Convergence rates of wavelet estimators in semiparametric regression models under NA samples

Abstract

Consider the following heteroscedastic semiparametric regression model: $y_i = X_i^T \beta + g\left( {t_i } \right) + \sigma _i e_i , 1 \leqslant i \leqslant n,$ where {X _i, 1 ≤ i ≤ n} are random design points, errors {e _i, 1 ≤ i ≤ n} are negatively associated (NA) random variables, σ _i ² = h(u _i), and {u _i} and {t _i} are two nonrandom sequences on [0, 1]. Some wavelet estimators of the parametric component β, the nonparametric component g (t) and the variance function h (u) are given. Under some general conditions, the strong convergence rate of these wavelet estimators is $O\left( {n^{ - \tfrac{1}{3}} \log n} \right)$. Hence our results are extensions of those results on independent random error settings.

Keywords

Semiparametric regression model / Wavelet estimate / Negatively associated random error / Strong convergence rate

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