Adaptive Maximum Likelihood Estimators of a Location Parameter

Consider estimators ₙ of the location parameter based on a sample of size n from + X, where the random variable X has an unknown distribution F which is symmetric about the origin but otherwise arbitrary. Let F denote the Fisher information on contained in + X. We show that there is a nonrandomized translation and scale invariant adaptive maximum likelihood estimator ₙ of which doe not depend on F such that L (n^1{2} (ₙ -) ) N (0, 1/J) as n for all symmetric F.