Estimation and deconvolution when the transfer function has zeros
type: Item
instance of: scholarly article
title: Estimation and deconvolution when the transfer function has zeros
zbMATH Open document ID: 0668.62071
DOI: 10.1007/BF01076289
author: Keh-Shin Lii
author: Murray Rosenblatt
published in: Journal of Theoretical Probability
publication date: 1988-01-01T00:00:00Z
review text: The authors study the problem of identifying the transfer function of a non-Gaussian linear process and of using a known or estimated transfer function to estimate the noise-innovations. That is, if \(X_ t=\sum^{\infty}_{j=-\infty}\alpha_ j\epsilon_{t-j}\) are observed random variables for \(t\in \{-n_ 2,...,n_ 1\}\), where \(0\leq n_ 1,n_ 2\leq \infty\), and where the \(\epsilon_ t's\) are independent and identically distributed random variables with mean 0 and variance 1, then it is desired to estimate the transfer function \[ \alpha (e^{- i\lambda})\equiv \sum_{j}\alpha_ je^{-ij\lambda} \] as well as the individual random variables \(\epsilon_ t\). After two simple examples illustrating the optimal order of estimation of \(\epsilon_ 0\) when \(\alpha\) (\(\cdot)\) is known, the authors rely on their previous joint work [Ann. Stat. 10, 1195-1208 (1982; Zbl 0512.62090)], to show under some general conditions on \(\alpha\) (\(\cdot)\), that if \(\{\epsilon_ t\}\) is non-Gaussian, \(\alpha (e^{-i\lambda})\) can be identified from \(\{X_ t: t\in {\mathbb{Z}}\}\) up to a factor \(\pm e^{im\lambda}\) for any integer m. They next show how to estimate the transfer function \(\alpha\) (\(\cdot)\), in settings including the possibility that \(\alpha\) (\(\cdot)\) has zeros. Using such consistent estimates of \(\alpha\) (\(\cdot)\), they conclude by giving a procedure utilizing the spectral representation of the (not necessarily stationary) process \(X_ t\) to provide consistent estimates of \(\{\epsilon_ t\}\). The estimation procedures are illustrated for some simulated linear process data.
Mathematics Subject Classification ID: 62M99
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 93E10
zbMATH DE Number: 4094633
zbMATH Keywords: deconvolution
zbMATH Keywords: unit circle
zbMATH Keywords: spectral density
zbMATH Keywords: asymptotics
zbMATH Keywords: bispectrum
zbMATH Keywords: identifiability
zbMATH Keywords: transfer function
zbMATH Keywords: non-Gaussian linear process
zbMATH Keywords: noise- innovations
zbMATH Keywords: optimal order of estimation
zbMATH Keywords: zeros
zbMATH Keywords: consistent estimates
zbMATH Keywords: spectral representation
reviewed by: Eric V. Slud
MaRDI profile type: Publication
cites work: Robust identification of a nonminimum phase system: Blind adjustment of a linear equalizer in data communications
cites work: Nonminimum phase non-Gaussian deconvolution
cites work: Deconvolution and estimation of transfer function phase and coefficients for nongaussian linear processes
cites work: Deconvolution of non-Gaussian linear processes with vanishing spectral values
full work available at URL: https://doi.org/10.1007/bf01076289
OpenAlex ID: W2061804622
description: scientific article; zbMATH DE number 4094633