Local estimators in multivariate generalized linear models with varying coefficients
In varying-coefficients models there are two types of regressors, the usual covariate \(x\) and the effect modifier \(u\). Let the data be given by \((y_i,x_i,u_i),\;i=1,\dots,n\), with the \(q\)-dimensional response \(y_i\) observed at covariates \((x_i,u_i)\). Let the varying-coefficient model be of the form \[ \mu_i=E(y_i |x_i,u_i)=h(Z_i\beta_i),\;\beta_i=\beta(u_i),\;Z_i=Z(x_i). \] For the estimation of \(\beta(u)\) the local likelihood function \[ l_{\gamma}^{(u)}(\beta(u))=\sum_{i=1}^n \omega_{\gamma}(u,u_i)[y'_i\theta_i(\beta(u))-k(\theta_i(\beta(u)))]/ \phi, \] is considered where \(\theta_i(\beta(u))\) is the natural parameter corresponding to the expectation \(\mu_i=h(Z_i\beta(u))\). Analogous to local likelihood estimates the locally weighted least squares estimate of \(\beta_i\) is based on the criterion \[ \sum_{s=1}^m \omega_{\gamma}(u_i,u_s) \sum_{j=1}^{m_s} [g(y_{js}-Z_{sj}\beta_i]' C_{sj}(\mu_{sj})^{-1}[g(y_{js}-Z_{sj}\beta_i]. \] Maximization of this expression yields the locally weighted least squares estimate. Extensions to the case of multivariate smoothing are considered. An example illustrates the usefulness of this approach.
scientific article; zbMATH DE number 1409727