Consistency of multivariate density estimators using random bandwidths

Asymptotic properties of a kernel density estimator using a random bandwidth are difficult to establish. Under some assumptions we prove the L-1 consistency of a class of multivariate kernel density estimators using different bandwidth vector selectors. The expected L-1 distance between such an estimator and the density is also shown to converge to zero. Our results hold even when the marginal densities are heavy-tailed. As a special case, we propose a simple estimator that depends on only one parameter, irrespective of the dimension. Its L-1 distance from the density goes to zero, exponentially. Simulations suggest that this estimator performs well in terms of the integrated squared error as well