Stochastic Models of Failure
4.3 THE PREDICTIVE FAILURE RATE FUNCTION OF A UNIVARIATE PROBABILITY DISTRIBUTION
A concept that plays a key role in reliability and survival analysis is that of the failure rate function, also known as the hazard rate function, or the force of mortality. The origins of the notion of a hazard rate function can be traced to the earliest work in actuarial sciences though much of the recent work on it has been spawned by problems in reliability. In view of the vast literature in applied probability that has been devoted to the nature and the crossing properties of the failure rate function, it is appropriate to make the claim that the notion of the failure rate is perhaps the main contribution of reliability to probability. In this section, I introduce and articulate the idea of the failure rate function and show its relationship to the survival function.
In the sequel, I also make the distinction between the predictive failure rate and the model failure rate. In what follows, I omit the background history argument, so that the Ft of the
PREDICTIVE FAILURE RATE FUNCTION 63 previous section is simplyFt, and similarlyRtisRt; furthermore, following a common convention, I denote the quantityRT=1−FtbyF T.
Suppose thatT is discrete, taking values 0 2 k , for some >0, and suppose that at time , the uncertainty about T is assessed via Fk, fork=012 ,. Then, the predictive failure rateofFk at some future timek, and as assessed at time (the now time) is defined as
rk=Pk < T≤k+1 PT > k
=F k−F k+1
F k (4.1)
Even though the failure rate is a property of the distribution functionFk, it is common to refer to it as the failure rate of T. Since rkcan be defined for all values ofk rk as a function ofk, fork=012 , is known as thepredictive failure rate functionofFk.
It is easy to verify that
1−rk=F k+1
F k (4.2)
thus, if we are able to specifyrk, fork=012 , then because of the fact thatF 0=1 we are able to obtainF k– the survival function – fork=012 , via the relationship
F k=
k−1 i=0
1−ri (4.3)
The above relationship, known as the multiplicative formula of reliability and survival, shows that there exists a one-to-one relationship between the failure rate functionrkand the survival functionF kfork=01 This one-to-one relationship can be advantageously exploited in the manner discussed below.
We first note that rk is a conditional probability (assessed at time ) that the item in question fails at the time pointk+1, were it to be surviving atk; bothkandk+1are future (subsequent to ) time points, andrt=0, for allt=0 2 Since probability is subjective, and since conditional probabilities are generally easier to assess than the unconditional ones,rkas a function ofk encapsulates one’s judgment about the aging or the non-aging features of an item. By ‘aging’ I mean the degradation of an item with use; for example, humans and mechanical devices experience some form of a deterioration and wear over time. The characteristic of non-aging is opposite to that of aging. Non-aging as a physical or a biological phenomenon is rare, and occurs for instance, in the context of work-hardening of materials, or with the development of a newborn’s immune system. With aging one would assess an increasing sequence of conditional failure probabilities over equal intervals of time. That is, rkwill be judged increasing ink. Conversely,rkwill be judged decreasing if the item experiences non-aging (section 4.4.2). If the physical state of an item does not change with age thenrkis assessed as being constant overk. For example, it has been claimed that electronic components neither degrade nor strengthen with age. For such devicesrkis constant ink.
Thus to summarize, the failure rate as a conditional probability can be subjectively specified by a subject matter specialist based on the aging characteristics of an item, and once this is done the survival functionF kcan be induced via (4.3). As an aside, suppose that we are to assume that the item has failed atkand are required to assessrk. Alternatively put, we are asked
‘what is the failure rate of a failed item?’ One is tempted to give zero for an answer, but this is not correct! To see why, we note from (4.2) that assuming failure at k1−rk=0/0 which is undefined; this is to be expected since probabilities are meaningful only for those events whose disposition is not known. Thus it does not make sense to talk about the failure rate of a failed item.
