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Doctor of Philosophy




Let X be a continuous real-valued random variable with a family of possible distributions indexed by the real parameter (theta)(ELEM)(THETA). For the triplet ((delta),(gamma),(pi)), let (delta) denote a real-valued function over the sample space, (gamma) a real-valued function over the parameter space (THETA), and (pi) a non-negative real-valued function over (THETA). We are interested in determining when for a given (delta) and (gamma) one can find a unique prior (pi), such that, (delta) is the Bayes estimator of (gamma) against (pi), for squared error loss. By imposing different sets of conditions on the form of (gamma) and (delta), positive answers to the question are obtained. The solutions can be considered as generalizations of the works of Diaconis and Ylvisaker (1979), Goldstein (1975), and Berger (1980). The implication of these results in the study of the admissibility of an estimator are noticed and other applications are discussed;A second area that is considered is the relationship between the Bayesness of an estimator and the unbiased property of an estimator. When the loss function is squared a dual relationship between the two is noticed. This suggest a general definition of unbiasedness for an arbitrary loss function which generalizes the notion of unbiasedness in the sense of Lehmann (1951). Some consequences of this general definition are noted;Bibliography;Berger, J. O., 1980. Statistical Decision Theory. Springer-Verlag, New York, N.Y. 425 pp;Diaconis, P., and Ylvisaker, D. 1979. Conjugate priors for exponential family. Ann. Statist. 7:269-281;Goldstein, M., 1975. Uniqueness relations for linear posterior expectations, J. Roy. Statist. Soc., Ser. B, 37:402-405;Lehmann, E. L., 1951. A general concept of unbiasedness. Ann. Math. Statist. 22:587-592.



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Siamak Noorbaloochi



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132 pages