Stochastic Models of Failure
4.7 PROBABILITY MODELS FOR INTERDEPENDENT LIFELENGTHS The notion of independent events was introduced in section 2.2.1, and that of independent
PROBABILITY MODELS FOR INTERDEPENDENT LIFELENGTHS 85 Similarly, corresponding to each term of (4.23), save the first, there exist random variables X2∗ X3∗ Xn−1∗ , independent of each other and also ofXn∗such that
PT1≥t1 Tn≥tn=PX1≥H1t1 PX2∗≥H2t2t1 PX∗n≥Hntnt1 tn−1
We now have, as a multivariate analogue to Theorem 4.1,
Theorem 4.4. Corresponding to every collection of nonnegative random variablesT1 Tn
having a survival functionRt1 tn, there exists a collection of n independent and exponen- tially distributed random variablesX1 X2∗ X∗n, (with scale parameter one), withX1indexed onH1t, andXi∗indexed onHitit1 ti−1, fori=2 nandn≥2.
4.7 PROBABILITY MODELS FOR INTERDEPENDENT LIFELENGTHS
Suppose that the partial derivative 2Ft1 t2/t1t2 exists (almost) everywhere, and let ft1 t2=2Ft1 t2/t1t2. Then, the bivariate distribution functionFt1 t2is said to have a bivariate densityft1 t2, and
Ft1 t2= t1 0
t2 0
fs1 s2ds1ds2
It is important to note that a bivariate distribution functionFt1 t2is not uniquely determined by its marginal distribution functionsF1t1andF2t2. There are an infinite number of solutions to the problem of determiningFt1 t2fromF1t1andF2t2. One such solution is given by theFréchet bounds(Fréchet, 1951).
maxF1t1+F2t2−10≤Ft1 t2≤minF1t1 F2t2 (4.27) the bounds are themselves bivariate distributions whose marginals areF1t1 and F2t2. An exception to the above occurs when for allt1 t2≥0, the eventsT1≤t1andT2≤t2are judged independent, because now
Ft1 t2=F1t1 F2t2 and so the marginals provide a unique joint distribution.
Another way to construct Ft1 t2 from F1t1 and F2t2 is due to Morgenstern (1956).
Specifically, for some 0≤≤1,
Ft1 t2=F1t1 F2t2 1+1−F1t11−F2t2 (4.28) A bivariate exponential distribution of Gumbel (1960) (section 4.7.2) is based on this con- struction. Yet another approach is the method of copulas described below. Singpurwalla and Kong (2004) have used this approach to construct a new family of bivariate distributions with exponential marginals (section 4.7.7).
The Method of Copulas
AcopulaCis a bivariate distribution on01×01, whose marginal distributions are uniform.
Copulas join (i.e. couple) univariate distribution functions to form multivariate distribution functions (Nelson, 1995). This feature is encapsulated in the following theorem.
Theorem 4.5. (Sklar, 1959). Let F be a two-dimensional distribution function with marginal distribution functions F1 and F2. Then there exists a copula C such that Ft1 t2= CF1t1 F2t2. Conversely, for any univariate distribution functionsF1andF2and any cop- ula C, the function Ft1 t2=CF1t1 F2t2 is a two-dimensional function with marginal distribution functionsF1andF2.
The notion of copulas and Sklar’s theorem generalize to the multivariate case. Sklar’s theorem enables us to generate copulas, and copulas can be used to characterize certain properties of sequences of random variables. Specifically, Ft1 t2=CF1t1 F2t2 implies that for 0≤u v≤1
Cu v=F
F1−1u F2−1v
PROBABILITY MODELS FOR INTERDEPENDENT LIFELENGTHS 87 so that knowing F F1 and F2, we are able to generate C. Thus, for example, if T1 and T2 are independent random variables with distribution functions F1 andF2, respectively, then Ft1 t2=F1t1 F2t2from which it follows thatCu v=uv. Conversely, by Sklar’s theorem, T1 andT2 are independent if Cu v=uv. Thus T1 and T2 are independent if, and only if, Cu v=uv. In a similar vein, we can argue that sinceT1andT2are exchangeable implies that the vectorsT1 T2andT2 T1have the same distribution,T1andT2are exchangeable if, and only if,F1=F2andCu v=Cu v.
Sklar’s theorem also enables us to obtain bounds on a copulaCu vvia the Fréchet bounds of (4.27); this is because the Fréchet bounds are themselves distributions. Specifically, for 0≤u v≤1,
maxu+v−10≤Cu v≤minu v
maxu+v−10 is a copula for maxF1t1+F2t2−10 and minu v is a copula for minF1t1 F2t2. Finally, Morgenstern’s distribution of (4.28) gives rise to the copula
Cu v=uv1+1−u1−v
An alternate method of generating copulas is due to Genest and MacKay (1986).
Whereas a copulaCjoins univariate distribution functions to form a multivariate distribution, a survival copula C joins univariate survival functions to form a multivariate survival func- tion. Thus in the bivariate case, we haveF t1 t2=CF1t1 F2t2, where it can be easily seen that
Cu v=u+v−1+C1−u1−v
LikeCCis a copula, butCis not to be confused withCu v, the bivariate survival function of two uniformly distributed random variables. This is because
Cu v=1−u−v+Cu v=C1−u1−v
Interest in survival copulas stems from the fact that for some multivariate distributions, such as Marshall and Olkin’s (1967) multivariate exponential, survival copulas take simpler forms than their corresponding copulas.
