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.^[1]

Definition

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 ${\displaystyle h}$:^[1]

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

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 ${\displaystyle \theta }$).

Examples of non-regular estimators

Both the Hodges' estimator^[1] and the James-Stein estimator^[2] are non-regular estimators when the population parameter ${\displaystyle \theta }$ is exactly 0.

See also

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

References

1. Vaart AW van der. Asymptotic Statistics. Cambridge University Press; 1998.
2. Beran, R. (1995). THE ROLE OF HAJEK'S CONVOLUTION THEOREM IN STATISTICAL THEORY