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December, 1964 On the Multivariate Analysis of Weakly Stationary Stochastic Processes
L. H. Koopmans
Ann. Math. Statist. 35(4): 1765-1780 (December, 1964). DOI: 10.1214/aoms/1177700398


The existence of the class of orthogonal projections which map an arbitrary $q$-variate weakly stationary stochastic process again into a $q$-variate process contained in the span of $p(\leqq q)$ of its component processes is established. Mimicking the definitions of the partial and multiple correlation coefficients (e.g., Anderson, 1958), these projections are used to define partial and multiple coefficients of coherence, thus providing the foundation for the multivariate covariance and correlation analyses for weakly stationary processes employed in special cases by Tick (1963) and Jenkins (1963). Some of the properties of the partial and multiple correlation coefficients are established for the corresponding coefficients of coherence. In particular, formulas are established for generating these parameters iteratively. When used for the sample coefficients of coherence, these formulas provide useful methods of defining and constructing estimates of the multiple and partial coefficients of coherence from the usual estimates of the ordinary coefficient of coherence. Results due to Goodman (1963) concerning the distributions of these estimators when the process is Gaussian are indicated.


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L. H. Koopmans. "On the Multivariate Analysis of Weakly Stationary Stochastic Processes." Ann. Math. Statist. 35 (4) 1765 - 1780, December, 1964.


Published: December, 1964
First available in Project Euclid: 27 April 2007

zbMATH: 0132.13304
MathSciNet: MR170380
Digital Object Identifier: 10.1214/aoms/1177700398

Rights: Copyright © 1964 Institute of Mathematical Statistics

Vol.35 • No. 4 • December, 1964
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