Shape-preserving wavelet-based multivariate density estimation

TL;DR

This paper introduces a wavelet-based multivariate density estimator that preserves the shape of the density by estimating its square root, ensuring non-negativity and proper normalization, and achieves optimal convergence rates.

Contribution

It generalizes shape-preserving density estimation to higher dimensions using wavelets and nearest-neighbour methods, providing theoretical guarantees and practical performance.

Findings

Estimator achieves optimal convergence rates.

Automatically produces valid density estimates.

Performs comparably to classical wavelet estimators in simulations.

Abstract

Wavelet estimators for a probability density f enjoy many good properties, however they are not "shape-preserving" in the sense that the final estimate may not be non-negative or integrate to unity. A solution to negativity issues may be to estimate first the square-root of f and then square this estimate up. This paper proposes and investigates such an estimation scheme, generalising to higher dimensions some previous constructions which are valid only in one dimension. The estimation is mainly based on 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. Simulations show that the new estimator performs as well in general as the classical wavelet estimator, while automatically producing estimates which are bona fide densities.