Multistage and sequential minimum risk point estimation procedures for the means of U-statistics

Date of Completion

January 1989






One of the fundamental problems in statistical estimation theory is the minimum risk point estimation of a parameter when the loss function is squared error plus cost of sampling. For this kind of situation an optimal fixed sample size solution may not be implementable because it may depend on the underlying population distribution which may either have an unknown form or involve some unknown parameters. Therefore, some appropriate sequential or multi-stage procedures are needed to approximate the otherwise fictitious fixed sample optimal solution.^ The main goal of this project is to estimate certain estimable parameter $\theta$, which is a functional of some unspecified distribution function (d.f.) F by means of a generalized U-statistic (Hoeffding (1948), Lehmann (1951)) approximately minimizing the associated risk. To attain this objective, two-stage, three stage and sequential procedures are developed for the case where F is the joint distribution of r($\ge$1) independent random variables each having (unknown) d.f. F$\sb{i}$, i = 1, ..., r. These sampling techniques and the estimators are shown to be asymptotically risk efficient. ^