and RA are the failure rate and the reliability functions, respectively, of the series system with components ∪j∈A Pj .
Remark 5.1. If the components are independent, then (5.2) can be replaced
by
lim inf r{i} (t)/r{1} (t) > 1.
t→∞
The path set satisfying (5.2) can be called, the leading path set. Theorem 3.1
in Block et al. (2003a) can be extended in a similar way by using Proposition
2.7.
Example 5.1. Consider a 2-out-of-3 system with minimal path sets P1 =
{1, 2}, P2 = {2, 3} and P3 = {1, 3}. Thus, the system reliability is given by
R(2,3) (t) = R{1} (t) + R{2} (t) + R{3} (t) − 2R(1,3) (t).
If the components have exchangeable joint distribution, then R(2,3) = 3R(1,2) −
2R(1,3) and the minimal signature is (0, 3, −2). If we suppose that the components are independent, then the system failure rate can be computed from
r(2,3) (t) =
+
R1 (t)R3 (t)
R1 (t)R2 (t)
(r1 (t) + r2 (t)) +
(r1 (t) + r3 (t))
R(2,3) (t)
R(2,3) (t)
R2 (t)R3 (t)
(r2 (t) + r3 (t))
R(2,3) (t)
−2
R1 (t)R2 (t)R3 (t)
(r1 (t) + r2 (t) + r3 (t)).
R(2,3) (t)
If we also suppose that they have exponential distributions with ri (t) = λi
for t > 0 and λ1 < λ2 < λ3 , then the system distribution is a negative
mixture of four exponential distributions. Moreover, from the preceding
proposition we have limt→∞ r(2,3) (t) = λ1 + λ2 . Note that, if λ1 = λ2 = λ3 ,
then it is a negative mixture of two exponential distributions and hence it
is IFR. If two components have the same distribution, then it is a negative
16
J. Navarro and P. J. Hernández
Figure 3: Failure rate r(2,3) (t) of a 2-out-of-3 system with three independent
exponential components and λ1 = λ2 = λ3 = 1.5, λ1 = 1, λ2 = λ3 = 2, λ1 =
λ2 = 1.5, λ3 = 2 and λ1 = 1, λ2 = 2, λ3 = 3 (from the bottom to the top at t = 1).
mixture of three exponential distributions and it can be studied from Baggs
and Nagaraja’s results. In Figure 3, we show the failure rate r(2,3) for
different values of λ1 , λ2 and λ3 . The dashed lines represent the asymptotic
behaviours of r(2,3) . In Figure 4, we show the failure rates of all the coherent
systems with three independent exponential components with λ1 = λ2 =
λ3 = 1. The definitions of these systems can be seen in Shaked and SuarezLlorens (2003) and its minimal signatures are (0,0,1), (0,2,-1), (0,3,-2),
(1,1,-1) and (3,-3,1).
Remark 5.2. We can also apply the results on mixtures to coherent systems with exchangeable components by using its representation as mixtures
of k-out-of-n systems based on Samaniego’s signature given in Navarro
et al. (2005) and Navarro and Rychlik (2006). The Samaniego’s signatures for coherent systems with 3 or 4 components can be seen in Shaked
and Suarez-Llorens (2003).
Reliability properties of mixtures
17
Figure 4: Failure rates of all the coherent systems with three independent exponential components and λ1 = λ2 = λ3 = 1. From the top to the bottom, the lines represent the failure rate functions of T = min(X1 , X2 , X3 )
(series system), T = min(X1 , max(X2 , X3 )), T = X(2,3) (2-out-of-3 system),
T = max(X1 , min(X2 , X3 )), and T = max(X1 , X2 , X3 ) (parallel system).
References
Baggs, G. E. and Nagaraja, H. N. (1996). Reliability properties of
order statistics from bivariate exponential distributions. Communication
in Statistics Stochastic Models, 12:611–631.
Barlow, R. E. and Proschan, F. (1975). Statistical theory of reliability
and life testing. Holt, Rinehart and Winston, New York.
Bartholomew, D. J. (1969). Sufficient conditions for a mixture of exponentials to be a probability density function. Annals of Mathematical
Statistics, 40:2183–2188.
Bekker, L. and Mi, J. (2003). Shape and crossing properties of mean
residual life functions. Statistics and Probability Letters, 64:225–234.
18
J. Navarro and P. J. Hernández
Block, H. W. and Joe, H. (1997). Tail behavior of the failure rate
functions of mixtures. Lifetime and Data Analysis, 3:269–288.
Block, H. W., Li, Y., and Savits, T. H. (2003a). Initial and final
behaviour of failure rate functions for mixtures and systems. Journal of
Applied Probability, 40:721–740.
Block, H. W., Mi, J., and Savits, T. H. (1993). Burn-in and mixed
populations. Journal of Applied Probability, 30:692–702.
Block, H. W., Savits, T. H., and Wondmagegnehu, E. T. (2003b).
Mixtures of distributions with increasing linear failure rates. Journal of
Applied Probability, 40:485–504.
Botta, R. F., Harris, C. M., and G., M. W. (1987). Characterizations
of generalized hyperexponential distribution functions. Communication
in Statistics Stochastic Models, 3:115–148.
Everitt, B. S. and Hand, D. J. (1981). Finite mixture distributions.
Chapman and Hall, New York.
Harris, C. M., Marchal, W. G., and Botta, R. F. (1992). A note
on generalized exponential distributions. Communications in Statistics
Theory & Methods, 8:179–191.
Li, Y. (2005). Asymptotic baseline of the hazard rate function of mixtures.
Journal of Applied Probability, 42:892–901.
Mi, J. (1995). Bathtub failure rate and upside-down bathtub mean residual
life. IEEE Transaction on Reliability, 44:388–391.
Mi, J. (2004). A general approach to the shape of failure rate and mrl
functions. Naval Research and Logistic, 51:543–556.
Navarro, J. and Hernandez, P. J. (2004). How to obtain bathtub
shaped failure rate models from normal mixtures. Probability in the
Engineering and Informational Sciences, 18:511–531.
Navarro, J., Ruiz, J. M., and Sandoval, C. J. (2005). A note on
comparisons among coherent systems with dependent components using
signatures. Statistics and Probability Letters, 72:179–185.
Reliability properties of mixtures
19
Navarro, J., Ruiz, J. M., and Sandoval, C. J. (2007). Modelling
coherent systems under dependence. To appear in Communications in
Statistics Theory & Methods, 36 (1).
Navarro, J. and Rychlik, T. (2006). Reliability and expectation bounds
for coherent systems with exchangeable components. To appear in Journal of Multivariate Analysis.
Savits, T. H. (2003). Preservation of generalized bathtub-shaped functions. Journal of Applied Probability, 40:473–484.
Shaked, M. and Spizzichino, F. (2001). Mixtures and monotonicity of
failure rate functions. In Advances in reliability, Handbook of Statististics,
pp. 185–198. North-Holland, Amsterdam.
Shaked, M. and Suarez-Llorens, A. (2003). On the comparison of
reliability experiments based on the convolution order. Journal of the
American Statistical Association, 98:693–702.
Wondmagegnehu, E. T. (2004). On the behavior and shape of mixture
failure rates from a family of ifr weibull distributions. Naval Research
Logistics, 51:491–500.
Wu, J. W. (2001). Characterizations of generalized mixtures of geometric
and exponential distributions based on upper record values. Statistical
Papers, 42:123–133.