We consider a reliability system, with an unknown number of faults, which is observed for a fixed length of time. Failures are assumed to follow an exponential distribution with an unknown mean, and we concentrate on estimation of the number of faults v. The likelihood function, considered as a function of v, has its maximum at one of the boundary points (the observed number of faults or infinity) with substantial probability. Likelihood interval estimates and improved point estimates are discussed. The harmonic mean of the endpoints of the likelihood interval turns out to be a reasonable point estimate. We comment on some extensions of this problem suggested in the software-reliability literature.
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Joe et al. (1985) studied this question.
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