THE ANALYSIS OF EXPONENTIALLY DISTRIBUTED LIFE-TIMES WITH Two TYPES OF FAILURE

SUMMARY A NUMBER of alternative probability models are considered for the interpretation of failure data when there are two or more types of failure. Some of the statistical techniques that can be used for such data are illustrated on an example discussed recently by Mendenhall and Hader. Mendenhall and Hader (1958) have recently given an interesting account of a model for the analysis of failure-time distributions when there are two, or more, types of failure. They illustrate their theory by analysing some data on the failure-times of radio transmitter receivers; the failures were classed into two types, those confirmed on arrival at the maintenance centre and those unconfirmed. In the present paper their example is used to illustrate and distinguish between a number of models that can be used for this type of data. The essential feature of the problem is that we have independent individuals exposed to risk, and that on failure an individual is withdrawn from risk. We observe, for example, that individual number one fails after life-time t, and that the failure is say of the first type: this means that we know the time at which failure of the first type occurs, but only that failure of the second and other types had not occurred by time t. In many applications, including Mendenhall and Hader's, the sample contains individuals that have not failed at the end of the period of the observation. Thus in their example no receivers were operated after 630 hr. Data like this arise in several fields in addition to industrial life-testing. For example, in medical and actuarial work the estimation and comparison of death rates from a particular cause requires corrections for deaths from other causes. In particular Seal (1954) and Elveback (1958) have discussed the more theoretical aspects of this in connection with actuarial work and given numerous references. Sampford (1954) has dealt with similar problems in bioassays. In tensile strength testing there may be two or more types of failure, for example jaw breaks and fractures in the centre of the test specimen; here the observation is load on failure, not life on failure. A further interesting application is in experimental psychology. Audley (1957) has interpreted latency measurements in learning experiments by postulating independent Poisson processes of A-responses and B-responses; the first process for which an event occurs is considered to determine the nature (A or B) of the response and the time at which it occurs.