Bayes and empirical bayes two-stage procedures for sampling inspection

A batch of M items is inspected for defectives. Suppose there are d defective items in the batch. Let d₀ be a given standard used to evaluate the quality of the population where 0 < d₀ < M. The problem of testing H₀: d < d₀ versus H₁d ≥ d₀ is considered. It is assumed that past observations are available when the current testing problem is considered. Accordingly, the empirical Bayes approach is employed. By using information obtained from the past data, an empirical Bayes two-stage testing procedure is developed. The associated asymptotic optimality is investigated. It is proved that the rate of convergence of the empirical Bayes two-stage testing procedure is of order O (exp(-c^* n)), for some constant c^* > 0, where n is the number of past observations at hand.