Bias reduction and negative regret in sequential point extimation

Abstract

Let X1,X2,... be independent random variables with common distribution Fo for which the mean, m =m(0), is a one-to-one function of the parameter qÎqÌ (-∞,∞). Suppose that X1,X2,...are to be taken one at a time up to stage t. If t is chosen before observing X1,…, Xt, then one may estimate µ by the sample mean Xt, which is an unbiased estimator of µ. However, if t is determined according to a stopping rule, then Xt may be biased for, µ. Letting t be a stopping time of the type proposed by Robbins (1959) and estimating m by a bias-reduction estimator, µt, subject to the loss function L;a = a2(µt -µ)2 +t,a > 0, we show that the asymptotic regret (as a®∞) of the sequential procedure (t, µt) can be negative if the bias-reduction function is chosen properly.