Statistica Sinica 17(2007), 177-197

ADAPTIVE VARYING-COEFFICIENT LINEAR MODELS

FOR STOCHASTIC PROCESSES: ASYMPTOTIC THEORY

Zudi Lu$^{1,2}$, Dag Tjøstheim$^3$ and Qiwei Yao$^{1,4}$

$^1$London School of Economics, $^2$Chinese Academy of Sciences

$^3$University of Bergen and $^4$Peking University

Abstract: We establish the asymptotic theory for the estimation of adaptive varying-coefficient linear models. More specifically, we show that the estimator of the index parameter is root-n-consistent. It differs from the locally optimal estimator that has been proposed in the literature with a prerequisite that the estimator is within a $n^{-\delta}$-distance of the true value. To this end, we establish two fundamental lemmas for the asymptotic properties of the estimators of parametric components in a general semiparametric setting. Furthermore, the estimation for the coefficient functions is asymptotically adaptive to the unknown index parameter. Asymptotic properties are derived using the empirical process theory for strictly stationary $\beta$-mixing processes.

Key words and phrases: Adaptive varying-coefficient model, asymptotic normality, β-mixing, empirical process, index parameter, root-n consistency, uniform convergence.