Weak conditions for shrinking multivariate nonparametric density estimators

Nonparametric density estimators on R K may fail to be consistent when the sample size n does not grow fast enough relative to reduction in smoothing. For example a Gaussian kernel estimator with bandwidths proportional to some sequence h n is not consistent if n h n K fails to diverge to infinity. The paper studies shrinkage estimators in this scenario and shows that we can still meaningfully use–in a sense to be specified in the paper–a nonparametric density estimator in high dimensions, even when it is not asymptotically consistent. Due to the “curse of dimensionality”, this framework is quite relevant to many practical problems. In this context, unlike other studies, the reason to shrink towards a possibly misspecified low dimensional parametric estimator is not to improve on the bias, but to reduce the estimation error.