On the bounded regret of empirical bayes estimators |
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Authors: | Kai F Yu |
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Institution: | Department of Statistics , University of South Carolina , Columbia, 29208, SC |
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Abstract: | Let (θ1,x1),…,(θn,xn) be independent and identically distributed random vectors with E(xθ) = θ and Var(x|θ) = a + bθ + cθ2. Let ti be the linear Bayes estimator of θi and θ~i be the linear empirical Bayes estimator of θi as proposed in Robbins (1983). When Ex and Var x are unknown to the statistician. The regret of using θ~i instead of ti because of ignorance of the mean and the variance is ri = E(θi ? θi)2 ?E(ti ?θi)2. Under appropriate conditions cumulative regret Rn = r1+…rn is shown to have a finite limit even when n tends to infinity. The limit can be explicitly computed in terms of a,b,c and the first four moments of x. |
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Keywords: | Empirical Bayes bounded regret uniform integrability |
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