Statistica Sinica 9(1999), 775-794

AUTOMATIC BANDWIDTH CHOICE IN A

SEMIPARAMETRIC REGRESSION MODEL

Sheng-Yan Hong

Northwestern University

Abstract: Speckman (1988) proposed a kernel smoothing method to estimate the parametric component $\beta$ in the semiparametric regression model $y=x^\tau \beta+g(t)+e$, and showed that this kernel smoothing estimator is $\sqrt n$-consistent for a certain deterministic bandwidth choice. However, the important issue of automatic bandwidth choice in this semiparametric setting has not been examined. This paper studies the asymptotic behavior of the bandwidth choice based on a general bandwidth selector which covers such well known data-driven methods as $GCV$ and $CV$. This automatic bandwidth choice is proved to be asymptotically optimal and its asymptotic normality is established. The resulting data-driven kernel smoothing estimator of $\beta$ is then showed to be still $\sqrt n$-consistent. A simulation study is performed to compare small sample behaviors of various commonly used bandwidth selectors in this semiparametric setting, and a real data example is given.

Key words and phrases: Asymptotic normality, automatic bandwidth choice, data-driven estimator, kernel smoothing, semiparametric regression model, √n-consist- ency.

tube method.