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

ABSTRACT

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

KEYWORDS:

• Berry-Esseen bounds
• asymptotic normality
• density function
• recursive kernel estimator

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Supported by the National Natural Science Foundation of China (NNSF) (11671012, 11871072, 11701004, 11701005), the Natural Science Foundation of Anhui Province (1508085J06), the Key Projects for Academic Talent of Anhui Province (gxbjZD2016005) and the Project on Reserve Candidates for Academic and Technical Leaders of Anhui Province (2017H123).