(especially that involving ordered failure times) can be produced, and the occasional "conservatism" of the exponential assumption, have also contributed to its popularity, in spite of its notorious nonrobustness. It is important to acknowledge that the exponential assumption is very special and highly restrictive, so that its use should be discouraged except in circumstances in which there is good physical, empirical and practical support for the model. In due course, we will review the basics of exponential life testing, both to make the present paper self-contained and to set the stage for the various comparisons we wish to make with alternative analyses. First, however, we will describe the type of problem—a sort of statistical hybrid—on which the present investigation is focused.
Suppose a statistician is faced with an application in which two hypotheses concerning the mean life µ of a new system are to be tested. He wishes to resolve the test of H0: µ = µ0 against the alternative H1 : µ = µ1, where µ1 < µ0 are fixed and known, with certain predetermined probabilities α and β for type I and type II errors (also often called the producer's and consumer's risks). Having no pressing reason to doubt exponentiality in the application at hand, the statistician determines (using the Department of Defense's Handbook H108, for example; U.S. Department of Defense, 1960) that these goals can be accomplished with an experimental design calling for some specific number of observed failures (say r), rejecting H0 in favor of H1 if the total time on test Tat the time of the rth failure is less than the threshold T0. Among the advantages afforded by an exponential life test plan is the fact the the resources required to perform the test (that is, the number of systems that must be placed on test and the maximum amount of testing time needed to resolve the rest) may be calculated in advance. The fact that the duration of the test, in real time, can be controlled and made suitably small by placing n > r systems on test while still resolving the test upon the rth failure is also an important advantage.
Consider, now, the analysis stage of this life testing experiment. Suppose that when the data have been collected, their characteristics suggest that they are definitely not exponential. It then falls upon the statistician to analyze the available data under some alternative model or, perhaps, nonparametrically. Let us suppose, as will be tacitly assumed in the sequel, that the two-parameter Weibull distribution is taken as an appropriate underlying model for the