Asymptotic properties of kernel density estimators when applying importance sampling

We study asymptotic properties of kernel estimators of an unknown density when applying importance sampling (IS). In particular, we provide conditions under which the estimators are consistent, both pointwise and uniformly, and are asymptotically normal. We also study the optimal bandwidth for minimizing the asymptotic mean square error (MSE) at a single point and the asymptotic mean integrated square error (MISE). We show that IS can improve the asymptotic MSE at a single point, but IS always increases the asymptotic MISE. We also give conditions ensuring the consistency of an IS kernel estimator of the sparsity function, which is the inverse of the density evaluated at a quantile. This is useful for constructing a confidence interval for a quantile when applying IS. We also provide conditions under which the IS kernel estimator of the sparsity function is asymptotically normal. We provide some empirical results from experiments with a small model.