PART V. Practices and Emerging Applications
7.4 Families of Bivariate Distributions that are Positive
Since the PQD concept is important in reliability applications, it is imperative for a reliability practitioner to know what kinds of PQD bivariate distribution are available for reliability modeling.
In this section, we list several well-known PQD distributions, some of which were originally derived from a reliability perspective. Most of these PQD bivariate distributions can be found, for example, in Hutchinson and Lai [1].
148 Statistical Reliability Theory
7.4.1 Positive Quadrant Dependent Bivariate Distributions with Simple Structures
The distributions whose PQD property can be established easily are now given below.
Example 1. The Farlie–Gumbel–Morgenstern bi- variate distribution [23]:
F (x, y)=FX(x)FY(y)
× {1+α[1−FX(x)][1−FY(y)]}
(7.15) For convenience, the above family may simply be denoted by FGM. This general system of bivariate distributions is widely studied in the literature. It is easy to verify thatXandYare PQD ifα >0.
Consider a special case of the FGM system where both marginals are exponential. The joint distribution function is then of the form (e.g.see Johnson and Kotz [24], p.262–3):
F (x, y)
=(1−e−λ1x)(1−e−λ2y)(1+αe−λ1x−λ2y) Clearly
w(x, y)=F (x, y)−FX(x)FY(y)
=αe−λ1x−λ2y(1−e−λ1x)(1−e−λ2x) 0< α≤1
satisfies the conditions in Equations 7.7–7.9, and henceXandY are PQD.
Mukerjee and Sasmal [25] have worked out the properties of a system of two exponential components having the FGM distribution. The properties are such things as the densities, means, moment-generating functions, and tail probabilities of min(X, Y ), max(X, Y ), and X+ Y, these being of relevance to series, parallel, and standby systems respectively.
Lingappaiah [26] was also concerned with properties of the FGM distribution relevant to the reliability context, but with gamma marginals.
Building upon a paper by Philips [27], Kotz and Johnson [28] considered a model in which
components 1 and 2 were subject to “revealed”
and “unrevealed” faults respectively, with (X, Y ) having an FGM distribution, whereXis the time between unrevealed faults andY is the time from an unrevealed fault to a revealed fault.
Example 2. The bivariate exponential distribu- tion
F (x, y)=1−e−x−e−y+(ex+ey−1)−1 This distribution is not well known, and we could not confirm its source. However, both marginals are exponential, which is used widely in a reliability context. This bivariate distribution function can be rewritten as
F (x, y)=1−e−x−e−y+e−(x+y) +(ex+ey−1)−1−e−(x+y)
=FX(x)FY(y)
+(ex+ey−1)−1−e−(x+y) Now
(ex+ey−1)−1−e−(x+y)
= (ex−1)(ey−1) (ex+ey−1)e(x+y)
=(1−e−x)(1−e−y) (ex+ey−1)
≥0 and thereforeF is PQD.
Example 3. The bivariate Pareto distribution F (x, y)¯ =(1+ax+by)−λ a, b, λ >0 (e.g. see Mardia, [29], p.91). Consider a system of two independent exponential components that share a common environment factorηthat can be described by a gamma distribution. Lindley and Singpurwalla [30] showed that the resulting joint distribution has a bivariate Pareto distribution. It is very easy to verify that this joint distribution is PQD. For a generalization to multivariate components, see Nayak [31].
Example 4. The Durling–Pareto distribution F (x, y)¯ =(1+x+y+kxy)−a
a >0, 0≤k≤a+1 (7.16)
Concepts of Stochastic Dependence in Reliability Analysis 149 Obviously, it is a generalization of Example 3
above.
Consider a system of two dependent exponen- tial components having a bivariate Gumbel distri- bution
F (x, y)=1−e−x−e−y+e−x−y−θ xy x, y≥0, 0≤θ≤1
and sharing a common environment that has a gamma distribution. Sankaran and Nair [32] have shown that the resulting bivariate distribution is specified by Equation 7.16. It follows from Equation 7.16 that
F (x, y)¯ − ¯FX(x)F¯Y(y)
= 1
(1+x+y+kxy)a− 1 [(1+x)(1+y)]a 0≤k≤(a+1)
= 1
(1+x+y+kxy)a− 1
(1+x+y+xy)a
≥0 0≤k≤1
Hence,F is PQD if0≤k≤1.
7.4.2 Positive Quadrant Dependent Bivariate Distributions with More Complicated Structures
Example 5. Marshall and Olkin’s bivariate expo- nential distribution [33]
P (X > x, Y > y)
=exp[−λ1x−λ2y−λ12max(x, y)] λ≥0 (7.17) This has become a widely used bivariate exponen- tial distribution over the last three decades. The Marshall and Olkin bivariate exponential distribu- tion was derived from a reliability context.
