Sequential Life Tests in the Exponential Case

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 {equation*}{1} f(x, θ) = e-x/θ/θ, x > 0.{equation*} The unknown parameter θ > 0 can be thought of physically as the mean life. Our primary aim is to test the simple hypothesis H₀: θ = θ₀ against the simple alternative H₁: θ = θ₁, where θ₁ < θ₀, with type I and II errors equal to preassigned values α and β, 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 H₀, or at any time between failures with acceptance of H₀. Since abnormally long intervals between failures furnish "information" in favor of H₀ and abnormally short intervals furnish "information" in favor of H₁, 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_θ(r), and for the expected waiting time E_θ(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_θ(t) = (θ/n)E_θ(r). A table giving approximate values of E_θ(r) for certain choices of θ₀/θ₁, α, and β is given for the replacement case. Some calculations of exact L(θ) and E_θ(r) values using formulae in [1] and [3] are reported. Several numerical examples are worked out.