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May, 1975 Rate of Strong Consistency of Two Nonparametric Density Estimators
B. B. Winter
Ann. Statist. 3(3): 759-766 (May, 1975). DOI: 10.1214/aos/1176343142


Two nonparametric density estimators, based on Fourier series and the Fejer kernel, are presented. One of them $(\tilde{f}_N)$ is appropriate when the unknown density $f$ vanishes outside a known bounded interval; the other $(f_N^\sharp)$ is applicable without any assumptions about the support of $f$. The estimator $f_N^\sharp$ is of the type studied by Watson and Leadbetter (Sankhya A 26, 1964) and $\tilde{f}_N$ is almost of that type: both may be said to be of the "$\delta$-sequence type". If $f$ satisfies a certain Lipschitz condition at $x$ and the "number of harmonics" used in $\tilde{f}_N$ is asymptotically proportional to $N^\frac{1}{3}$, and $\rho_N/\log N \rightarrow \infty$, then $(N^{\frac{1}{3}}/\rho_N) \cdot |\tilde{f}_N(x) - f(x)| \rightarrow 0$ a.s.; a similar result holds for $f_N^\sharp$.


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B. B. Winter. "Rate of Strong Consistency of Two Nonparametric Density Estimators." Ann. Statist. 3 (3) 759 - 766, May, 1975.


Published: May, 1975
First available in Project Euclid: 12 April 2007

zbMATH: 0308.62027
MathSciNet: MR375612
Digital Object Identifier: 10.1214/aos/1176343142

Primary: 60F15
Secondary: 42A08 , 42A20 , 62G05 , 65D10

Keywords: Fourier series , Nonparametric density estimation , rate of convergence , strong consistency

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 3 • May, 1975
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