Suppose we have a two-component system subjected to shocks that are always fatal. These shocks are assumed to be governed by three independent Poisson processes with parameters λ1,λ2, and λ12, according to whether the shock applies to component 1 only, component 2 only, or to both components. Then the joint survival function is given by Equation 7.17.
Barlow and Proschan [3], p.129, show that X andY are PQD.
Example 6. Bivariate distribution of Block and of Basu [34]
F (x, y)¯ =2+θ
2 exp[−x−y−θmax(x, y)]
−θ
2 exp[−(2+θ )max(x, y)] θ , x, y≥0
This was constructed to modify Marshall and Olkin’s bivariate exponential, which has a singular part. It is, in fact, a reparameterization of a special case of Freund’s [35] bivariate exponential distribution. The marginal is
F¯X(x)=1+θ
2 exp[−(1+θ )x] −θ
2 exp[(1+θ )x] and a similar expression for F¯Y(y). It is easy to show that this distribution is PQD.
Example 7. Kibble’s bivariate gamma distribu- tion. The joint density function is
fρ(x, y;α)=fX(x)fY(y)
×exp[−ρ(x+y)/(1−ρ)](α) 1−ρ
×(xyρ)−(α−1)/2Iα−1
2√ xyρ 1−ρ
0≤ρ <1
with,fX, fY being the marginal gamma probabil- ity density function with shape parameterα. Here, Iα(ã)is the modified Bessel function of the first kind and theαth order.
Lai and Moore [36] show that the distribution function is given by
F (x, y;ρ)=FX(x)FX(y) +α
2 ρ
0
ft(x, y;α+1)dt
≥FX(x)FY(y) because u(x, y)=3ρ
0 ft(x, y;α+1)dt is obvi- ously positive.
For the special case of when α=1, Kibble’s gamma becomes the well-known Moran–
Downton bivariate exponential distribution.
150 Statistical Reliability Theory
Downton [37] presented a construction from a reliability perspective. He assumed that the two componentsC1andC2 receive shocks occurring in independent Poisson streams at rates λ1 and λ2 respectively, and that the numbers N1
and N2 shocks needed to cause failure of C1
and C2 respectively have a bivariate geometric distribution.
For applications of Kibble’s bivariate gamma, see, for example, Hutchinson and Lai [1].
Example 8. The bivariate exponential distribu- tion of Sarmanov. Sarmanov [38] introduced a family of bivariate densities of the form
f (x, y)=fX(x)fY(y)[1+ωφ1(x)φ2(y)] (7.18) where
2 ∞
−∞φ1(x)fX(x)dx=0 2 ∞
−∞φ2(y)fY(y)dy=0
and ω satisfies the condition that 1+ωφ1(x)φ2(y)≥0for allxandy.
Lee [39] discussed the properties of the Sarmanov family; in particular, she derived the bivariate exponential distribution given below:
f (x, y)=λ2e−(x+y)
1+ω
e−x+ λ1
1+λ1
×
e−y+ λ2
1+λ2
(7.19) where
−(1+λ1)(1+λ2)
λ1λ2 ≤ω≤(1+λ1)(1+λ2) max(λ1, λ2) (Here, φ1(x)=e−x− [λ1/(1+λ1)] and φ2(y)= e−y− [λ2/(1+λ2)].)
It is easy to see that, forω >0:
F (x, y)=(1−e−λx)(1−e−λy)+ω λ
1+λ 2
× [e−λx−e−(λ+1)x][e−λy−e−(λ+1)y]
≥FX(x)FY(y)
whenceXandY are shown to be PQD if 0≤ω≤ (1+λ1)(1+λ2)
max(λ1, λ2)
Example 9. The bivariate normal distribution has a density function given by
f (x, y)=(2π 7
1−ρ2)−1
×exp
− 1
2(1−ρ2)(x2−2ρxy+y2)
−1< ρ <1
X and Y are PQD for 0≤ρ <1 and NQD for
−1< ρ≤0. This result follows straightaway from the following lemma:
Lemma 1. Let (X1, Y1) and (X2, Y2) be two standard bivariate normal distributions, with correlation coefficients ρ1 and ρ2 respectively. If ρ1≥ρ2, then
Pr(X1> x, Y1> y)≥Pr(X2> x, Y2> y) The above is known as the Slepian inequality [40, p.805].
By letting ρ2=0 (thus ρ1≥0), we establish thatXandY are PQD. On the other hand, letting ρ1=0(thusρ2≤0),XandY are then NQD.
7.4.3 Positive Quadrant Dependent Bivariate Uniform Distributions
A copulaC(u, v)is simply the uniform represen- tation of a bivariate distribution. Hence a copula is just a bivariate uniform distribution. For a formal definition of a copula, see, for example, Nelsen [41]. By a simple marginal transformation, a cop- ula becomes a bivariate distribution with specified marginals. There are many examples of copulas that are PQD, such as that given in Example 10.
