Shape-preserving wavelet-based multivariate density estimation

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, generalizing to higher dimensions some previous constructions which are valid only in one dimension. The estimation is mainly based on nearest-neighbor-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 terms of Mean Integrated Squared Error as the classical wavelet estimator and better than it in terms of Mean Squared Hellinger Distance between the estimator and the truth, while automatically producing estimates which are bona fide densities.