Regression Models with Incidental Parameters

Abstract

SUMMARY It is known that iterative weighted least squares (IWLS) does not merely yield a convenient way of fitting the general class of regression models considered by Green but is also a useful means of unifying techniques of model checking. However, an extension of these results to models containing nuisance or incidental parameters has made little progress beyond some special cases. Unfortunately, regression models with nuisance parameters that are not dispersion parameters arise very commonly in applications. The present paper presents a unified treatment of regression models containing incidental parameters that are permitted to contain covariates as well as being functionally dependent on the predictor or regression parameters. For the general regression model considered it is shown that Fisher's scoring algorithm can also be implemented as IWLS. Unlike the commonly suggested fitting strategy which updates the regression parameters with the nuisance parameters assumed known and set at the current values, the present approach yields the correct asymptotic covariance matrix for the regression parameter estimates. By considering simultaneous inference, residuals and influence measures which accommodate the effect of estimating the incidental parameters are outlined.