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 [math]\displaystyle{ \hat{\theta}_n }[/math] of [math]\displaystyle{ \psi(\theta) }[/math] based on a sample of size [math]\displaystyle{ n }[/math] is said to be regular if for every [math]\displaystyle{ h }[/math]:[1]
[math]\displaystyle{ \sqrt n \left ( \theta_n - \psi (\theta + h/\sqrt n) \right ) \stackrel{\theta+h/\sqrt n} {\rightarrow} L_\theta }[/math]
where the convergence is in distribution under the law of [math]\displaystyle{ \theta + h/\sqrt n }[/math].
Examples of non-regular estimators
Both the Hodges' estimator[1] and the James-Stein estimator[2] are non-regular estimators when the population parameter [math]\displaystyle{ \theta }[/math] is exactly 0.
See also
- Estimator
- Cramér-Rao bound
- Hodges' estimator
- James-Stein estimator
References
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