4.5 Large-Sample Theory for Estimators

Anscombe (1949) provided a large-sample theory for sequential estimators when there is only one unknown parameter. He showed, using a heuristic argument, that an estimation formula valid for fixed-sample size remained valid when the sample size was determined by a sequential stopping rule. An alternative proof was given by Cox (1952a) which suggests that fixed-sample size formulas might be valid generally, for sequential sampling, provided the sample size is large. Anscombe (1952) simplified his previous work by introducing the concept "uniform continuity in probability" of the statistic employed. Towards this assume that there exists a real number ?, a sequence of positive numbers {w _n }, and a distribution function G(x), such that the following conditions are satisfied:

(C1) Convergence of {Y _n }: For any x such that G(x) is continuous (a continuity point of G(x)),

(C2) Uniform continuity in probability of {Y _n }: Given any small ? and ? there exists a large ? and a small positive c such that, for any n> ?,

(4.5.1)

Note that, as n ? ?, Y _n ? ? in probability if w _n ?0.

In most applications, G(x) is continuous, and usually is the normal distribution function, w _n is a linear measure of dispersion of Y _n , and for example, is the standard deviation or the quartile range. The term "uniform continuity"...