ForT continuous, one is tempted to define the failure rate as the limit of the right-hand side of (4.1) as↓0, andk→t. However, such a temptation does not lead us to a satisfactory limiting operation. Instead forT continuous, the instantaneous predictive failure rateatt(when the following limit exists) is defined as
rt=lim
dt↓0
Pt < T≤t+dtT > t
dt (4.4)
andrtas a function oftis called the predictive failure rate function of the distribution function Ft=PT≤t, or equivalently ofT. Informally,rtis a measure of the risk that a failure will occur att.
Note that for small values of dt rtdtcan be interpreted as, approximately, the probability that an item of agetwill fail int t+dt. Thus, likerk, the instantaneous failure rate is a conditional probability, the probability assessment made at time. It is important to note that the above interpretation ofrtdtis not appropriate ifPT≤tis singular, or is otherwise not absolutely continuous. SinceFt=1−F t, We may rewrite (4.4) as
rt=lim
dt↓0
Ft+dt−Ft
F tdt (4.5)
provided thatF t >0.
Clearly, the above limit will exist ifFtis absolutely continuous, in which case rtF t= d
dtFt= −d
dtF t (4.6)
If dFt/dt=ft, the probability density generated by Ft at t, then (4.6) leads us to the relationship
rt= ft
F t (4.7)
which with F t >0 is often taken as the definition of the predictive failure rate and is the starting point of a mathematical theory of reliability (Barlow and Proschan (1965)).
Since the limit in (4.5) need not always exist for allt≥0, the predictive failure rate at tis sometimes also defined as
rt=dFt
F t (4.8)
where dFtisftdtwhereverFtis differentiable, and is otherwiseFt+−Ft−, the jump inFtatt. WhenFtis the distribution function of Figure 4.1(c), its failure rate att=1 does not exist.
Equation (4.6), which comes into play when Ftis absolutely continuous, is a differential equation withF 0=1 as the initial condition; indeedF 0=1 is a defining feature of failure time distributions. The solution of this differential equation is the famous exponentiation formula
PREDICTIVE FAILURE RATE FUNCTION 65 of reliability and survival (not to be confused with the product integral formula which will be described soon). Specifically, fort≥0
F t=exp− t 0
rudu (4.9)
The quantityHt=t
0ruduis important. As a function oft, it is known as the cumulative hazard functionofFt(or equivalently ofT). SinceHt= −logF t– a classic relationship in reliability and survival analysis – it is easy to see thatftthe probability density generated byFtis given as
ft=exp− t 0
rudu rt fort≥0 (4.10) Finally, using the fact that sinceHt=Ht1+Ht−Ht1, for anyt1, 0≤t1≤t, we have
F t=e−Ht1e−Ht−Ht1 from which the well-known fact that
F t=PT≥t=PT≥t1 PT≥tT≥t1 follows.
The one-to-one relationship between the failure rate function and the survival function of (4.9) parallels that given by (4.3) forT discrete. To see this parallel, we may rewrite (4.3) as
F k=exp
−
k−1
i=0
ln 1
1−ri 4.3.1 The Case of Discontinuity
WhenFtis not absolutely continuous, it does not have a probability density at all values of t, and thus it is (4.8) that comes into play. We may use this equation to define the cumulative hazard function as
Ht= t 0
dFs/F s
Note thatHtwill be non-decreasing and right-continuous on0; it will be interpreted as dHs=PT∈s s+dsT≥s=dFs/F s
Thus
dFs=dHsF s
so that for any 0≤a≤b <
b a
dFs= b a
dHsF s or thatF a b=b
a dHsF s.
The solution to the above differential equation is the famous product integral formula;
namely, fort≥0
F t=
0t
1−dHs
see (4.3) for a parallel. It can be shown (Hjort (1990), p. 1268) that the right-hand side of the above equation equals exp−ha bif and only ifHis continuous. This means that the classic relationshipHt= −logF tis true only whenF is absolutely continuous.
4.4 INTERPRETATION AND USES OF THE FAILURE RATE