Namespaces
Variants
Actions

Regression spectrum

From Encyclopedia of Mathematics
Revision as of 17:14, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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

(1)

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

(2)

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.

References

[1] U. Grenander, M. Rosenblatt, "Statistical analysis of stationary time series" , Wiley (1957)
How to Cite This Entry:
Regression spectrum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regression_spectrum&oldid=15954
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article