A Berry-Esseen Type Bound of Regression Estimator Based on Linear Process Errors

Abstract

Abstract. Consider the nonparametric regression modelis an unknown regression function, x ni are known fixed design points, and the correlated errors {ϵ ni , 1 ≤ i ≤ n} have the same distribution as {V i , 1 ≤ i ≤ n}, here Vt = P ∞ j=−∞ ψ j e t−j with P ∞ j=−∞ |ψ j | < ∞ and {et} are negatively associated random variables. Under appropriate conditions, we derive a Berry-Esseen type bound for the estimator of g(·). As corollary, by choice of the weights, the BerryEsseen type bound can attain O(n −1/4 (log n) 3/4 ).

Berry–Esseen bounds for wavelet estimator in a regression model with linear process errors

“…In addition, under some regularity conditions, γ 3n → 0 holds with the usual AR, MA and ARMA processes which are extensively used to model serially correlated data (cf. Remark 2.1 (c) in Liang and Li (2008)). …”

“…Following the method of the proof of Theorem 2.1 in Liang and Li (2008) and applying Lemmas 3.2-3.4, we obtain…”

“…Applying Lemma A.4(iv) in the Appendix and Ee 2 0 < ∞ and following the procedure for ES 2 2n in Liang and Li (2008), we obtain ES 2 2n ≤ C γ 3n . Therefore the proof of Lemma 3.1(1) is completed.…”

“…For instance, Georgiev (1985) proposed a general weighted regression estimator of g(·) and subsequently it has been studied by many scholars. One can refer to Georgiev (1988) under independent random errors, Roussas et al (1992) and Yang and Li (2006) under strong mixing random errors, Yang (2003) under negatively associated errors, Tran et al (1996) for asymptotic normality under the assumption that the errors form a weakly stationary linear process based on a martingale difference sequence, and Liang and Li (2008) for the Berry-Esseen type bound based on linear process errors under negatively associated random variables. Since the 1990s, some authors have considered using wavelet methods in statistics.…”

The Berry-Esseen bounds of wavelet estimator for regression model whose errors form a linear process with a ρ-mixing

“…China. 2 School of Mathematics and Computer Science, Shangrao Normal University, Shangrao, 334001, P.R. China.…”