Semiparametric versus nonparametric estimation in single index regression model: a computational approach (Q1965926)

Let \((x_i,y_i)\) be an i.i.d. random sample, \(x_i\in R^p\), \(y\in R\), \(m(x)= E(y |x_i=x)\). The authors consider the model \(m(x)=r({}^tx\theta)\), where \(r\) is an unknown function, \(\theta=(\theta_1,\dots,\theta_p)\) is an unknown parameter vector with \(\theta_1=1\). Two different methods to estimate \(r\) and \(\theta\) are considered. The first one is a pseudo maximum likelihood estimate (PMLE). The pseudo log-likelihood is obtained by replacing in the log-likelihood the true function \(r\) by some kernel estimator of \(r.\) The second one is a fully nonparametric estimator based on the average derivative estimation method (ADE). The vector \(\delta=(\delta_1,\dots,\delta_p) = E(\nabla m(x_i))\) is estimated by \[ \hat\delta=-n^{-1}\sum_{i=1}^n y_i\nabla\hat f(x_i)/\hat f(x_i), \] where \(\hat f\) is the kernel estimator of the density of \(x_i.\) Then the ADE estimator for \(\theta_k\) is \(\hat\theta_k=\hat\delta_k/\hat\delta_1\). The performances of these two estimators are compared on data simulated from a homoscedastic binary model.