Asymptotic properties of wavelet estimators in semiparametric regression models under dependent errors

Consider the semiparametric regression model yi=xiTβ+g(ti)+εi for i=1,…,ni=1,…,n, where xi∈Rpxi∈Rp are the random design vectors, titi are the constant sequences on [0,1][0,1], β∈Rpβ∈Rp is an unknown vector of the slop parameter, gg is an unknown real-valued function defined on the closed interval [0,1][0,1], and the error random variables εiεi are coming from a stationary stochastic process, satisfying the strong mixing condition in some results. Under suitable conditions, we obtain expansions for the bias and the variance of wavelet estimators βˆn and gˆn(⋅) of ββ and g(⋅)g(⋅) respectively, prove their weak consistency, and establish the asymptotic normality and the Berry–Esseen bound of βˆn.