Stochastic Models of Failure
4.6 THE HAZARD POTENTIAL OF ITEMS AND INDIVIDUALS
THE HAZARD POTENTIAL OF ITEMS AND INDIVIDUALS 79 To conclude this section, we note thatRt1 tncan 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.
Cumulative hazard H(t) b
a
0 Ta Tb (=Time to failure
when X=b) Time t The (unknown)
hazard potential, X
Figure 4.7 Illustration of resource depletion by the cumulative hazard function.
nondecreasing int) depicts the rate at which the resource gets consumed (Figure 4.7). This is one interpretation ofHt; its role in explaining lifelengths having a decreasing failure rate function has been articulated by Kotz and Singpurwalla (1999).
An alternate interpretation ofHtis motivated by the fact that the exponential distribution of Xis indexed byHt. That is,Htmay be viewed as a change of time scale from the natural clock timet to a transformed time Ht. Under this interpretation ofHt, we may, de facto, make the claim that the lifelengths of any and all items are always exponentially distributed (with a scale parameter one) on a suitably chosen scale,Ht. The choice of the scaleHt is subjective, and is determined by an assessment of the item’s inherent quality and its operating environment. For example, two different components operating in the same environment may not necessarily have the sameHt; similarly, changing the environment from sayE1toE2, will generally change the cumulative hazard function fromH1ttoH2t (Figure 4.8). In general Ht > twould reflect a harsh environment (e.g. an accelerated test) whereas Ht < t would
The (unknown) hazard potential, X
a
0
Time t T1 (=Time to failure under environment E2) T1
H1(t) H2(t)
Figure 4.8 Effect of changing the environment on the cumulative hazard function.
THE HAZARD POTENTIAL OF ITEMS AND INDIVIDUALS 81 correspond to a benign environment. The change of a time scale interpretation ofHthas also been recognized by Çinlar and Ozekici (1987).
The above material can be summarized via the following two assertions, the second of which may be interpreted as a type of an indifference principle of survival.
Theorem 4.1. Corresponding to every nonnegative random variable T having distribution function F(t), there exists an exponentially distributed random variable X (with a scale parameter one) that is indexed byHt=t
0dFt/F t. Equivalently,
Corollary 4.1. All lifelengths can be regarded as having an exponential distribution (with a scale parameter one) on a suitably chosen time scale.
Regarding the exponentially distributed random variableX, it is evident that, in general, it will not possess the lack of memory property unlessHt=t. A distribution functionFx=1−F x is said to possess the lack of memory property if for allx t≥0 F x+t=F xF t; the only univariate distribution function possessing this property is the exponential distribution indexed by Ht=t. Continuing along this vein, it is also easy to verify that the entropy of the exponentially distributed random variableXis one, only whenHt=t. The entropy of a random variableX having an absolutely continuous distribution functionFxis−
0 fxlogfxdx, wherefx is the probability density generated byFxatx.
We close this discussion by noting that three quantities have been introduced here, a lifelength T that can be observed (i.e. measured), its cumulative hazard function Ht and the hazard potentialX; bothXandHtare not observable. Given any two of these three entities we can obtain the third. Finally, we are able to interpretXas a resource, andHtas the rate at which this resource gets consumed.
4.6.1 Hazard Potentials and Dependent Lifelengths
The likes of Figure 4.8 suggest thatT1 andT2, the lifelengths of two items having a common (unknown) hazard potential X, will be dependent. This is because a knowledge of T1 (orT2) tells us something about X, and this in turn changes our assessment ofT2 (orT1). The notion of dependent lifelengths is further articulated in section 4.7. Similarly, if the hazard potentials X1andX2are dependent, then their lifelengthsT1andT2will be dependent, but only if we can specify bothH1tandH2t, fort≥0, or if only one of these can be specified, then we should be able to say something about the relationship betweenH1tandH2t. Such assertions are best summarized via the following two theorems (and also Theorem 4.4, of section 4.6.2).
Theorem 4.2. LifetimesT1and T2 are independent if and only if their hazard potentials X1
andX2are indepenent, and ifH1t H2t t≥0are assumed known.
