论文标题
Dirichlet内核密度估计器的渐近性能
Asymptotic properties of Dirichlet kernel density estimators
论文作者
论文摘要
从理论上讲，我们首次研究了Aitchison和Lauder（1985）引入的Dirichlet内核估计器，以估算$ D $ d $ dimemensional Simplex支持的多元密度。单纯形是一个重要的案例，因为它是组成数据的自然领域，并且在不对称核文献中被忽略了。 DIRICHLET内核估计器概括了Chen（1999）的（非修饰）单维β内核估计器，它没有边界偏见和单纯偏见。我们表明，它实现了平均平方误差和平均集成平方误差的最佳收敛速率$ o（n^{ - 4/（d+4）}）$，我们证明其渐近正态性和均匀的强一致性，我们还发现平均集成绝对误差的渐近表达式。为了说明DIRICHLET内核法及其有利的边界特性，我们提出了有关矿物处理的案例研究。
We study theoretically, for the first time, the Dirichlet kernel estimator introduced by Aitchison and Lauder (1985) for the estimation of multivariate densities supported on the $d$-dimensional simplex. The simplex is an important case as it is the natural domain of compositional data and has been neglected in the literature on asymmetric kernels. The Dirichlet kernel estimator, which generalizes the (non-modified) unidimensional Beta kernel estimator from Chen (1999), is free of boundary bias and non-negative everywhere on the simplex. We show that it achieves the optimal convergence rate $O(n^{-4/(d+4)})$ for the mean squared error and the mean integrated squared error, we prove its asymptotic normality and uniform strong consistency, and we also find an asymptotic expression for the mean integrated absolute error. To illustrate the Dirichlet kernel method and its favorable boundary properties, we present a case study on minerals processing.
