A partially linear single‐index transformation model and its nonparametric estimation

Abstract

In this paper, we consider the nonparametric estimation of the partially linear single‐index transformation model, where the transformation function, single‐index function and error distribution are all completely unknown. We first use the minimum average variance estimation method to estimate the regression coefficients, and then propose a new incorporated local linear regression estimator for the derivative function of the single‐index function. Accordingly by integration we can obtain the estimator of the single‐index function. Finally we propose a constrained least square estimator for the transformation function, where basis function approximation is employed and cross validation method is proposed to select suitable sets of basis functions. Asymptotical properties of the estimators are established. Simulation studies show that our proposed estimators work well. A real‐world data analysis of total health care charges was used to illustrate the proposed procedure. The Canadian Journal of Statistics 43: 97–117; 2015 © 2015 Statistical Society of Canada

Partially linear estimation using sufficient dimension reduction

“…When the number of rows of B is d = 1, the model can be regarded as a partial linear single index model. For this case, Carroll et al [2], Yu and Ruppert [39], Xia and Härdle [37] and Ding et al [8] studied the estimation methods and properties of mean regression. Wang et al [35] proposed the SDR method for partial linear single index models.…”