An ARMA type probability density estimator. (English) Zbl 0645.62049

Hart, Jeffrey D.

Ann. Stat. 16, No. 2, 842-855 (1988).

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) then the best tapered Fourier series estimator. The most interesting cases are those in which the Fourier coefficients \(\phi_ j\) are asymptotic to \(Kj^{-\rho}\) as \(j\to \infty\), where \(\rho >.\)
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 [0,\(\pi\) ], the condition \(\rho =2\) occurs whenever \(f'(0+)\neq 0\), \(f'(\pi -)=0\) and f” is square integrable.

MSC:

62G05	Nonparametric estimation
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G20	Asymptotic properties of nonparametric inference

Keywords:

generalized jackknife; regularly varying function; density estimator; rational form of an ARMA spectrum; Fourier coefficients; mean integrated squared error; best tapered Fourier series estimator