Lower boundaries for parametric estimations in different norms

TL;DR

This paper derives new non-asymptotic lower bounds for the deviation of regular unbiased estimators in various norms, extending classical inequalities and analyzing convergence rates of estimators under different norm conditions.

Contribution

It introduces novel lower bounds for estimator deviations in different norms and characterizes convergence rates of estimators based on the norm strength.

Findings

Lower bounds similar to Rao-Kramer's inequality are established.

Convergence rate of $1/\sqrt{n}$ is shown for weaker norms.

MLE convergence rate remains $1/\sqrt{n}$ under stronger norms.

Abstract

We establish some new non-asymptotical lower bounds for deviation of regular unbiased estimation of unknown parameter from its true value in different norms, alike the classical Rao-Kramer's inequality. We show that if the new norm is weaker that ordinary Hilbertian norm, that the rate of convergence of arbitrary regular unbiased estimate does not exceed $ 1/\sqrt{n}, $ and if the new norm is stronger that one, the rate of convergence of the well-known Maximal Likelihood Estimate (MLE) is also equal to $ 1/\sqrt{n}.