Asymptotic properties of wavelet estimators in heteroscedastic semiparametric model based on negatively associated innovations

Abstract

Abstract Consider the heteroscedastic semiparametric regression model yi=xiβ+g(ti)+εi $y_{i}=x_{i}\beta+g(t_{i})+\varepsilon_{i}$, i=1,2,…,n $i=1, 2, \ldots, n$, where β is an unknown slope parameter, εi=σiei $\varepsilon_{i}=\sigma_{i}e_{i}$, σi2=f(ui) $\sigma^{2}_{i}=f(u_{i})$, (xi,ti,ui) $(x_{i},t_{i},u_{i})$ are nonrandom design points, yi $y_{i}$ are the response variables, f and g are unknown functions defined on the closed interval [0,1] $[0,1]$, random errors {ei} $\{e_{i} \}$ are negatively associated (NA) random variables with zero means. Whereas kernel estimators of β, g, and f have attracted a lot of attention in the literature, in this paper, we investigate their wavelet estimators and derive the strong consistency of these estimators under NA error assumption. At the same time, we also obtain the Berry–Esséen type bounds of the wavelet estimators of β and g.

Keywords

• Semiparametric regression model
• Wavelet estimator
• Berry–Esséen bound
• Negatively associated random error
• Consistency