ARMA (Autoregressive Moving Average) Estimators of Probability Densities with Exponential or Regularly Varying Fourier Coefficients

摘要

Properties of a probability density estimator having the rational form of a 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 window-type Fourier series estimator. The most interesting cases are those in which the Fourier coefficients are regularly varying with index-p,p > 1/2. For example, when p=2 the asymptotic MISE of a certain ARMA estimator is only about 75% of that for the optimum window estimator. For a density f with support in 0, PI, the condition p=2 occurs whenever f'(0+) does not equal to 0, f' (pi-) =0, and f is square integrable. Keywords: Generalized jackknife; Regularly varying function.