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Efficient recursive algorithms for functionals based on higher order derivatives of the multivariate Gaussian density

机译：基于多元高斯密度高阶导数的函数有效递归算法

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摘要

Many developments in Mathematics involve the computation of higher order derivatives of Gaussian density functions. The analysis of univariate Gaussian random variables is a well-established field whereas the analysis of their multivariate counterparts consists of a body of results which are more dispersed. These latter results generally fall into two main categories: theoretical expressions which reveal the deep structure of the problem, or computational algorithms which can mask the connections with closely related problems. In this paper, we unify existing results and develop new results in a framework which is both conceptually cogent and computationally efficient. We focus on the underlying connections between higher order derivatives of Gaussian density functions, the expected value of products of quadratic forms in Gaussian random variables, and -statistics of degree two based on Gaussian density functions. These three sets of results are combined into an analysis of non-parametric data smoothers.
机译：数学的许多发展涉及高斯密度函数的高阶导数的计算。对单变量高斯随机变量的分析是一个公认的领域，而对它们的多变量对应变量的分析则包含大量分散的结果。后面的结果通常分为两大类：揭示问题深层结构的理论表达式，或可以掩盖与密切相关的问题的联系的计算算法。在本文中，我们统一了现有结果，并在概念上有效且计算效率高的框架中开发了新结果。我们关注于高斯密度函数的高阶导数，高斯随机变量中二次形式的乘积的期望值以及基于高斯密度函数的二阶统计量之间的潜在联系。将这三组结果组合成对非参数数据平滑器的分析。

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