Difference-based Estimates Of Parameters In A Semiparametric Regression Model And Minimax Properties

Partial linear model, as one of the class of semiparametric regression models, has the form y=zβ+f(x)+ε, where for simplicity all variables(y,z,x) are assumed to be scalars. The response variable, y, is modeled as the sum of three components:the nonparametric component f(·), the linear component zβ, and random errorε β and f are unknown. Usually β is the parametric of interest, with the function f treated as a nuisance parametric.What this paper concern is to estimate the parameter β and discuss the properties. The paper introduces conditions of difference-based minimax estimates of the regression parameters β in a partial linear model. Considering the ordinary least squares estimator βdiff based on higher order differences of the observations, we get the minimax linear estimator of β at the same time. Then discussing the difference-based ridge regression estimator βdiff(k) that used in the presence of multicollinearity in partial linear model, we can get the conditions of βdiff(k) as a minimax linear estimator.Furthermore, we consider another kind of semiparametric regression models, that is, semivarying-coefficient model. Get the difference based minimax estimator of the linear components and the inference which is similar to partial linear model.