On the rates of asymptotic normality for recursive kernel density estimators under ϕ-mixing assumptions

Mengmei Xi and Xuejun Wang

Journal of Nonparametric Statistics, 2019, vol. 31, issue 2, 340-363

Abstract: In this paper, we mainly consider two kinds of recursive kernel estimators of $ f(x) $ f(x), which is the probability density function of a sequence of ϕ-mixing random variables $ \{X_i, i\geq 1\} $ {Xi,i≥1}. Under some suitable conditions, we establish the convergence rates of asymptotic normality for the two recursive kernel estimators $ \hat {f}_n(x)=({1}/{n\sqrt {b_n}})\sum _{j=1}^nb_j^{-{1}/{2}}K({(x-X_j)}/{b_j}) $ fˆn(x)=(1/nbn)∑j=1nbj−1/2K((x−Xj)/bj) and $ \tilde {f}_n(x)=({1}/{n})\sum _{j=1}^n({1}/{b_j})K ({(x-X_j)}/{b_j}) $ f~n(x)=(1/n)∑j=1n(1/bj)K((x−Xj)/bj). In particular, by the choice of the bandwidths, the convergence rates of asymptotic normality for the estimators $ \hat {f}_n(x) $ fˆn(x) and $ \tilde {f}_n(x) $ f~n(x) can attain $ O(n^{-{1}/{8}}\log ^{{1}/{3}}n) $ O(n−1/8log1/3⁡n) and $ O(n^{-{1}/{6}}\log ^{{1}/{3}}n), $ O(n−1/6log1/3⁡n), respectively. Besides, the simulation study and a real data analysis are presented to verify the validity of the theoretical results.

Date: 2019
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DOI: 10.1080/10485252.2019.1566542

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