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.

Definition

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

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

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

Examples of non-regular estimators

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

See also

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