# The asymptotic properties of the multichannel autoregressive spectral estimates

The asymptotic properties of the multichannel autoregressive spectral estimates If we fit aγ-vector stationary time series using observationsx(1), ...,x(T) with AR models $$x(t) + a_k^{(T)} (1)x(t - 1) + \cdots + a_k^{(T)} (k)x(t - k) = \tilde \varepsilon (t)$$ , then the spectral densityf(λ) of {x(t)} can be estimated byf k (T) (λ)=(2π)− A k (T) (e −λ)−1∑ k (T) A k (T) (e −iλ)−⋆, where $$A_k^{(T)} (z) = \sum\limits_0^k {a_k^{(T)} (j)z^j ,\Sigma _k^{(T)} }$$ are estimates of the variance matrixΣ ofε(t), the residuals of the best linear prediction. By extending some results for the scalar case, this paper treats the asymptotic properties of the estimates in the multichannel case. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# The asymptotic properties of the multichannel autoregressive spectral estimates

, Volume 4 (1) – Jul 13, 2005
12 pages

/lp/springer-journals/the-asymptotic-properties-of-the-multichannel-autoregressive-spectral-yJtCV6FQ3O
Publisher
Springer Journals
Copyright © 1988 by Science Press, Beijing, China and Allerton Press, Inc. New York, U.S.A.
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/BF02018708
Publisher site
See Article on Publisher Site

### Abstract

If we fit aγ-vector stationary time series using observationsx(1), ...,x(T) with AR models $$x(t) + a_k^{(T)} (1)x(t - 1) + \cdots + a_k^{(T)} (k)x(t - k) = \tilde \varepsilon (t)$$ , then the spectral densityf(λ) of {x(t)} can be estimated byf k (T) (λ)=(2π)− A k (T) (e −λ)−1∑ k (T) A k (T) (e −iλ)−⋆, where $$A_k^{(T)} (z) = \sum\limits_0^k {a_k^{(T)} (j)z^j ,\Sigma _k^{(T)} }$$ are estimates of the variance matrixΣ ofε(t), the residuals of the best linear prediction. By extending some results for the scalar case, this paper treats the asymptotic properties of the estimates in the multichannel case.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jul 13, 2005

### References

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