Hermite Series Estimates of a Probability Density and its Derivatives

The following estimate of the pth derivative of a probability density function is examined: [Sigma]k = 0Nâkhk(x), where hk is the kth Hermite function and âk = ((-1)p/n)[Sigma]i = 1n hk(p)(Xi) is calculated from a sequence X1,..., Xn of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n-[alpha]) and O(n-[alpha] log n), respectively, where [alpha] = 2(r - p)/(2r + 1). Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n-[beta]) and O(n-[beta]/2 log n), respectively, where [beta] = (2(r - p) - 1)/(2r + 1).