Statistical estimation of the multivariate parameter of spectral density. I
type: Item
instance of: scholarly article
title: Statistical estimation of the multivariate parameter of spectral density. I
zbMATH Open document ID: 0678.62091
DOI: 10.1007/BF01027188
author: R. Yu. Bentkus
author: R. R. Malyukyavichyus
published in: Lithuanian Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: Let \(X_ t\), \(t=...,-1,0,1,...\), be a real random sequence with mean E \(X_ t=0\) and spectral density (s.d.) \(f(\lambda)\), which is stationary in the broad sense. That is \[ X_ t=\int^{\pi}_{- \pi}e^{it\lambda} d\zeta (\lambda),\quad t=...,-1,0,1,..., \] where \(\zeta(\lambda)\), \(-\pi\leq \lambda \leq \pi\), is a stochastic process with uncorrelated increments such that E \(d\zeta(\lambda)=0\) and \(E| d\zeta (\lambda)|^ 2=f(\lambda)d\lambda.\) Let us assume that we are interested in the values of the s.d. \(f(\lambda)\) on a fixed interval \(\lambda_ 1\leq \lambda \leq \lambda_ 2\), where \(0\leq \lambda_ 1<\lambda_ 2\leq \pi\) and that these values are uniquely determined by the multivariate parameter \(\theta =(\theta_ 1,\theta_ 2,...,\theta_ n)\in R^ n\) according to a rule which is known to us. Our problem will be to estimate this parameter from the ``observation'' \(X_ 1,X_ 2,...,X_ N\). Since the s.d. is an even function, \(f(-\lambda)=f(\lambda)\), it is sufficient to study it for \(\lambda\geq 0\) and so we restrict ourselves to the case \([\lambda_ 1,\lambda_ 2]\subseteq [0,\pi]\).
Mathematics Subject Classification ID: 62M15
zbMATH DE Number: 4111863
zbMATH Keywords: moments
zbMATH Keywords: semi-invariants
zbMATH Keywords: asymptotic properties of Whittle statistic
zbMATH Keywords: limiting Gaussian distribution
zbMATH Keywords: stationary in the broad sense
zbMATH Keywords: stochastic process with uncorrelated increments
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4111863