Multiscale inference for multivariate deconvolution

Abstract

In this paper we provide new methodology for inference of the geometric features of a multivariate density in deconvolution. Our approach is based on multiscale tests to detect significant directional derivatives of the unknown density at arbitrary points in arbitrary directions. The multiscale method is used to identify regions of monotonicity and to construct a general procedure for the detection of modes of the multivariate density. Moreover, as an important application a significance test for the presence of a local maximum at a pre-specified point is proposed. The performance of the new methods is investigated from a theoretical point of view and the finite sample properties are illustrated by means of a small simulation study.

Multiscale Inference and Long-Run Variance Estimation in Non-Parametric Regression with Time Series Errors

“…As an alternative to strong approximation theory, Eckle et al . () and Proksch et al . () have recently used Gaussian approximation results that were derived in Chernozhukov et al .…”

“…We rather calibrate the statisticsψ T .u, h/=σ that correspond to the bandwidth h by subtracting the additive correction term λ.h/. This approach was pioneered by Dümbgen and Spokoiny (2001) and has been used in numerous other studies since then; see for example Dümbgen (2002), Rohde (2008), Dümbgen and Walther (2008), Rufibach and Walther (2010), Schmidt-Hieber et al (2013) and Eckle et al (2017).…”

“…However, it has not been combined with anticoncentration results to approximate the quantiles of the multiscale statistic. As an alternative to strong approximation theory, Eckle et al (2017) and Proksch et al (2018) have recently used Gaussian approximation results that were derived in Chernozhukov et al (2014Chernozhukov et al ( , 2017 to analyse multiscale tests for independent data. Even though it might be possible to adapt these techniques to the case of dependent data, this is not trivial at all as part of the technical arguments and the Gaussian approximation tools strongly rely on the assumption of independence.…”

“…More recently, Proksch et al (2018) have constructed multiscale tests for inverse regression models. In the context of density estimation, multiscale tests have been investigated in Dümbgen and Walther (2008), Rufibach and Walther (2010), Schmidt-Hieber et al (2013) and Eckle et al (2017) among others.…”

Sparse covariance matrix estimation in high-dimensional deconvolution

“…Hence, our study contributes to the deconvolution theory by treating the multivariate case; in particular, our techniques for the lower bounds might be of interest. To our knowledge, only Masry [39], Eckle et al [20], and Lepski and Willer [35,36] have studied the setting of multivariate deconvolution. They deal with a different problem, namely that of nonparametric estimation of the density of X j or its geometric features when the distribution of ε j is known.…”