Poisson intensity estimation for the Spektor–Lord–Willis problem using a wavelet shrinkage approach

Abstract

In this paper, we focus on nonparametric estimation in the stereological problem of unfolding sphere size distribution from linear sections. Using a Wavelet–Vaguelette Decomposition (WVD), we construct a rate minimax estimator of the intensity function of a Poisson process that describes the problem. This paper builds upon recent results by the same author concerning the model with a minimal detection radius and shows that this restriction is not necessary to obtain the minimax risk. The proposed adaptive estimator achieves the optimal rate of convergence over Besov balls to within logarithmic factors. Additionally, a construction of a new method of selection of a smoothing parameter by empirical risk minimization is discussed in detail. This paper also demonstrates finite sample behavior of this estimator in a numerical experiment, by using a discrete version of the wavelet algorithm.