Measures of Interdependence
The dependence of T1 on T2 (or vice versa) is best described via the conditional survival functionPT1≥t1T2=t2, and/or theconditional meanET1T2=t2, where
PT1≥t1T2=t2=PT1≥t1 T2=t2
PT2=t2 and (4.29)
ET1T2=t2= 0
1−PT1≥t1T2=t2dt1
It follows from the above that when (T1≥t1andT2=t2are judged independent,PT1≥ t1T2=t2=PT1≥t1, andET1T2=t2=ET1. The conditional meanET1T2=t2is also known as theregressionofT1onT2.
If T1andT2 are judged dependent, then the extend to whichT1 andT2 experience a linear relationship is measured by the product momentET1T2, where (assuming thatft1 t2exists),
ET1T2= 0
0
t1t2ft1 t2dt1dt2 (4.30)
= 0
t1ET2T1=t1f1t1dt1 f1t1is the probability density generated byF1t1.
A normalization of the product moment to yield values between−1 and+1 results in Pearson’s coefficient of correlationT1 T2, where
T1 T2=ET1T2−ET1ET2 VT1VT21/2
VTiis the variance ofFiti i=12. The numerator,ET1T2−ET1ET2, is known as the covarianceofT1andT2; it is denoted CovT1 T2.
It can be verified that−1≤T1 T2≤+1, and thatT1 T2=0 ifT1andT2are independent.
However, since T1 T2 provides an assessment of only the extent of a linear relationship betweenT1andT2 T1 T2=0, does not necessarily imply thatT1andT2are independent; one exception is the case whereinT1andT2have a bivariate Gaussian distribution. Another exception is the BVE of section 4.7.4. A stronger result pertaining to independence under uncorrelatedness is due to Joag-Dev (1983), who shows that whenT1andT2are ‘associated’,T1 T2=0 implies thatT1andT2are independent. The notion of association, as a measure of dependence, is due to Esary, Proschan and Walkup (1967). It says that a random vectorT=T1 Tnis associated if for every (co-ordinatewise) non-decreasing functionf andg,CovfT gT≥0.
Values of T1 T2 > <0 suggest that the events T1≥t1 and T2≥t2 are positively (negatively) dependent, for all values oft1andt2. Under positive dependence the failure of one component increases the probability of failure of the surviving components. Positive dependence manifests itself when components share a common load or when components operate in a common environment. With negative dependence, the failure of one component increases (decreases) the probability of survival (failure) of the other component. For physical systems, the judgment of negative dependence is difficult to foresee. With certain biological systems, negative dependence is justified on grounds that for entities that compete for limited resources, such as food, the failure of one unit increases the available resources for the surviving unit, and thus its probability of failure decreases.
The departure ofT1 from the regression ofT1 on T2 is measured by the squared correla- tion ratio
2T1T2= 1 VT2
0
ET1−ET1T2=t22f2t2dt2
Like the conditional mean and the coefficient of correlation, the squared correlation ratio is also a measure of the dependence ofT1onT2.
By making the transformationsu=F1t1 v=F2t2andcu v=FF1−1u F2−1v, it can be seen that in terms of copulas, the coefficient of correlation takes the form
T1 T2=VT1 VT2−12 1
0
1 0
Cu v−uvdF1−1udF2−1v so thatT1 T2=0, ifT1andT2are independent.
PROBABILITY MODELS FOR INTERDEPENDENT LIFELENGTHS 89 The above representation ofT1 T2suggests some alternate measures of dependence ofT1 onT2. These areSpearman’s Rho, given by
rT1 T2=12 1
0
1 0
Cu v−u vdudv andKendall’s Taugiven by
T1 T2=4 1
0
1 0
Cu vdCu v−1 for 0≤u v≤1
Finally, there is a measure of positive dependence that is suitable for lifelengths sharing a common environment. This is apositive quadrant dependence, wherein for allt1andt2 PT1≤ t1 T2≤t2≥PT1≥t1 PT2≥t2. According to this definition,T1andT2are positively quadrant dependent if, and only if, Cu v≥uv, for all 0≤u v≤1, or equivalently Cu v≥uv.
Geometrically, the above inequality suggests that for a positive quadrant dependence, the graph ofCu vmust lie on or above that line ofuv. Thus copulas provide a graphical way to portray the positive dependence between two lifetimes T1andT2, the strength of dependence being a function of the deviation ofCu vfromuv. SinceCu vchanges withuandv, the dependence portrayed byCu v≥uvis ‘local’. Spearman’s Rho,rT1 T2, provides a more global measure of positive quadrant dependence.
4.7.2 The Bivariate Exponential Distributions of Gumbel
Gumbel (1960) has proposed a system of bivariate distributions whose marginal distributions are exponential – thus the name ‘bivariate exponential’. This system may be viewed as the very first family of probability models for describing dependent lifelengths. A drawback of Gumbel’s system is that the proposed models lack a physical motivation, and that a generalization to the multivariate case is not obvious. Its advantages are that unlike many of the other multivariate lifelength distributions, both positive and negative dependence can be represented; also the marginal distributions are unit exponential (i.e. an exponential distribution with a scale parameter one). This latter feature makes the system a natural choice for generating other families of distributions for dependent lifelengths via hazard potentials (Theorem 4.3). This is one of the main reasons for introducing the Gumbel family.
Under Gumbel’s system, there are two versions of the bivariate exponential distribution. Under the first version the correlation is always negative, and the largest value that it can take is
−0 4036. Under the second version, both positive and negative values of the correlation can be had, and the maximum value of the correlation is±0 25. I overview below each of the two versions.