Large Deviations of Estimators

The performance of a sequence of estimators $\{T_n\}$ of $g(\theta)$ can be measured by its inaccuracy rate $-\lim \inf_{n\rightarrow\infty} n^{-1} \log \mathbb{P}_\theta(\|T_n - g(\theta)\| > \varepsilon)$. For fixed $\varepsilon > 0$ optimality of consistent estimators $\operatorname{wrt}$ the inaccuracy rate is investigated. It is shown that for exponential families in standard representation with a convex parameter space the maximum likelihood estimator is optimal. If the parameter space is not convex, which occurs for instance in curved exponential families, in general no optimal estimator exists. For the location problem the inaccuracy rate of $M$-estimators is established. If the underlying density is sufficiently smooth an optimal $M$-estimator is obtained within the class of translation equivariant estimators. Tail-behaviour of location estimators is studied. A connection is made between gross error and inaccuracy rate optimality.