Truncated Life Tests in the Exponential Case

1954/09/01 by Benjamin Epstein · 20 citations
Physics and Astronomy · #Radioactive Decay and Measurement Techniques

paper · pdf · doi:10.1214/aoms/1177728723

Abstract

It is frequently desirable on practical grounds to terminate a life test by a preassigned time T₀. 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, T₀), where X_r0,n is a random variable equal to the time at which the r₀th failure occurs and T₀ is a truncation time, beyond which the experiment will not be run. Both r₀ and T₀ are assigned before experimentation starts. If the experiment is terminated at X_r0,n (that is, if r₀ failures occur before time T₀), then the action in terms of hypothesis testing is the rejection of some specified null-hypothesis. If the experiment is terminated at time T₀ (that is, if the r₀th failure does not occur before time T₀), 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; θ) = θ^-1e^-x/θ, x > 0, θ > 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_θ(r), the expected number of observations to reach a decision; for E_θ(T), the expected waiting time to reach a decision; and for L(θ), the probability of accepting the hypothesis that θ = θ₀, the value associated with the null-hypothesis, when θ 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 T₀. In Section 3 we give exact formulae for E_θ(r) and E_θ(T) for a decision procedure given in [2]. There is a close relation between these results and those in Section 2.

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