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The benefits of the wavelet approach for density estimation are wellestablished in the literature, especially when the density to estimate isirregular or heterogeneous in smoothness. However, wavelet density estimatesare typically not bona fide densities. In Aya-Moreno et al (2018), a`shape-preserving' wavelet density estimator was introduced, including as mainstep the estimation of the square-root of the density. A natural conceptinvolving square-root of densities is the Hellinger distance - or equivalently,the Bhattacharyya affinity coefficient. In this paper, we deliver a fullydata-driven version of the above 'shape-preserving' wavelet density estimator,where all user-defined parameters, such as resolution level or thresholdingspecifications, are selected by optimising an original leave-one-out version ofthe Hellinger-Bhattacharyya criterion. The theoretical optimality of theproposed procedure is established, while simulations show the strong practicalperformance of the estimator. Within that framework, we also propose a novelbut natural 'jackknife thresholding' scheme, which proves superior to other,more classical thresholding options.

Hellinger-Bhattacharyya cross-validation for shape-preserving multivariate wavelet thresholding