Asymptotic bounds for estimators without limit distribution

Abstract

Let$\mathfrak{P}$ be a general family of probability measures,κ :$\mathfrak{P} \to \mathbb{R}$ a functional, and$N_{(0,\sigma ^2 (P))}$ the optimal limit distribution for regular estimator sequences of κ. On intervals symmetric about 0, the concentration of this optimal limit distribution can be surpassed by the asymptotic concentration of an arbitrary estimator sequence only forP in a “small” subset of$\mathfrak{P}$. For asymptotically median unbiased estimator sequences the same is true for arbitrary intervals containing 0. The emphasis of the paper is on “pointwise” conditions for$P \in \mathfrak{P}$, as opposed to conditions on shrinking neighbourhoods, and on “general” rather than parametric families.