Multiscale inference for a multivariate density with applications to X-ray astronomy

TL;DR

This paper introduces multiscale statistical methods for analyzing the geometric features of multivariate densities, with applications to identifying high-energy sources in X-ray astronomy.

Contribution

It develops new multiscale inference techniques for density features, including mode detection and monotonicity testing, with proven optimal localization rates.

Findings

Methods effectively identify modes in simulated data

Application to X-ray data verifies source positions

Theoretical results demonstrate optimal localization rates

Abstract

In this paper we propose methods for inference of the geometric features of a multivariate density. Our approach uses multiscale tests for the monotonicity of the density at arbitrary points in arbitrary directions. In particular, a significance test for a mode at a specific point is constructed. Moreover, we develop multiscale methods for identifying regions of monotonicity and a general procedure for detecting the modes of a multivariate density. It is is shown that the latter method localizes the modes with an effectively optimal rate. The theoretical results are illustrated by means of a simulation study and a data example. The new method is applied to and motivated by the determination and verification of the position of high-energy sources from X-ray observations by the Swift satellite which is important for a multiwavelength analysis of objects such as Active Galactic Nuclei.