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

Abstract

Local general depth (LGD) functions 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 with density f(⋅), the scaled version of LGD (referred to as τ-approximation) converges, uniformly and in Ld(Rp) to f(⋅) when τ 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 HG, a class of functions yielding LGD. Third, using the sample version of the τ-approximation (SτA) 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 SτA. For this reason, we establish 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.