Open Access
Translator Disclaimer
June, 1988 An ARMA Type Probability Density Estimator
Jeffrey D. Hart
Ann. Statist. 16(2): 842-855 (June, 1988). DOI: 10.1214/aos/1176350839


Properties of a probability density estimator having the rational form of an ARMA spectrum are investigated. Under various conditions on the underlying density's Fourier coefficients, the ARMA estimator is shown to have asymptotically smaller mean integrated squared error (MISE) than the best tapered Fourier series estimator. The most interesting cases are those in which the Fourier coefficients $\phi_j$ are asymptotic to $Kj^{-p}$ as $j \rightarrow \infty$, where $\rho > 1/2$. For example, when $\rho = 2$ the asymptotic MISE of a certain ARMA estimator is only about 63% of that for the optimum series estimator. For a density $f$ with support in $\lbrack 0, \pi \rbrack$, the condition $\rho = 2$ occurs whenever $f'(0 +) \neq 0, f'(\pi -) = 0$ and $f"$ is square integrable.


Download Citation

Jeffrey D. Hart. "An ARMA Type Probability Density Estimator." Ann. Statist. 16 (2) 842 - 855, June, 1988.


Published: June, 1988
First available in Project Euclid: 12 April 2007

zbMATH: 0645.62049
MathSciNet: MR947581
Digital Object Identifier: 10.1214/aos/1176350839

Primary: 62G05
Secondary: 62G20, 62P10

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 2 • June, 1988
Back to Top