Unbiased estimation when sampling from renewal processes: The single sample and k-sample random means cases

Suppose random variables are sampled sequentially until the sum of associated non-negative random variables is greater than or equal to a predetermined constant. When no distributional assumptions are made, we derive uniform minimum variance unbiased estimators for functions which are U-estimable in the fixed sample size case. We also consider estimation for some special cases when distributional assumptions are made. When estimating the mean and variance of random means, it is often assumed that sample sizes are independent of populations selected for sampling, and hence independent of the random means associated with the populations sampled. When sampling from renewal processes sample size is dependent on the distribution of the population selected for sampling. Thus, in addition to the single sample case we consider the k-sample problem and the estimation of the expectation and variance of random means.