Proof. WhenX1andX2are indepenent,
P X1≥H1t1 X2≥H2t2=PX1≥H1t1·PX2≥H2t2 for anyH1t1andH2t2. Consequently,
PT1≥t1 T2≥t2 H1t H2t t≥0
=P X1≥H1t1 X2≥H2t2
=PX1≥H1t1·PX2≥H2t2
=PT1≥t1 H1t t≥0·PT2≥t2 H2t t≥0
Thus knowingH1tandH2t T1andT2are independent. Similarly the reverse.
WhenHit i=12 or both i=1 and 2, fort≥0 cannot be specified, Theorem 4.2 gets weakened in the sense that only the ‘if’ part of the theorem will hold. Specifically,T1andT2will be independent even whenX1andX2are dependent. The intuitive argument proceeds as follows.
ObservingT1 provides no added insight about X1 since H1tis not specified. Consequently, there is no added insight aboutX2either, and thence aboutT2. Similarly, whenT2is observed and H2tnot specified. ThusT1andT2are independent (unless of course one is willing to make other assumptions aboutH1tandH2t t≥0). Mathematically, without knowingHit i=12, we are unable to relatePT1≥t1 T2≥t2 with the distribution of X1 and X2. The above is summarized via
Corollary 4.2.1. LifetimesT1andT2are independent wheneverH1tand/orH2t t≥0are unknown.
The converse of Theorem 4.2 is
Corollary 4.2.2. LifetimesT1 andT2are dependent if and only if their hazard potentialsX1 andX2are depenent, and ifH1tandH2tare specified.
Corollary 4.2.2 puts aside the often expressed view that the lifetimes of items sharing a common environment are necessarily dependent (cf. Marshall, 1975; Lindley and Singpurwalla, 1986).
That is, it is a common environment that causes dependence among lifetimes. Corollary 4.2.2 asserts that it is the commanilities in the (HP)s, that is the cause of interdependent lifetimes.
Dependent (HP)s are a manifestation of similarities in design, manufacture, or a genetic make-up.
In the language of probabilistic causality of Suppes (1970), the common environment can be interpreted as a spurious cause of dependent lifetimes, whereas dependent (or identical) (HP)s as their prima facie (or genuine) cause.
Theorem 4.2 and Corollary 4.2.1 pertain to the two extreme cases wherein theHits i=12 are either known or are unknown. An intermediate case is wherein one of the Hits, say H1t t≥0 is known, and the other unknown, save for the fact thatH1t > H2t. For such scenarios we are able to show that
Theorem 4.3. Suppose that H1t > ≤H2t, and eitherH1torH2t t≥0is specified;
thenX1andX2dependent implies thatT1andT2are also dependent.
Proof. The proof is by contradiction. For this, suppose that X1 and X2 have a bivariate exponential distribution of Marshall and Olkin (1967). Specifically, for1 2and12>0,
P X1≥x X2≥y=exp−1x−2y−12maxx y
=exp−1+12x−2y ifx > y
The marginal distribution ofXi PXi≥x=exp−i+12 x i=12. For theXis to be dependent on (HP)s we need to havei+12=1, fori=12, and12>0; this would imply that1=2=. Thus
P X1≥x X2≥y=exp−x+2y
If we setx=H1t1andy=H2t2, for somet1 t2≥0, thenx > y would imply thatH2t2= H1t2−, for some unknown >0 Consequently
P X1≥x X2≥y=P X1≥H1t1 X2≥H1t2−
=exp−H1t1+2H1t2−
THE HAZARD POTENTIAL OF ITEMS AND INDIVIDUALS 83
Given the above we need to show thatT1andT2are dependent. Suppose not; then P T1≥t1 T2≥t2 H1t1 H2t2 t1 t2≥0
=P T1≥t1 H1t1 t1≥0 P T2≥t2 H2t2 t2≥0
=P X1≥H1t1 P X2≥H2t2
=exp−H1t1exp−H1t2−
=P X1≥H1t1 X2≥H1t2−
since the first term of the above equation does not entail elements of the second. Thus I have P X1≥H1t1 X2≥H1t2−=exp−H1t1+H1t2−
The expressions P X1≥x X2≥yandP X1≥H1t1 X2≥H1t2−will agree with each other if2=1. However,2=1 would imply that12=0, which would contradict the hypothesis thatX1andX2are dependent. The proof whenH1t≤H2twill follow along similar lines.
