Hellinger-Bhattacharyya Cross-Validation for Shape-Preserving Multivariate Wavelet Thresholding

The benefits of the wavelet approach for density estimation are well established in the literature, especially when the density to estimate is irregular or heterogeneous in smoothness. However, wavelet density estimates are typically not bona fide densities. In previous work, we introduced a ‘shape-preserving’ wavelet density estimator, including as main step the estimation of the square-root of the density using k-th nearest neighbours. A natural concept involving square-root of densities is the Hellinger distance—or equivalently, the Bhattacharyya affinity coefficient. In this paper, we deliver a fully data-driven version of the above ‘shape-preserving’ wavelet density estimator, where all user-defined parameters, such as resolution level or thresholding specifications, are selected by optimising an original leave-one-out version of the Hellinger-Bhattacharyya criterion. The theoretical optimality of the proposed procedure is established, while simulations show the strong practical performance of the estimator. Within that framework, we also propose a novel but natural ‘jackknife thresholding’ scheme, which proves superior to other, more classical thresholding options.