ABSTRACT In this paper, we mainly consider two kinds of recursive kernel estimators of , which is the probability density function of a sequence of ϕ-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.

中文翻译：

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关于在 ϕ 混合假设下递归核密度估计量的渐近正态率

摘要 在本文中，我们主要考虑 的两种递归核估计量，它是 φ 混合随机变量序列的概率密度函数。在一些合适的条件下，我们建立了两个递归核估计量 和 的渐近正态收敛率。特别是，通过选择带宽，估计器的渐近正态性收敛速度可以分别达到 和 。此外，通过仿真研究和实际数据分析来验证理论结果的有效性。