Adaptive circular deconvolution by model selection under unknown error distribution

Abstract

We consider a circular deconvolution problem, in which the density f of a circular random variable X must be estimated nonparametrically based on an i.i.d. sample from a noisy observation Y of X. The additive measurement error is supposed to be independent of X. The objective of this work was to construct a fully data-driven estimation procedure when the error density ϕ is unknown. We assume that in addition to the i.i.d. sample from Y , we have at our disposal an additional i.i.d. sample drawn independently from the error distribution. We first develop a minimax theory in terms of both sample sizes. We propose an orthogonal series estimator attaining the minimax rates but requiring optimal choice of a dimension parameter depending on certain characteristics of f and ϕ, which are not known in practice. The main issue addressed in this work is the adaptive choice of this dimension parameter using a model selection approach. In a first step, we develop a penalized minimum contrast estimator assuming that the error density is known. We show that this partially adaptive estimator can attain the lower risk bound up to a constant in both sample sizes n and m. Finally, by randomizing the penalty and the collection of models, we modify the estimator such that it no longer requires any previous knowledge of the error distribution. Even when dispensing with any hypotheses on ϕ, this fully data-driven estimator still preserves minimax optimality in almost the same cases as the partially adaptive estimator. We illustrate our results by computing minimal rates under classical smoothness assumptions.

Nonparametric intensity estimation from noisy observations of a Poisson process under unknown error distribution

“…The rates in terms of n coincide formally with the classical rates for nonparametric inverse problems (see [Fan91,Lac06], for instance). The rates in m are of the same order as those that have already been obtained in the related model of (circular) density deconvolution with unknown error density in [Joh09, CL11,JS13a]. They can also be compared with the rates in the indirect Gaussian sequence model with partially known operator [JS13b], which provides a benchmark model for a variety of nonparametric inverse problems.…”

“…Upper bound for 3 : The term 3 is bounded analogously to the bound established for 3 in the proof of Theorem 5 (here, we do not have to exploit the additional Assumption 2), and we get The proof follows along the lines of the proof of Lemma A5 in [JS13a] and is thus omitted.…”

Adaptive procedure for Fourier estimators: application to deconvolution and decompounding

“…Deconvolution with unknown error distribution has also been studied (see e.g. [47,20,35,45], if an additional error sample is available, or [17,21,36,15,38] under other set of assumptions).…”

Laguerre deconvolution with unknown matrix operator

“…The rigorous study of adaptive procedures in a deconvolution model with unknown errors has only recently been addressed. We are aware of the work by Comte and Lacour (2011) and Kappus and Mabon (2014) who extended it to the adaptive strategy, by Johannes and Schwarz (2013) who consider a model of circular deconvolution and by Dattner et al (2016), who deal with adaptive quantile estimation via Lespki's method.…”