Nonparametric density estimates consistent of the order of n^−1/2 in the L₁–norm

Abstract.

We introduce an approximate minimum Kolmogorov distance density estimate of a probability density f₀ on the real line and study its rate of consistency for n→∞. We define a degree of variations of a nonparametric family of densities containing the unknown f₀. If this degree is finite then the approximate minimum Kolmogorov distance estimate is consistent of the order of n^−1/2 in the L₁-norm and also in the expected L₁-norm. Comparisons with two other criteria leading to the same order of consistency are given.