Regular estimator

Regular estimators are a class of statistical estimators that satisfy certain regularity conditions which make them amenable to asymptotic analysis. The convergence of a regular estimator's distribution is, in a sense, locally uniform. This is often considered desirable and leads to the convenient property that a small change in the parameter does not dramatically change the distribution of the estimator.

An estimator ${\displaystyle {\hat {\theta }}_{n}}$of ${\displaystyle \psi (\theta )}$based on a sample of size ${\displaystyle n}$is said to be regular if for every ${\displaystyle h}$:

$${\displaystyle {\sqrt {n}}\left({\hat {\theta }}_{n}-\psi (\theta +h/{\sqrt {n}})\right){\stackrel {\theta +h/{\sqrt {n}}}{\rightarrow }}L_{\theta }}$$

where the convergence is in distribution under the law of ${\displaystyle \theta +h/{\sqrt {n}}}$. ${\displaystyle L_{\theta }}$is some asymptotic distribution (usually this is a normal distribution with mean zero and variance which may depend on ${\displaystyle \theta }$).

Both the Hodges' estimator and the James-Stein estimator are non-regular estimators when the population parameter ${\displaystyle \theta }$is exactly 0.

• Estimator
• Cramér-Rao bound
• Hodges' estimator
• James-Stein estimator