Bias reduction and negative regret in sequential point estimation

Let X-1,X-2,... be independent random variables with common distribution Fg for which the mean, mu=mu(theta). is a one-to-one function of the paramete r theta is an element of Theta subset of(- infinity,infinity). Suppose that XI,X2, are to be taken one at a time up to stage t. If t is chosen before observing X-1,...,X-t, then one may estimate mu by the sample mean (X) over bar (t), which is an unbiased estimator of CL. However, if t is determined according to a stopping rule, then (X) over bar (t), may be biased for, mu . Letting t be a stopping time of the type proposed by Robbins (1959) and e stimating mu by a bias-reduction estimator, <(<mu>)over cap>(t), subject to the loss function L-a = a(2)(<(<mu>)over cap>(t) - mu)(2) + t, a > 0, we s how that the asymptotic regret (as a-->infinity) of the sequential procedur e (t, <(<mu>)over cap>(t)) can be negative if the bias-reduction function i s chosen properly.