Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series A, Pt. 1 , pp. 96--105

A COMPARISON OF BIASED AND BIAS-CORRECTED ESTIMATORS

By

THOMAS A. SEVERINI, *Northwestern University*

SUMMARY. Let $\hat{\theta}$
denote an estimator of a scalar parameter ${\theta}$ based on a sample of size n. Suppose that the expected value of $\hat{\theta}$ maybe written q
+a(q
)n^{-1}+O(n^{-2}) , where a is a known function. In this case a bias-corrected estimator may be constructed, given by $\tilde{\theta}={\hat{\theta}}-a(\hat{\theta})n^{-1}$. In this paper $\hat{\theta}$ and $tilde{\theta}$ are compared by considering the large-sample properties P{$|\tilde{\theta}-\theta|<|\hat{\theta}-\theta|$}. It is shown that if the bias of $\hat{\theta}$ and the skewness of $\hat{\theta}$ are of opposite signs then $\tilde{\theta}$ is preferable to $\hat{\theta}$; if the bias and skewness of $\hat{\theta}$ have the same sign then $\tilde{\theta}$ is preferable only if the bias is large relative to the skewness.

*AMS (1980) subject classification.* 62F10, 62F12.

*Key words and phrases.* Asymptotic theory, maximum likelihood estimation, Pitman nearness, unbiased estimation.