Sequential Life Tests in the Exponential Case

Abstract

This paper describes sequential life test procedures, considering, as in a recent paper [4] devoted to nonsequential methods, the special case in which the underlying distribution of the length of life is given by the exponential density \begin{equation*}\tag{1} f(x, \theta) = e^{-x/\theta}/\theta,\quad x > 0.\end{equation*} The unknown parameter $\theta > 0$ can be thought of physically as the mean life. Our primary aim is to test the simple hypothesis $H_0: \theta = \theta_0$ against the simple alternative $H_1: \theta = \theta_1,$ where $\theta_1 < \theta_0,$ with type I and II errors equal to preassigned values $\alpha$ and $\beta,$ respectively. The test is carried out by drawing $n$ items at random from the population and placing them all on a life test. We consider both the replacement case, in which failed items are immediately replaced by new items, and the nonreplacement case. The test can be terminated either at failure times with rejection of $H_0$, or at any time between failures with acceptance of $H_0$. Since abnormally long intervals between failures furnish "information" in favor of $H_0$ and abnormally short intervals furnish "information" in favor of $H_1$, these features are not only reasonable but actually desirable. Similar problems involving a continuous time parameter have recently appeared [3], [5]. In this paper we obtain likelihood ratio tests and give approximate formulae for the O.C. (operating characteristic) curve, for the expected number of failures $E_\theta(r)$, and for the expected waiting time $E_\theta(t)$ before a decision is reached. In the replacement case where the number of items on test throughout the experiment is the same, namely $n$, it is shown that $E_\theta(t) = (\theta/n)E_\theta(r)$. A table giving approximate values of $E_\theta(r)$ for certain choices of $\theta_0/\theta_1, \alpha,$ and $\beta$ is given for the replacement case. Some calculations of exact $L(\theta)$ and $E_\theta(r)$ values using formulae in [1] and [3] are reported. Several numerical examples are worked out.