Asymptotic Normality of the Stopping Times of Some Sequential Procedures

Abstract

Two problems of sequential estimation, viz. the estimation of the mean of a normal distribution with unknown variance and the estimation of a binomial proportion are studied as the cost per observation tends to 0. For the first problem the asymptotic distribution of the stopping time of a procedure due to Robbins (1959) is shown to be normal. For the second problem the stopping time of a modification of Wald's (1951) procedure is asymptotically normal when the parameter is different from $\frac{1}{2}$. When the parameter is $\frac{1}{2}$, this stopping time does not enjoy asymptotic normality. The method employed is to first prove the convergence in probability of the stopping time which is then converted to convergence in distribution by using a theorem of Wittenberg (1964). This method also yields a new proof of a theorem of Siegmund (1968).