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\footnotesize \noindent Statistica Sinica {\bf 22} (2012), 1075-1101

\footnotesize \noindent

\vspace{3pc} \normalsize \centerline{\large\bf NEW EFFICIENT AND
ROBUST ESTIMATION IN } \vspace{2pt} \centerline{\large\bf
VARYING-COEFFICIENT MODELS} \vspace{2pt} \centerline{\large\bf WITH
HETEROSCEDASTICITY} \vspace{.25cm}
\author{Jie Guo$^1$, Maozai Tian$^1$ and Kai Zhu$^2$}
\vspace{.4cm} \centerline{\it $^1$Renmin University of China and
$^2$Chinese Academy of Sciences} \vspace{.55cm} 
\begin{quotation}
\noindent {\it Abstract:} Varying-coefficient models with
heteroscedasticity are considered in this paper. Based on local
composite quantile regression, we propose a new estimation method to
estimate the coefficient functions and heteroscedasticity
simultaneously. Moreover, we can get the estimated conditional
quantile curves of the error part. The conditional biases,
variances, and asymptotic normalities of these estimators are
studied explicitly. A simple and quick plug-in bandwidth selector is
employed to select the optimal bandwidth. The estimators of the
coefficient functions perform efficiently and robustly regardless of
the error distributions. When the error $\varepsilon$ follows a
non-normal distribution, the proposed estimators of the coefficient
functions are much more efficient than local polynomial weighted
least squares estimators and almost as efficient for normal random
errors. The estimator of heteroscedasticity also outperforms other
classical estimators in the literature. A goodness-of-fit test based
on a bootstrap procedure is proposed to test whether the coefficient
functions are actually varying. Both simulations and data analysis
are used to illustrate the proposed method.\par \vspace{9pt}
\noindent {\it Key words and phrases:} Goodness-of-fit test,
heteroscedasticity, local composite quantile regression, plug-in
bandwidth selector, varying-coefficient models. \par
\end{quotation}

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