Estimation and deconvolution when the transfer function has zeros
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.
scientific article; zbMATH DE number 4094633