Large-sample theory of sequential estimation

Abstract

In a previous large-sample treatment of sequential estimation (1), it was shown that in certain circumstances, when there was only one unknown parameter in the distribution of the observations, an estimation formula valid for fixed sample sizes remained valid when the sample size was determined by a sequential stopping rule. The proof was heuristic, in that it depended on an application of the central limit theorem of which the justification was not obvious. Another proof has recently been given by Cox (2) (in the course of deriving a correction term to my result). Dr Cox pointed out to me that this work suggested that fixed-sample-size formulae might be valid generally for sequential sampling, provided the sample size was large. In establishing a proposition to that effect, I have now been able to by-pass some of the complexity of my previous approach by concentrating attention on a condition of ‘uniform continuity in probability’ to be satisfied by the statistic used. Theorem 1 is the basic result, which is applied in Theorem 2 to determine a sequential stopping rule giving required accuracy of estimation of an unknown parameter. Theorems 3–6 indicate some situations in which the uniform continuity condition postulated in Theorems 1 and 2 is satisfied. A few examples are discussed.