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2015 High-dimensional autocovariance matrices and optimal linear prediction
Timothy L. McMurry, Dimitris N. Politis
Electron. J. Statist. 9(1): 753-788 (2015). DOI: 10.1214/15-EJS1000

Abstract

A new methodology for optimal linear prediction of a stationary time series is introduced. Given a sample $X_{1},\ldots,X_{n}$, the optimal linear predictor of $X_{n+1}$ is $\tilde{X}_{n+1}=\phi_{1}(n)X_{n}+\phi_{2}(n)X_{n-1}+\cdots+\phi_{n}(n)X_{1}$. In practice, the coefficient vector $\phi(n)\equiv(\phi_{1}(n),\phi_{2}(n),\ldots,\phi_{n}(n))'$ is routinely truncated to its first $p$ components in order to be consistently estimated. By contrast, we employ a consistent estimator of the $n\times n$ autocovariance matrix $\Gamma_{n}$ in order to construct a consistent estimator of the optimal, full-length coefficient vector $\phi(n)$. Asymptotic convergence of the proposed predictor to the oracle is established, and finite sample simulations are provided to support the applicability of the new method. As a by-product, new insights are gained on the subject of estimating $\Gamma_{n}$ via a positive definite matrix, and four ways to impose positivity are introduced and compared. The closely related problem of spectral density estimation is also addressed.

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Timothy L. McMurry. Dimitris N. Politis. "High-dimensional autocovariance matrices and optimal linear prediction." Electron. J. Statist. 9 (1) 753 - 788, 2015. https://doi.org/10.1214/15-EJS1000

Information

Received: 1 July 2014; Published: 2015
First available in Project Euclid: 2 April 2015

zbMATH: 1309.62155
MathSciNet: MR3331856
Digital Object Identifier: 10.1214/15-EJS1000

Subjects:
Primary: 62M20
Secondary: 62M10

Keywords: Autocovariance matrix , prediction , Spectral density , time series

Rights: Copyright © 2015 The Institute of Mathematical Statistics and the Bernoulli Society

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Vol.9 • No. 1 • 2015
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