  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.
