In this paper we propose a new class of Hermite series-based direct plug-in bandwidth selectors for kernel density estimation and we describe their asymptotic and finite sample behaviours. Unlike the direct plug-in bandwidth selectors considered in the literature, the proposed methodology does not involve multistage strategies and reference distributions are no longer needed. The new bandwidth selectors show a good finite sample performance when the underlying probability density function presents not only ‘easy-to-estimate’ but also ‘hard-to-estimate’ distribution features. This quality, that is not shared by other widely used bandwidth selectors as the classical plug-in or the least-square cross-validation methods, is the most significant aspect of the Hermite series-based direct plug-in approach to bandwidth selection.

中文翻译：

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核密度估计的带宽选择：基于 Hermite 系列的直接插件方法

在本文中，我们提出了一类新的基于 Hermite 级数的直接插入式带宽选择器，用于核密度估计，并描述了它们的渐近和有限样本行为。与文献中考虑的直接插入式带宽选择器不同，所提出的方法不涉及多级策略，并且不再需要参考分布。当基础概率密度函数不仅呈现“易于估计”而且呈现“难以估计”的分布特征时，新的带宽选择器显示出良好的有限样本性能。这种质量是其他广泛使用的带宽选择器作为经典插件或最小二乘交叉验证方法所不具备的，是基于 Hermite 系列的直接插件方法进行带宽选择的最重要方面。