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

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.

Multiscale inference for multivariate deconvolution

“…ch ≥ x 0 − t ≥ 2 √ dh for some c > 2 √ d and angle(x 0 − t, s) → 0 for n → ∞. Following the line of arguments presented in the proof of Theorem 3.3 in Eckle et al (2016), one can prove that, under the given assumptions, ∂ s f (x) −h for all x ∈ suppφ t,h . Hence,…”

Optimal inference with a multidimensional multiscale statistic

“…Several interesting applications of the multiscale approach exist when d = 1 (following the seminal paper of Dümbgen and Spokoiny [11]): In Dümbgen and Walther [12] the authors propose a multiscale test statistic to make inference about a probability density on the real line given i.i.d. observations; Schmidt-Hieber et al [44] use multiscale methods to make inference in a deconvolution problem; Rivera and Walther [42] use multiscale methods to detect a jump in the intensity of a Poisson process; Eckle et al [13] and Eckle et al [14] use multiscale approaches to make inferences about multivariate densities in deconvolution problems, etc. We believe that our extension beyond d = 1 will also lead to several interesting multidimensional applications.…”

Tests for qualitative features in the random coefficients model

“…The boundedness of sup (t,h,v)∈T β h |X t,h,v | σ t,h,v − α h β h follows by an application of Theorem 6.1 and Remark 1 in Dümbgen and Spokoiny (2001). C.4 Replacing the true densities in the limit process by estimators Hence, the hypotheses (3.5) are rejected simultaneously on the set of scales T n with asymptotic probability one. Moreover, for a mode b 0 in (a 1 , a 2 ) of f β and any triple (t, h, v) ∈ T b 0 n , one can prove that ∂ v f β (b) −h for all b ∈ suppφ t,h by following the arguments in the proof of Theorem 10 in Eckle et al (2017b). Consequently,…”