MULTIVARIATE ANALOGUES OF THE FAILURE RATE FUNCTION Multivariate analogues of the failure rate function have been considered, among others, by

Stochastic Models of Failure

4.5 MULTIVARIATE ANALOGUES OF THE FAILURE RATE FUNCTION Multivariate analogues of the failure rate function have been considered, among others, by

Basu (1971), Cox (1972), Johnson and Kotz (1975), Marshall (1975) and by Puri and Rubin (1974). One such analogue is the ‘hazard gradient’ discussed by Marshall (1975); this will be considered first.

4.5.1 The Hazard Gradient

A good starting point for introducing the hazard gradient is the cumulative hazard function Ht= −lnF t= t

0

rudu t≥0 (4.11)

mentioned in section (4.3).

ConsiderRt₁ t_n=PT₁≥t₁ T_n≥t_n, the joint survival function ofnlifelengths T₁ T_n, where, for convenience, we have omitted the. Lett=t₁ t_nbe such that Rt1 tn >0. Then, analogous to (4.11), we define Ht, the multivariate cumulative hazard functionasHt= −lnRt1 tn. If then-dimensional functionHthas a gradient, sayrt=Ht, wherert=r₁t r_nt, with

rit=

tHt i=1 n (4.12)

thenrtis called thehazard gradientofRt₁ t_n.

Johnson and Kotz (1975) interpretr_itas the conditional failure rate of T_i evaluated at t_i, were it to be so thatT_j> t_j, for allj=i. That is

rit= fitiTj> tj for allj=i

PTi> tiTj> tj for allj=i (4.13) wheref_it_iT_j> t_j, for allj=iis the conditional probability density function ofT_i, given that T_j> t_j, for allj=i.

From elementary calculus, it now follows that Ht= t 0

rudu (4.14)

MULTIVARIATE ANALOGUES OF THE FAILURE RATE FUNCTION 77

from which we have as a multivariate analogue of (4.9) the result that PT₁≥t₁ T_n≥t_n=exp− t

0

rudu (4.15)

the role of the hazard gradient is now apparent. Equation (4.14) holds as long as ruexists almost everywhere and that along the path of integration0t Htis absolutely continuous.

The following (additive) decomposition of Htgiven by Marshall (1975) is noteworthy; it parallels the (additive) decomposition ofHtin the univariate case. Specifically,

Ht= t1 0

r₁u₁0 0du₁+ t2 0

r₂t₁ u₂0 0du₂+ + tn 0

r_nt₁ t_n−1 u_ndu_n (4.16) where r₁u₁0 0 is the failure rate of T₁ at u₁, and r_it₁ t_i−1 u_i0 0 is the conditional failure rate of T_i at u_i, were it be such that T₁≥t₁ T₂≥t₂ T_i−1≥t_i−1. An interpretation of the decomposition of (4.16) will be given in section 4.6.2; however, a conse- quence of this decomposition is the well-known (multiplicative) decomposition ofRt1 tn. Specifically,

PT1≥t1 Tn≥tn=PT1≥t1 PT2≥t2T1≥t1 PTn≥tnT1≥t1 Tn−1≥tn−1 see Marshall (1975).

The univariate analogues of the additive and the multiplicative decompositions given above appear in section 4.3, following Equation (4.10).

4.5.2 The Multivariate Failure Rate Function

A more natural, though perhaps less useful, multivariate analogue of the univariate failure rate function is the multivariate failure rate function introduced by Basu (1971). I give below its bivariate version, and state some of its interesting characteristics.

Suppose that an absolutely continuous bivariate distribution functionFt₁ t₂withF00=0 generates a probability density function ft₁ t₂ at t₁ t₂. Then the bivariate failure rate of Ft₁ t₂at some_{1 2}is, forPT₁≥₁ T₂≥₂ >0, given as

r_{1 2}= f1 2

PT1≥1 T2≥2 (4.17)

= f_{1 2}

1+F_{1 2}−F₁−F 2

Clearly, when T₁ andT₂ are mutually independent,r_{1 2}=r₁r₂, where r₁and r2are the corresponding univariate failure rates at 1 and2, respectively (equation (4.7)).

