A Statistical Method In Single-index Models

This paper is mainly concerned on the Single-Index Model: In this model,g is the ttnkown link function, 6) is the p dimension parametric vector, x~ are nonra.ndom designed p dimension vectors and Yi are the ohserved values of the link ft inc ti on. The paper mainly study its quality in large sample situation while the variate x~ is nonra.ndom. Single-Index model is developed in 1980s progressively, which is widely used in economics and biometrics. The article is based on the assumption that the variate vec- t()r x~ is fixed designed. we use kernel regression estimation and quasi-likelihood fttnction~ method to construct the estimation of regression fitncion g and parameter vector 6). The main idea in the paper is to construct the preliminary estimation of the regression function and its derivatives at first,; and secondly apply quasi-likelihood to obtain 6),which is the estimation nfl); a.nd at. last, substitute it. to the preliminary estima.t.ion,t.hen get the fina.l estimation of regression function, we also obtain the consistent speed of the estimation of 6) and the fina.l estimation of regression function.