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Suppose X = (X 1 , • • • , X p), (p ≥ 2), where X i represents the mean of a random sample of size n i drawn from binomial bin(1, θ i) population. Assume the parameters θ 1 , • • • , θ p are unknown and the populations bin(1, θ 1), • • • , bin(1, θ p) are independent. A subset of random size is selected using Gupta's (Gupta, S. S. (1965). On some multiple decision(selection and ranking) rules. Technometrics 7,225-245) subset selection procedure. In this paper, we estimate of the average worth of the parameters for the selected subset under squared error loss and normalized squared error loss functions. First, we show that neither the unbiased estimator nor the riskunbiased estimator of the average worth (corresponding to the normalized squared error loss function) exist based on a single-stage sample. Second, when additional observations are available from the selected populations, we derive an unbiased and risk-unbiased estimators of the average worth and also prove that the natural estimator of the average worth is positively biased. Finally, the bias and risk of the natural, unbiased and risk-unbiased estimators are computed and compared using Monti Carlo simulation method.

Throughout this section, we define the empirical process P n and P indexed by a function f (V ) asWe also define M n (θ) = P n l n (θ|V ) and M (θ) = P l n (θ|V ), where l n is given in (5). Let K, K 1 , K 2 and K 3 be generic constants. Without loss of generality, we assume in this section that a n = b n (the number of basis functions in constructing H and φ j , respectively).