Summary
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-sizerandom 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 thatall density estimators have the convergence rate ceiling possessed by kernel estimators.
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The research of Dr. Hart was done while he was visiting the Australian National University, and was supported in part by ONR Contract N00014-85-K-0723
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Hall, P., Hart, J.D. Convergence rates in density estimation for data from infinite-order moving average processes. Probab. Th. Rel. Fields 87, 253–274 (1990). https://doi.org/10.1007/BF01198432
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DOI: https://doi.org/10.1007/BF01198432