Suppose that a statistician is faced with a decision problem involving an unknown parameter. Before making his decision he can observe, possibly at random, one of k possible experiments. For this problem, a decision procedure for the statistician is a pair ((gamma),(delta)) where (gamma) = ((gamma)(,1),...,(gamma)(,k)), (delta) = ((delta)(,1),...,(delta)(,k)); (gamma)(,i) is the probability of observing experiment i and (delta)(,i) is the decision function to be used in connection with experiment i. When (gamma) (ELEM) (GAMMA), the class of all probability distributions on 1,...,k, a characterization of the class of admissible pairs relative to (GAMMA) for this problem was given by Meeden and Ghosh (1983). This thesis deals with this problem in the case when, for some reason like cost or time limitations, the statistician is restricted in choosing among those experiments. That is, a characterization of the class of admissible pairs ((gamma),(delta)) relative to an arbitrary subclass of (GAMMA) is given;This characterization is then used to give some uniform admissibility results for some problems in infinite population sampling. Other applications in nonparametric problems are also discussed;Following Meeden and Ghosh (1981), an admissible estimator of a population U-statistic is constructed. This estimator turns out to be a constant times the corresponding sample U-statistic where the constant is some positive number less than one;References;Meeden, G. and Ghosh, M. 1981. Admissibility in finite problems, Ann. Statist. 9:846-852. Meeden, G. and Ghosh, M. 1983. Choosing between experiments: Applications to finite population sampling. Ann. Statist. 11:296-305.