Shape-preserving wavelet-based density estimation with applications to image analysis

Abstract

Wavelet estimators for a probability density enjoy many good properties; however, they are not shape-preserving in the sense that the final estimate may be negative nor integrate to unity. A solution to negativity issues may be to estimate first the square-root of the density and then square this estimate up. In this thesis, we propose and investigate such an estimation scheme, generalising to higher dimensions a previous construction of Penev and Dechevsky (1997}, which is valid only in one dimension, using nearest-neighbour balls. The theoretical properties of the proposed estimator are obtained, and it is shown to reach the optimal rate of convergence uniformly over large classes of densities under mild conditions. For spatially inhomogeneous densities and in general, there is a need to threshold the empirical wavelet coefficients in order to avoid over-fitting. In the case of density estimation, the most common approach is to use cross-validation over a likelihood function. Aligned with our results, we provide a principled alternative using a cross-validation type approach over an empirical approximation to the Bhattacharyya coefficient and the associated Hellinger distance, which is suitable when the square-root of the density is estimated. The effectiveness of these data-driven algorithms is demonstrated via Monte Carlo simulations and a thorough review of their usage in the traditional Old Faithful geyser dataset. Finally, we aim to extend these tools and applications to the raising field of intrinsic statistics in Riemannian manifolds and present an example on how techniques based on k-th nearest neighbours can be applied in image analysis using the MNIST and Fashion-MNIST datasets.