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

In this paper, we mainly consider two kinds of recursive kernel estimators of , which is the probability density function of a sequence of phi-mixing random variables . Under some suitable conditions, we establish the convergence rates of asymptotic normality for the two recursive kernel estimators and . In particular, by the choice of the bandwidths, the convergence rates of asymptotic normality for the estimators and can attain and respectively. Besides, the simulation study and a real data analysis are presented to verify the validity of the theoretical results.