Confidence Interval Estimation in the Inverse Power Law Model

Abstract

SUMMARY Estimation and prediction procedures are discussed for the inverse power law model, when the time to failure follows an exponential distribution. In the context of life test experiments, procedures are given for estimating the parameters in the power law model, and for estimating mean life at a given future stress level. The procedures given are conditional confidence interval procedures, obtained by conditioning on ancillary statistics. A comparison is made of these procedures and procedures based on asymptotic properties of the maximum likelihood estimates. IN studies concerning the length of life of certain types of manufactured items, it is often wished to consider the relationship between length of life and one or more concomitant variables. Thus, for example, in an experiment to study the lifetimes of a certain type of electrical insulation, the relationship between length of life and environmental temperature was studied (Nelson, 1970). Sometimes the main problem of interest involves determining the effect of environmental factors (concomitant variables) on the life distribution of the items in question, and in incorporating this into a useful statistical model. In other situations, the general form of the model may be considered determined, and it may be wished to estimate various parameters in the model. The estimated relationship between length of life and the concomitant variables allows the prediction of item life under specified environmental condi- tions. This latter situation commonly arises in life testing where, on the basis of tests run at accelerated test conditions, it is desired to predict item life under standard operating conditions. This paper deals with the second type of problem: we discuss estimation and prediction procedures for a model which is commonly used in reliability and life testing work, the so-called inverse power law model, with exponential time to fail data. This model has been discussed a number of times in the statistical literature, and a number of estimation procedures have been proposed for it, mainly based on large sample theory. Our purpose here is to describe confi- dence interval estimation procedures for this model, and to illustrate their use. The procedures do not involve the use of any asymptotic approximations and so all distributions given are exact for any sample size. Before describing the model, we remark that an excellent survey of work on the inverse power law model and its application in life testing is given by Nelson (1970), who discusses the more general situation in which the time to failure follows the two-parameter Weibull distribution. We consider the inverse power law in the following form: let the lifetime of an item under environmental condition i have an exponential distribution with mean Oi. In this model the environmental conditions are specified by means of a single covariate vi (which we will call the stress), and the relationship Oi = c/vg? is assumed, where c,p are (unknown) constants.