The bias and skewness of M-estimators in regression

Abstract

We consider M estimation of a regression model with a nuisance parameter and a vector of other parameters. The unknown distribution of the residuals is not assumed to be normal or symmetric. Simple and easily estimated formulas are given for the dominant terms of the bias and skewness of the parameter estimates. For the linear model these are proportional to the skewness of the ‘independent’ variables. For a nonlinear model, its linear component plays the role of these independent variables, and a second term must be added proportional to the covariance of its linear and quadratic components. For the least squares estimate with normal errors this term was derived by Box [1]. We also consider the effect of a large number of parameters, and the case of random independent variables.