Local rates of convergence for consistent estimators of location of translation families

We consider the estimation of a location parameter θ in a one‐sample problem. A measure of the asymptotic performance of an estimator sequence { T n } = T is given by the exponential rate of convergence to zero of the tail probability,\documentclass{article}\pagestyle{empty}\begin{document}$$ \beta \left({T,\theta,\varepsilon } \right) = \lim \limits_{n \to \infty } \left({ - \frac{1} {n}\log P\left({\left| {T_n - \theta } \right|} \right) \ge \varepsilon } \right) $$\end{document}which for consistent estimator sequences is bounded by a constant, B (θ, ϵ), called the Bahadur bound. We consider two consistent estimators: the maximum‐likelihood estimator (mle) and a consistent estimator based on a likelihood‐ratio statistic, which we call the probability‐ratio estimator (pre). In order to compare the local behaviour of these estimators, we obtain Taylor series expansions in ϵ for B (θ, ϵ) and the exponential rates of the mle and pre. Finally, some numerical work is presented in which we consider a variety of underlying distributions.