An ARMA Type Probability Density Estimator

Abstract

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.