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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[edit]

An estimator ${\hat {\theta }}_{n}$ of $\psi (\theta )$ based on a sample of size $n$ is said to be regular if for every $h$ :^[1]

{\sqrt {n}}\left(\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 $\theta +h/{\sqrt {n}}$ .

Examples of non-regular estimators[edit]

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

References[edit]

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

[vdv-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

[1]

[2]

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Definition[edit]

Examples of non-regular estimators[edit]

See also[edit]

References[edit]

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