Computerassisted semiparametric generalized linear models
A model is considered which is referred to as a generalized partially linear model with the following structure: \(E(Y |x,t)=G\{x^T\beta+m(t)\}\), where \(\beta\in R^p\) is a finite-dimensional parameter, \(m(t)\) is a smooth function and the explanatory variable \(z\) is decomposed into two vectors \(x\) and \(t\) which denote realizations of a \(p\)-variate random vector \(X\) with discrete covariates and realizations of a \(q\)-variate random vector \(T\) of continuous covariates. From \(n\) i.i.d. observations \((x_i,t_i,y_i)\) the usual likelihood \(L(\beta)=\sum_{i=1}^n Q(\beta^Tx_i +m_{\beta}(t_i);y_i)\) is used to obtain \(\hat{\beta}\) and a smoothed likelihood \(LS(\beta)=\sum_{i=1}^n K_h(t-t_i) Q(\beta^Tx_i +\eta;y_i)\) to obtain the nonparametric smooth function \(\hat{m}_{\beta}(t)=\eta\) at point \(t\). The computational algorithm consists in finding the maximum of both likelihoods simultaneously. The application to a data set on East-West German migration illustrates the use of this technique. The practical implementation of a new version of the statistical computing environment XploRe is presented.
scientific article; zbMATH DE number 1409725