Local likelihood estimation in varying-coefficient models including additive bias correction

Varying coefficient models result from generalized linear models by allowing the parameter of the linear predictor to vary across some additional explanatory quantity called effect modifier. While Hastie & Tibshirani (1993) have used spline smoothing techniques in varying-coefficient models with univariate response here the local likelihood approach is considered within the framework of multivariate generalized models. The local likelihood approach has several advantages. It allows the derivation of asymptotic properties under weak assumptions, consistency and asymptotic normality of the estimates are shown under rather general conditions. The estimation procedure may be performed with standard software. This holds even for the additive bias reduction method which is proposed and investigated. The results are given for discrete as well as for continuous effect modifiers and asymptotically optimal rates of smoothing are derived. An alternative normalization of weights is proposed which corresponds to the augmentation of the information supplied by the observation. The normalization results from theoretical considerations and is supported by simulations which show improved variance properties for finite sample size. A real data example demonstrates the applicability of the results.