Abstract
An approach to construct M-estimators is discussed. It is known, in various cases, that influence functions of maximum likelihood estimators are given in terms of certain orthogonal polynomials. In such cases, since such polynomials tend to infinity as x→±∞, the estimators are by no means robust against occurrence of gross errors. The method adopted in this paper is based on robustification of the influence functions which are given as functions of orthogonal polynomials corresponding to the density function. The present method has an advantage that it can automatically provide Fisher-consistent M-estimators to various problems as long as suitable orthogonal polynomials are available.
Applications of the present method to various estimation problems are also shown. Robust estimation of location and scale parameters of normal distributions is discussed in some depth. Other applications include estimation in life-time distributions (Gamma and Weibull distributions), estimation of correlation coefficient in bivariate normal distributions, and centering and sphering in projection pursuit.
