Analytical and statistical properties of local depth functions motivated by clustering applications

General local depth functions ($LGD$) are used for describing the local geometric features and mode(s) in multivariate distributions. In this paper, we undertake a rigorous systematic study of $LGD$ and establish several analytical and statistical properties. First, we show that, when the underlying probability distribution is absolutely continuous, scaled version of $LGD$ (referred to as $\tau$-approximation) converges, uniformly and in $L^d(\mathbb{R}^p)$, to the density, when $\tau$ converges to zero. Second, we establish that, as the sample size diverges to infinity the centered and scaled sample $LGD$ converge in distribution to a centered Gaussian process uniformly in the space of bounded functions on $\mathcal{H}_G$, a class of functions yielding $LGD$. Third, using the sample version of the $\tau$-approximation ($S \tau \hspace{-0.06cm} A$) and the gradient system analysis, we develop a new clustering algorithm. The validity of this algorithm requires several results concerning the uniform finite difference approximation of the gradient system associated with $S \tau \hspace{-0.06cm} A$. For this reason, we establish \emph{Bernstein}-type inequality for deviations between the centered and scaled sample $LGD$, which is also of independent interest. Finally, invoking the above results, we establish consistency of the clustering algorithm. Applications of the proposed methods to mode estimation and upper level set estimation are also provided. Finite sample performance of the methodology are evaluated using numerical experiments and data analysis.

翻译：通用本地深度函数 (LGD$) 用于描述本地几何特征和多变量分布中的模式。在本文中, 我们对美元进行严格的系统化研究, 并建立若干分析和统计属性。首先, 我们显示, 当基本概率分布绝对连续时, 基概率分布( 称为$tau$- accolomation) 的缩放版本 $LGD$ (称为$\tau$- accolation) 统一并用 $\d( mathb{R ⁇ p) 来描述当美元接近于零时的密度。第二, 我们确定, 当样本大小大小大小的样本大小使美元GD$( $GD$ ) 变得不完全相同时, 当样本化的样本中位值和比例化的样本值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值