Example 10. The Ali–Mikhail–Haq family
C(u, v)= uv
1−θ (1−u)(1−v) θ∈ [0,1]
It is clear that the copula is PQD. In fact, Bairamov et al. [42] have shown that it is a copula that corresponds to the Durling–Pareto distribution given in Example 4.
Nelsen [41], p.152, has pointed out that if X andY are PQD, then their copulaCis also PQD.
Nelsen’s book provides a comprehensive treatment
Concepts of Stochastic Dependence in Reliability Analysis 151 on copulas and a number of examples of PQD
copulas can be found therein.
7.4.3.1 Generalized
Farlie–Gumbel–Morgenstern Family of Copulas
The so-called bivariate FGM distribution given in Example 1 was originally introduced by Mor- genstern [43] for Cauchy marginals. Gumbel [44]
investigated the same structure for exponential marginals.
It is easy to show that the FGM copula is given by
Cα(u, v)=uv[1+α(1−u)(1−v)] 0≤u, v≤1, −1≤α≤1 (7.20) It is clear that the FGM copula is PQD for 0≤α
≤1.
It was Farlie [23] who extended the construc- tion by Morgenstern and Gumbel to
Cα(u, v)=uv[1+αA(u)B(v)] 0≤u, v≤1 (7.21) whereA(u)→0andB(v)→0asu, v→1,A(u) and B(v) satisfy certain regularity conditions ensuring thatC is a copula. Here, the admissible range ofαdepends on the functionsAandB.
IfA(u)=B(v)=1−u, we then have the clas- sical one-parameter FGM family Equation 7.20.
Huang and Kotz [45] consider the two types:
(i) A(u)=(1−u)p, B(v)=(1−v)p, p >1, −1≤α≤
p+1 p−1
p−1
(ii) A(u)=(1−up), B(v)=(1−vp), p >0, −(max{1, p})−2≤α≤p−1 We note that copula (ii) was investigated earlier by Woodworth [46].
Bairamov and Kotz [47] introduce further generalizations such that:
(iii) A(u)=(1−u)p, B(v)=(1−v)q, p >1, q >1(p=q),
−min
1, 1+p
p−1 p−1
1+q q−1
q−11
≤α≤min
1+p p−1
p−1
, 1+q
q−1 q−11
(iv) A(u)=(1−un)p, B(v)=(1−vn)q, p≥1; n≥1,
−min
1 n2
1+np n(p−1)
2(p−1)
,1 1
≤α≤ 1 n
1+np n(p−1)
p−1
Recently, Bairamovet al.[48] considered a more general model:
(v) A(u)=(1−up1)q1, B(v)=(1−vp2)q2, p1, p2≥1; q1, q2>1,
−min
1, 1 p1p2
1+p1q1 p1(q1−1)
q1−1
×
1+p2q2
p2(q2−1) q2−11
≤α
≤min
1 p1
1+p1q1
p1(q1−1) q1−1
, 1
p2
1+p2q2
p2(q2−1) q2−11
Motivated by a desire to construct PQDs, Lai and Xie [16] derived a new family of FGM copulas that possess the PQD property with:
(vi) A(u)=ub−1(1−u)a,
B(v)=vb−1(1−v)a, a, b≥1; 0≤α≤1 so that
Cα(u, v)=uv+αubvb(1−u)a(1−v)a a, b≥1, 0≤α≤1 (7.22)
152 Statistical Reliability Theory
Table 7.1. Range of dependence parameterαfor some positive quadrant dependent FGM copulas Copula type αrange for which copula is PQD
(i) 0≤α≤
p+1 p−1
p−1
(ii) 0≤α≤p−1
(iii) 0≤α≤min
1+p p−1
p−1 ,
1+q q−1
q−11
, p >1, q >1
(iv) 0≤α≤ 1
n
1+np n(p−1)
p−1
(v) 0≤α≤min
1 p1
1+p1q1 p1(q1−1)
q1−1 , 1
p2
1+p2q2 p2(q2−1)
q2−11
(vi) 0≤α≤ 1
B+(a, b)B−(a, b), B+, B−are some functions ofaandb
Bairamov and Kotz [49] have shown that the range ofαin Equation 7.22 can be extended and they also provide the ranges of α for which the copulas (i)–(v) are PQD. These are summarized in Table 7.1.
In concluding this section, we note that Joe [2], p.19, considered the concepts of PQD and the concordance ordering (more PQD) that are discussed in Section 7.6 as being basic to the parametric families of copulas in determining whether a multivariate parameter is a dependence parameter.