定义随机函数分布的概率密度
Defining probability density for a distribution of random functions
摘要：
随机函数的概率密度的概念不像有限维情形那样简单。虽然密度函数一般不存在的功能数据，我们表明，它是可能的发展密度的概念时，功能数据被认为是在空间中确定的特征函数的主成分分析。这导致了一个透明的和有意义的替代密度定义在一个给定的维度的主成分的分布的密度的平均值的多项式。这种密度近似值很容易从数据中估计出来。它以单调的方式准确地表示了密度的小球近似的关键特征。我们的主成分得分的密度估计的结果也是独立的兴趣，他们揭示了有趣的形状差异，以前没有被考虑。这些结果和属性的统计影响进行了识别和讨论，并在数值计算中说明了实际的后果。
英文摘要：
The notion of probability density for a random function is not as straightforward as in finite-dimensional cases. While a probability density function generally does not exist for functional data, we show that it is possible to develop the notion of density when functional data are considered in the space determined by the eigenfunctions of principal component analysis. This leads to a transparent and meaningful surrogate for density defined in terms of the average value of the logarithms of the densities of the distributions of principal components for a given dimension. This density approximation is estimable readily from data. It accurately represents, in a monotone way, key features of small-ball approximations to density. Our results on estimators of the densities of principal component scores are also of independent interest; they reveal interesting shape differences that have not previously been considered. The statistical implications of these results and properties are identified and discussed, and practical ramifications are illustrated in numerical work.
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