Abstract

: In some applications, the mean or median response is linearly related to some variables but the relation to additional variables are not easily parameterized. Partly linear models arise naturally in such circumstances. Suppose that a random sample f(T i ; X i ; Y i ); i = 1; 2; \Delta \Delta \Delta ; ng is modeled by Y i = X T i fi 0 + g 0 (T i ) + error i , where Y i is a real-valued response, X i 2 R p and T i ranges over a unit square, and g 0 is an unknown function with a certain degree of smoothness. We make use of bivariate tensor-product B-splines as an approximation of the function g 0 and consider M-type regression splines by minimization of P n i=1 ae(Y i \Gamma X T i fi \Gamma g n (T i )) for some convex function ae. Mean, median and quantile regressions are included in this class. We show under appropriate conditions that the parameter estimate of fi achieves its information bound asymptotically and the function estimate of g 0 attains the optimal rate of convergen...

Citations