Functional approach for excess mass estimation in the density model

TL;DR

This paper introduces a new method for estimating the excess mass of an unknown density function at a specified level, with theoretical analysis and practical implementation, applicable to various statistical tasks.

Contribution

It proposes the first estimator of excess mass as an integrated functional of the density and evaluates its convergence rates over Besov classes.

Findings

The estimator achieves improved convergence rates over existing plug-in methods.

Numerical studies demonstrate practical effectiveness and implementation advantages.

The approach is applicable to tasks like multimodality testing and anomaly detection.

Abstract

We consider a multivariate density model where we estimate the excess mass of the unknown probability density $f$ at a given level $\nu>0$ from $n$ i.i.d. observed random variables. This problem has several applications such as multimodality testing, density contour clustering, anomaly detection, classification and so on. For the first time in the literature we estimate the excess mass as an integrated functional of the unknown density $f$. We suggest an estimator and evaluate its rate of convergence, when $f$ belongs to general Besov smoothness classes, for several risk measures. A particular care is devoted to implementation and numerical study of the studied procedure. It appears that our procedure improves the plug-in estimator of the excess mass.