Abstract

The approximation properties of finite mixtures of location-scale distributions on Euclidean space have been well studied. It has been shown that mixtures of location-scale distributions can approximate arbitrary probability density function up to any desired level of accuracy provided the number of mixture components is sufficiently large. However, analogous results are not available for probability density functions defined on the unit hypersphere. The von-Mises-Fisher distribution, defined on the unit hypersphere S^m in ${\mathbb{R}}^{m+1},$ plays the central role in directional statistics. We prove that any continuous probability density function on S^m can be approximated to arbitrary degrees of accuracy in sup norm by a finite mixture of von-Mises-Fisher distributions. Our proof strategy and result are also useful in studying the approximation properties of other finite mixtures of directional distributions.

摘要

欧几里得空间上位置尺度分布的有限混合的近似性质已经得到很好的研究。已经表明，只要混合分量的数量足够大，位置尺度分布的混合可以近似任意概率密度函数，达到任何所需的精度水平。然而，类似的结果不适用于在单位超球面上定义的概率密度函数。von-Mises-Fisher 分布，在单位超球面S ^m上定义${\mathbb{R}}^{米+1},$在定向统计中起着核心作用。我们证明了S ^m上的任何连续概率密度函数都可以通过 von-Mises-Fisher 分布的有限混合在sup norm中近似为任意准确度。我们的证明策略和结果对于研究其他有限混合方向分布的近似特性也很有用。