Signal extraction from nonstationary time series
type: Item
instance of: scholarly article
title: Signal extraction from nonstationary time series
zbMATH Open document ID: 0542.62081
author: N. E. Zubov
published in: The Annals of Statistics
publication date: 1984-01-01T00:00:00Z
review text: The paper deals with the extraction problem \(z_ t=s_ t+n_ t\), where \(z_ t\) is an observable time series, \(s_ t\) is an unobservable signal which is to be estimated and \(n_ t\) is an unobservable noise. The considered extraction problem is nonstationary so that either \(s_ t\) or \(n_ t\) or both are nonstationary. Certain assumptions on the generation of the time series \(z_ t\), \(s_ t\) and \(n_ t\) and their starting values are considered in the paper. E.g. it is assumed that processes \(s_ t\) and \(n_ t\) can be modelled by means of \(\delta_ s(B)s_ t=u_ t\) and \(\delta_ n(B)=v_ t\), where \(u_ t\) and \(v_ t\) are independent stationary time series and \(\delta_ s(B)\) and \(\delta_ n(B)\) are polynomials whose zeros lie on or inside the unit circle (it implies the form of the model for \(z_ t).\) Two sets of alternative assumptions regarding the generation of \(z_ t\), \(s_ t\) and \(n_ t\) including starting values are considered in the paper. These assumptions play an important role in the extraction and should not be in any case ignored. The paper gives the solution of the extraction problem in such a way that the exact formulae for the conditional expectation \(E(s_ t| \{z_ t\})\) and for the conditional variance \(var(s_ t| \{z_ t\})\) are derived. Extensions to signal extraction with a finite number of observations, to the non Gaussian case and to the multivariate case are discussed. No practical experiences and applications are reported in the paper.
Mathematics Subject Classification ID: 62M15
Mathematics Subject Classification ID: 60G35
zbMATH DE Number: 3862287
zbMATH Keywords: nonstationary time series
zbMATH Keywords: conditional variance
zbMATH Keywords: starting values
zbMATH Keywords: conditional expectation
zbMATH Keywords: signal extraction
zbMATH Keywords: non Gaussian case
zbMATH Keywords: multivariate case
reviewed by: Tomáš Cipra
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176346512
OpenAlex ID: W1966696278
DOI: 10.1214/AOS/1176346512
description: scientific article; zbMATH DE number 3862287