Properties of Hermite Series Estimation of Probability Density

An unknown density function $f(x)$, its derivatives, and its characteristic function are estimated by means of Hermite functions ⱼ\. The estimates use the partial sums of series of Hermite functions with coefficients âⱼₙ = (1/n) ∑ⁿᵢ₌₁ hⱼ(Xᵢ) where X₁⋯ Xₙ represent a sequence of i.i.d. random variables with the unknown density function f. The integrated mean square rate of convergence of the pth derivative of the estimate is O(n(p/r) + (5/6r)-1). The same is true for the Fourier transform of the estimate to the characteristic function. Here the assumption is made that (x - D)ʳ f ∈ L² and $p < r$. Similar results are obtained for other conditions on f and uniform mean square convergence.