Sequential Optimum Procedures for Unbiased Estimation of a Binomial Parameter
1964; Taylor & Francis; Volume: 6; Issue: 3 Linguagem: Inglês
10.1080/00401706.1964.10490183
ISSN1537-2723
Autores Tópico(s)Stochastic processes and statistical mechanics
ResumoLet x 1, x 2, … x n, … be a sequence of independent random variables with a common density function P(x = 1) = p, P(x = 0) = 1 – p, 0 < p < 1. This paper considers the non-randomized sequential procedures δ's for estimating p and the following three problems on choice of δ. (i) Choose δ to minimize E v (Z δ – P)2 subject to E p N δ ≤ m where m ≥ 1 and Z δ is an unbiased estimate of p; (ii) Choose δ to minimize Ep N δ subject to E p (Z δ – p)2 ≤ α where α is a real positive number; (iii) choose δ to minimize C Ep N δ + E p (Z δ – p)2 where C is the cost of an observation. In each case the minimization is to be done uniformly in p if possible; otherwise the supremum over p of the risk in question is to be minimized. A procedure is constructed for problem (i) when m is not an integer. A fixed sample size procedure is shown to be admissible and minimax for problem (ii). A procedure is constructed which is asymptotically uniformly better than the fixed sample size for problem (ii). Furthermore, for problem (iii) some optimum procedures are constructed.
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