Pointwise consistency of the Hermite series density estimate
The Hermite series estimate of a density \(f\in L_ p\), \(p>1\), converges in the mean square to f(x) for almost all \(x\in {\mathbb{R}}\), if N(n)\(\to \infty\) and \(N(n)/n^ 2\to 0\) as \(n\to \infty\), where N is the number of the Hermite functions in the estimate while n is the number of observations. Moreover, the mean square and weak consistency are equivalent. For m times differentiable densities, the mean squares convergence rate is \(O(n^{-(2m-1)/2m})\). Results for complete convergence are also given.
scientific article; zbMATH DE number 3919563
scientific article