The Estimation of Parameters From Sequentially Sampled Data on a Discrete Distribution

It is supposed that a constant finite discrete population (which may have arisen through grouping another population) is sampled, with replacement, until some linear function of the frequencies reaches a predetermined value. The generating function of the final frequencies is found exactly when each coefficient in the linear function is either 0 or + 1, and the asymptotic means, variances and covariances in more general cases by using an extension of Wald's identity. The approximate first and second moments of estimators of the parameters determining the population based on these final frequencies, can accordingly be found. The information matrix is considered, first as regards the effect on it of grouping, and then as regards the effect of the stop rule used. Different stop rules provide estimators whose second moments depend on the parameters in a variety of different ways, although the average sample size needed for a given minimum amount of information is almost independent of the stop rule. Some simple examples are given.It is supposed that a constant finite discrete population (which may have arisen through grouping another population) is sampled, with replacement, until some linear function of the frequencies reaches a predetermined value. The generating function of the final frequencies is found exactly when each coefficient in the linear function is either 0 or + 1, and the asymptotic means, variances and covariances in more general cases by using an extension of Wald's identity. The approximate first and second moments of estimators of the parameters determining the population based on these final frequencies, can accordingly be found. The information matrix is considered, first as regards the effect on it of grouping, and then as regards the effect of the stop rule used. Different stop rules provide estimators whose second moments depend on the parameters in a variety of different ways, although the average sample size needed for a given minimum amount of information is almost independent of the stop rule. Some simple examples are given.