Truncated Life Tests in the Exponential Case

Abstract

It is frequently desirable on practical grounds to terminate a life test by a preassigned time $T_0$. In this paper we consider life tests which are truncated as follows. With $n$ items placed on test, it is decided in advance that the experiment will be terminated at $\min (X_{r0,n}, T_0)$, where $X_{r0,n}$ is a random variable equal to the time at which the $r_0$th failure occurs and $T_0$ is a truncation time, beyond which the experiment will not be run. Both $r_0$ and $T_0$ are assigned before experimentation starts. If the experiment is terminated at $X_{r0,n}$ (that is, if $r_0$ failures occur before time $T_0$), then the action in terms of hypothesis testing is the rejection of some specified null-hypothesis. If the experiment is terminated at time $T_0$ (that is, if the $r_0$th failure does not occur before time $T_0$), then the action in terms of hypothesis testing is the acceptance of some specified null-hypothesis. While truncated procedures can be considered for any life distribution, we limit ourselves here to the case where the underlying life distribution is specified by a p.d.f. of the exponential form, $f(x; \theta) = \theta^{-1}e^{-x/\theta}, x > 0, \theta > 0$. The practical justification for using this kind of distribution as a first approximation to a number of test situations is discussed in a recent paper by Davis [1]. It is a common assumption for electron tube life. Two situations are considered. The first is the nonreplacement case in which a failure occurring during the test is not replaced by a new item. The second is the replacement case where failed items are replaced at once by new items drawn at random from the same p.d.f. as the original $n$ items. Formulae are given for $E_\theta(r)$, the expected number of observations to reach a decision; for $E_\theta(T)$, the expected waiting time to reach a decision; and for $L(\theta)$, the probability of accepting the hypothesis that $\theta = \theta_0$, the value associated with the null-hypothesis, when $\theta$ is the true value. Some procedures are worked out for finding truncated tests meeting specified conditions, and practical illustrations are given. It is an intrinsic feature of all life test decision procedures that they are in some sense truncated, although not necessarily by a fixed time $T_0$. In Section 3 we give exact formulae for $E_\theta(r)$ and $E_\theta(T)$ for a decision procedure given in [2]. There is a close relation between these results and those in Section 2.