## The Annals of Statistics

### Properties of Hermite Series Estimation of Probability Density

Gilbert G. Walter

#### Abstract

An unknown density function $f(x)$, its derivatives, and its characteristic function are estimated by means of Hermite functions $\{h_j\}$. The estimates use the partial sums of series of Hermite functions with coefficients $\hat{a}_{jn} = (1/n) \sum^n_{i=1} h_j(X_i)$ where $X_1\cdots X_n$ represent a sequence of i.i.d. random variables with the unknown density function $f$. The integrated mean square rate of convergence of the $p$th 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 $(x - D)^r f \in L^2$ and $p < r$. Similar results are obtained for other conditions on $f$ and uniform mean square convergence.

#### Article information

Source
Ann. Statist., Volume 5, Number 6 (1977), 1258-1264.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176344013

Digital Object Identifier
doi:10.1214/aos/1176344013

Mathematical Reviews number (MathSciNet)
MR448692

Zentralblatt MATH identifier
0375.62041

JSTOR