Estimation and testing in an imperfect-inspection model

An imperfect inspection model in which failures can only be detected with probability β is considered. The lifetimes of all the components are IID exponential with rate λ. Using the approximate likelihood filtering method, a first order binomial autoregressive model is obtained when 1-β is small. When λ is known, a locally most powerful test is proposed for testing the hypothesis that β=1 against the alternative that β<1. The asymptotic distribution of the test statistic is obtained under the null hypothesis as well as under the alternative. The test statistic is modified to handle the case of unknown λ. Its asymptotic distribution is also given. The bias of the maximum likelihood estimators of β and λ obtained from the approximated likelihood function is studied by simulation