Estimating invariant laws of linear processes by U-statistics

• 成果类型: Article
• 署名作者: Schick, A; Wefelmeyer, W
• 署名单位: State University of New York (SUNY) System; Binghamton University, SUNY; University of Cologne
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 2004
• 页码: 603-632
• 关键词: time-series models; Adaptive estimation; nonparametric density; efficient estimation; marginal density; markov-chains

摘要

Suppose we observe an invertible linear process with independent mean-zero innovations and with coefficients depending on a finite-dimensional parameter, and we want to estimate the expectation of some function under the stationary distribution of the process. The usual estimator would be the empirical estimator. It can be improved using the fact that the innovations are centered. We construct an even better estimator using the representation of the observations as infinite-order moving averages of the innovations. Then the expectation of the function under the stationary distribution can be written as the expectation under the distribution of an infinite series in terms of the innovations, and it can be estimated by a U-statistic of increasing order (also called an infinite-order U-statistic) in terms of the estimated innovations. The estimator can be further improved using the fact that the innovations are centered. This improved estimator is optimal if the coefficients of the linear process are estimated optimally. The variance reduction of our estimator over the empirical estimator can be considerable.