On the rates of asymptotic normality for recursive kernel density estimators under ϕ-mixing assumptions. Issue 2 (3rd April 2019)

• Record Type: Journal Article
• Authors: Xi, Mengmei; Wang, Xuejun
• Is Part Of: Journal of nonparametric statistics. Volume 31:Issue 2(2019)
• Page Start: 340
• Page End: 363
• Publication Date: 2019-04-03
• Subjects: Berry-Esseen bounds -- asymptotic normality -- density function -- recursive kernel estimator; Nonparametric statistics -- Periodicals; 519.5
• DOI: 10.1080/10485252.2019.1566542
• Languages: English
• ISSNs: 1048-5252

Abstract

ABSTRACT: In this paper, we mainly consider two kinds of recursive kernel estimators off ( x ), which is the probability density function of a sequence of ϕ -mixing random variables{ X i, i ≥ 1 } . Under some suitable conditions, we establish the convergence rates of asymptotic normality for the two recursive kernel estimatorsf ˆ n ( x ) = ( 1 / n b n ) ∑ j = 1 n b j − 1 / 2 K ( ( x − X j ) / b j ) andf ~ n ( x ) = ( 1 / n ) ∑ j = 1 n ( 1 / b j ) K ( ( x − X j ) / b j ) . In particular, by the choice of the bandwidths, the convergence rates of asymptotic normality for the estimatorsf ˆ n ( x ) andf ~ n ( x ) can attainO ( n − 1 / 8 log 1 / 3 ⁡ n ) andO ( n − 1 / 6 log 1 / 3 ⁡ n ), respectively. Besides, the simulation study and a real data analysis are presented to verify the validity of the theoretical results.