EMPIRICAL BAYES ESTIMATION OF THE SCALE PARAMETER IN A PARETO DISTRIBUTION

Let f(x{double up tack}θ) = αθαx-(α+1)I(x>θ) be the pdf of a Pareto distribution with known shape parameter α>0, and unknown scale parameter θ. Let {(Xi, θi)} be a sequence of independent random pairs, where Xi's are independent with pdf f(x{double up tack}αi), and θi are iid according to an unknown distribution G in a class G of distributions whose supports are included in an interval (0, m), where m is a positive finite number. Under some assumption on the class G and squared error loss, at (n + 1)th stage we construct a sequence of empirical Bayes estimators of θn+1 based on the past n independent observations X1,..., Xn and the present observation Xn+1. This empirical Bayes estimator is shown to be asymptotically optimal with rate of convergence O(n-1/2). It is also exhibited that this convergence rate cannot be improved beyond n-1/2 for the priors in class G. © 1990.