3. Statistical Models for Biomedical Research

PARTIAL LINEAR MODELS WITH HETEROSCEDASTIC VARIANCES

Abstract

Consider the partial linear heteroscedastic model Y_i=X^T_iβ+g(T_i)+σ_ie_i, 1≤i≤n with random variables (X_i, T_i) and response variables Y_i and unknown regression function g(·). We assume that the errors are heteroscedastic, i.e., σ²_i≠const. e_i are i.i.d. random errors with mean zero and variance 1. In this partial linear heteroscedastic model, we consider the situations that the variance is an unknown smooth function of exogenous variables, or of nonlinear variables T_i, or of the mean response X^T_iβ+g(T_i). Under the general assumptions, we construct an estimator of the regression parameter vector β which is asymptotically equivalent to the weighted least squares estimators with known variance. In procedure of constructing the estimators, the technique of splitting-sample is adopted.