The Berry–Esseen-type bound for the G-M estimator in a nonparametric regression model with α-mixing errors. Issue 1 (2nd January 2022)

• Record Type: Journal Article
• Authors: Wang, Yan; Yu, Wei; Li, Xiaoqin; Wang, Xuejun
• Is Part Of: Statistics. Volume 56:Issue 1(2022)
• Page Start: 97
• Page End: 120
• Publication Date: 2022-01-02
• Subjects: Nonparametric regression model -- α-mixing random variables -- G-M estimator -- Berry–Esseen-type bound; 62G20 -- 62E20; Mathematical statistics -- Periodicals; 519.505
• DOI: 10.1080/02331888.2022.2038600
• Languages: English
• ISSNs: 0233-1888

Abstract

Abstract : Consider the fixed design nonparametric regression model: Y i = g ( x i ) + ε i, i = 1, 2, …, n, n ≥ 1, where x i are known fixed design points from [ 0, 1 ] satisfying 0 = x 0 ≤ x 1 ≤ ⋯ ≤ x n − 1 ≤ x n = 1 and g ( x ) is an unknown function defined on the closed interval [ 0, 1 ], Y i are the response variables, and ε i are zero mean α -mixing random variables with α ( n ) = O ( n − λ ) for some λ > ( 2 + δ ) ( 1 + δ ) / δ, where δ > 0 . We derive the Berry–Esseen-type bound for the G-M estimator of g ( ⋅ ) under some mild conditions, which approximates to O ( n − 1 4 ) provided that weights and moments are appropriate. Our results generalize some known ones in the literature. In addition, a simulation study is given to illustrate the validity of the theoretical result.