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April 2004 Estimating invariant laws of linear processesby U-statistics
Anton Schick, Wolfgang Wefelmeyer
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Ann. Statist. 32(2): 603-632 (April 2004). DOI: 10.1214/009053604000000111


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.


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Anton Schick. Wolfgang Wefelmeyer. "Estimating invariant laws of linear processesby U-statistics." Ann. Statist. 32 (2) 603 - 632, April 2004.


Published: April 2004
First available in Project Euclid: 28 April 2004

zbMATH: 1091.62066
MathSciNet: MR2060171
Digital Object Identifier: 10.1214/009053604000000111

Primary: 62M09 , 62M10
Secondary: 62G05 , 62G20

Keywords: constrained model , Efficient estimator , least dispersed estimator , Plug-in estimator , Semiparametric model , time series

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 2 • April 2004
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