Regression spectrum

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The spectrum of a stochastic process occurring in the regression scheme for a stationary time series. Thus, let a stochastic process which is observable for be represented in the form


where is a stationary stochastic process with , and let the mean value be expressed in the form of a linear regression


where , , are known regression vectors and are unknown regression coefficients (cf. Regression coefficient). Let be the spectral distribution function of the regression vectors (cf. Spectral analysis of a stationary stochastic process). The regression spectrum for is the set of all such that for any interval containing , .

The regression spectrum plays an important role in problems of estimating the regression coefficients in the scheme (1)–(2). For example, the elements of a regression spectrum can be used to express a necessary and sufficient condition for the asymptotic efficiency of an estimator for by the method of least squares.


[1] U. Grenander, M. Rosenblatt, "Statistical analysis of stationary time series" , Wiley (1957)
How to Cite This Entry:
Regression spectrum. A.V. Prokhorov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098