The relationship of (4.17) can be extended to the multivariate case.

As seen before, in section 4.4, the only univariate distribution whose failure rate is a constant is the exponential. We now ask if there is an analog to this result when ther_{1 2}of (4.17) is a constant, say >0. Basu (1971) shows that the only absolutely continuous bivariate distribution with marginal distributions that are exponential and whose bivariate failure rate function is a constant, is the one that is obtained whenT1andT2 are exponentially distributed.

The search for multivariate distributions with exponential marginals is motivated by the fact that univariate exponential distributions, because of their simplicity and ease of use, are very popular in engineering reliability. The attractiveness of absolute continuity stems from the fact

that statistical inference involving such distributions is relatively easy (cf. Proschan and Sullo, 1974). If Basu’s (1971) requirement of exponential marginals is relaxed, then according to a theorem by Puri and Rubin (1974), the only absolutely continuous multivariate distribution whose multivariate failure rate (in the sense of (4.17)) is a constant is the one given by a mixture of exponential distributions.

Thus to summarize, it appears that the main value of the multivariate analogue of the failure rate function considered in this subsection is a characterization of the multivariate failure rate function that is a constant.

4.5.3 The Conditional Failure Rate Functions

Another multivariate analogue of the univariate failure rate function is the one proposed by Cox (1972) in his classic paper, and under the subheading ‘bivariate life tables’. In the bivariate case involving lifelengthsT1andT2we define, analogous to (4.4) of section 4.3, the following four univariate conditional failure rate functions:

r_p0t=lim

dt↓0

Pt≤T_p≤t+dtT₁≥t T₂≥t

dt forp=12

r₂₁tu=lim

dt↓0

Pt≤T₂≤t+dtT₂≥t T₁=u

dt foru < t and r₁₂tu=lim

dt↓0

Pt≤T₁≤t+dtT₁≥t T₂=u

dt foru < t (4.18) In the latter two expressions,T_i=u i=12, is to be interpreted asT_i∈u u+du, where duis infinitesimal.

A motivation for introducing the conditional failure rate functions parallels that for introducing the failure rate function of (4.4) and (4.5). Namely, that a subject matter specialist can subjectively specify the functionsr_p0t, forp=12, based on the aging characteristics of an item, and the functionsr_ijt u i j=12 based on the aging as well as the load sharing features of the surviving item; for example, Freund (1961). IfFt₁ t₂denotes the joint distribution function ofT1 andT2, and ft1 t2the probability density at t1 t2generated byFt1 t2, then using arguments that parallel to those leading to (4.10), it can be shown that fort1≤t2

ft₁ t₂=exp− t₁ 0

r₁₀u+r₂₀udu− t₂ t₁

r₂₁ut₁dur₁₀t₁r₂₁t₂t₁ (4.19) analogously for the caset₂≤t₁.

Consequently, as was the case with the univariate failure rate function, a specification of the conditional failure rate functions of (4.18) enables us to obtain ft₁ t₂and hence Ft₁ t₂. Alternatively, if we are to knowRt₁ t₂=PT₁≥t₁ T₂≥t₂, then we may obtainr₁₀tand r_ijtu i j=12, via relationships of the type:

r₁₀t= − 1 Rt t

tRt u

u=t

and

r12tu= −²Rt u

t u /Rt u u

Finally, it is relatively easy to show thatT₁is independent ofT₂, if and only ifr₁₂tu=r₁₀t, andr₂₁tu=r₂₀t.

THE HAZARD POTENTIAL OF ITEMS AND INDIVIDUALS 79 To conclude this section, we note thatRt₁ t_ncan be obtained through (4.15) via the hazard gradientruor through (4.19) via the conditional failure rate function. The former is of mathematical interest, especially for the additive decomposition of (4.16); the latter has an operational appeal since it upholds the original spirit of introducing the failure rate function.