CONVERGENCE-RATES IN DENSITY-ESTIMATION FOR DATA FROM INFINITE-ORDER MOVING AVERAGE PROCESSES

The effect of long-range dependence in nonparametric probability density estimation is investigated under the assumption that the observed data are a sample from a stationary, infinite-order moving average process. It is shown that to first order, the mean integrated squared error (MISE) of a kernel estimator for moving average data may be expanded as the sum of MISE of the kernel estimator for a same-size random sample, plus a term proportional to the variance of the moving average sample mean. The latter term does not depend on bandwidth, and so imposes a ceiling on the convergence rate of a kernel estimator regardless of how bandwidth is chosen. This ceiling can be quite significant in the case of long-range dependence. We show that all density estimators have the convergence rate ceiling possessed by kernel estimators. © 1990 Springer-Verlag.