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November, 1977 Properties of Hermite Series Estimation of Probability Density
Gilbert G. Walter
Ann. Statist. 5(6): 1258-1264 (November, 1977). DOI: 10.1214/aos/1176344013


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.


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Gilbert G. Walter. "Properties of Hermite Series Estimation of Probability Density." Ann. Statist. 5 (6) 1258 - 1264, November, 1977.


Published: November, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0375.62041
MathSciNet: MR448692
Digital Object Identifier: 10.1214/aos/1176344013

Primary: 62G05

Keywords: Density estimates , ‎Hermite functions

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 6 • November, 1977
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