Sequential Tolerance Regions

Abstract

Consider a measurable space with a linear ordering on the space and the family of all probability measures which assign measure zero to each equivalence class induced by the ordering. For such a space and family of probability distributions sequential tolerance regions are defined. The procedure assigns for each finite sample a Borel set with boundaries determined by the order observations. The sampling terminates when the region remains unchanged for a certain number of observations. The coverage of the region thus sequentially determined is distribution free with respect to that family of distributions. Some relationships are derived between the distribution of the coverage and the generating function of the random sample size, which permit the determination of one in terms of the other. This paper includes as a special case the previous results of Jirina on the distribution of coverage for his sequential procedure. Also, formulae are obtained for the expected sample sizes of the Jirina procedure which were previously unknown. The results of Wilks for fixed sample tolerance limits are obtained as a limiting case and comparisons are made with sequential procedures in terms of coverage and expected sample size. For example it is shown that for one-sided tolerance limits no sequential procedure is as good as Wilks fixed sample procedure in the sense that if the expected sample sizes are the same the coverage of the Wilks procedure is stochastically greater than the coverage of the sequential procedure.