Lower Bounds for Nonparametric Density Estimation Rates

In Wegman's paper [5] on nonparametric density estimation, he states that it would be interesting to show that there is no density estimator which has mean integrated square rate better than O(n-1). The object of this note is to prove such a result, making no arbitrary assumptions about the specific form of the estimator. This proof is given in Section 2. Our method applies to some other measures of error, as we point out in Section 3.