Normalized estimating equation for robust parameter estimation

Abstract

Robust parameter estimation has been discussed as a method for reducing a bias caused by outliers. An estimating equation using a weighted score function is often used. A typical estimating equation is non-normalized, but this paper considers a normalized estimating equation, which is corrected to ensure that the mean of the weight is one. In robust parameter estimation, it is important to control the difference between the target parameter and the limit of the robust estimator, which is referred to as the latent bias in this paper. The latent bias is usually discussed in terms of influence function and breakdown point. It is illustrated by some examples that the latent bias can be close to zero for the normalized estimating equation even if the proportion of outliers is not small, but not close to zero for the non-normalized estimating equation. Furthermore, this behavior of the normalized estimating equation can be proved under mild conditions. The asymptotic normality of the robust estimator is also presented and then it is shown that the outliers are naturally ignored with an appropriate proportion of outliers from the viewpoint of asymptotic variance. The results can be extended to the regression case. The behaviors of the latent bias and mean squared error are investigated by numerical studies.