A consequence of Corollary 4.2.2 is that we are able to generate families of dependent lifelengths using a multivariate distribution with exponential marginals as a seed. The multivariate exponential distribution of Marshall and Olkin (1967) can be one such seed; others could be the one proposed by Singpurwalla and Youngren (1993) and those referred to in Kotz and Singpurwalla (1999). Details are in Singpurwalla (2006a).
4.6.2 The Hazard Gradient and Conditional Hazard Potentials
In this section, we obtain a converse to Theorem 4.3. Specifically, we start with dependent lifelengths and explore the nature of their dependence on the hazard potentials. The hazard gradient plays a role here, and in the sequel, it motivates the introduction of another notion, namely that of the ‘conditional hazard potential’. The main result of this section is that a collection of dependent lifelengths can be replaced by a collection of independent exponentially distributed random variables indexed on suitably chosen scales. The independent random variables are the conditional hazard potentials.
To put matters in perspective, we recall from section 4.5.1 that if Ht= −lnRt, where Rt=PT1≥t1 Tn≥tn, thenruis the hazard gradient of Ht. Furthermore Ht= t
0rudu, and that Hthas an additive decomposition given by (4.16). The first term of this decomposition is the integral of r1u10 0 – the failure rate of T1 at u1. This integral, which is the cumulative hazard rate function ofT1att1will be denoted byH1t1; that is
H1t1= t1 0
r1u10 0du1
Similarly, the second term of the decomposition is denoted by H2t2t1= t2
0
r2t1 u20 0du2
where the integrand r2t1 u20 0 is the conditional failure rate of T2 at u2 given that T1≥t1. Continuing in this vein, the last term of the decomposition is
Hntnt1 tn−1= tn 0
rnt1 tn−1 undun
Thus we have
Ht=H1t1+H2t2t1+ +Hntnt1 tn−1 (4.22) Since the cumulative hazard function has no interpretive content within the calculus of prob- ability,Htand its components will also not have any interpretive content. However, since Rt=exp−Ht, we may write
P T1≥t1 Tn≥tn=e−H1t1+H2t2t1+ +Hntn t1 tn−1
=e−H1t1e−H2t2t1 e−Hntnt1 tn−1 (4.23) Clearly, exp−H1t1=PT1≥t1, since H1t1 is the cumulative hazard function of T1. Similarly, by following arguments analogous to those leading to (4.9), we can see that exp−H2t2t1=PT2≥t2T1≥t1, and in general
PTn≥tnT1≥t1 Tn−1≥tn−1=e−Hntnt1 tn−1 (4.24) As a consequence of the above analogies we have, from (4.23), the relationships
P T1≥t1 Tn≥tn=e−H1t1e−H2t2t1 e−Hntnt1 tn−1
=PT1≥t1 PT2≥t2T1≥t1 PTn≥tnT1≥t1 Tn−1≥tn−1
the latter equality also being true by virtue of the multiplicative decomposition of PT1≥ t1 Tn≥tn.
LetX1 Xn be the hazard potentials corresponding to the lifelengthsT1 Tn, and the cumulative hazard functionsH1t1 Hntn, respectively. Then a consequence of (4.24) is the result that
P Tn≥tnT1≥t1 Tn−1≥tn−1=PXn≥HntnX1≥H1t1 Xn−1≥Hn−1tn−1
=e−Hntnt1 tn−1 (4.25) SinceT1 Tnare not assumed to be independent, the hazard potentialsX1 Xnare, by virtue of Theorem 2, not independent. However, exp−Hntnt1 tn−1is the distribution function of an exponentially distributed random variable, sayX∗n, with a scale parameter one, evaluated atHntnt1 tn−1. Thus, a consequence of (4.25) is the result that for alln≥2
PXn≥HntnX1≥H1t1 Xn−1≥Hn−1tn−1=PX∗n≥Hntnt1 tn−1 The random variable X∗n is called the conditional hazard potential of the n-th item; i.e.
the item whose lifelength is Tn. Its (unit) exponential distribution function is indexed by Hntnt1 tn−1. By contrast,Xn, the hazard potential of then-th item has a (unit) exponential distribution that is indexed byHntn.
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