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.
Estimating Graph Dimension with Cross-validated Eigenvalues
Department of Statistics, University of Wisconsin–Madison（统计学系，威斯康星大学麦迪逊分校）；Department of Mathematics, University of Wisconsin–Madison（数学系，威斯康星大学麦迪逊分校）
