Locally adaptive fitting of semiparametric models to nonstationary time series.
type: Item
instance of: scholarly article
title: Locally adaptive fitting of semiparametric models to nonstationary time series.
zbMATH Open document ID: 1046.62032
DOI: 10.1016/S0304-4149(00)00060-0
author: Michael H. Neumann
author: R. Dahlhaus
published in: Stochastic Processes and their Applications
publication date: 2004-09-22T00:00:00Z
review text: A sequence of stochastic processes \(\{X_ {t,T}\}_ {t=1,\dots , T}\) is called locally stationary with transfer function \(A^ {0}\) and (smooth) trend \(\mu \) if \[ X_ {t,T}=\mu (t/T)+ \int _ {-\pi }^ {\pi } A_ {t,T}^ {0}(\omega ) \exp \{i\omega t\}\, d\xi (\omega ) \] under the conditions that (i) \(\xi (\omega )\) is a process with orthonormal increments and (ii) there exist a smooth function \(A(u,\omega )\) and a constant \(K\) such that \[ \sup _ {t, \omega } | A_ {t,T}^ {0}(\omega ) - A (t/T, \omega)| \leq KT^ {-1}. \] The function \(f(u,\omega )=| A(u, \omega )| ^ {2}\) is called the time-varying spectral density of \(\{X_ {t,T}\}\). It is assumed that \(f(u, \omega )\) depends on a \(p\)-dimensional parameter function \(\theta (u)\), \(u\in [0,1]\). The mean function is estimated by a usual kernel estimator whereas locally adaptive estimation by wavelet thresholding is applied to the components of \(\theta \). The fundamental tool in this procedure is the preperiodogram \(J_ T(t/T, \lambda)\), which uses the single product \(X_ {[t+0.5-k/2],T} X_ {[t+0.5+k/2],T}\) as a kind of local estimator of the covariance of lag \(k\) at time \(t\). The preperiodogram is connected with the ordinary periodogram \(I_ T (\lambda )\) by the formula \(I_ T (\lambda )= T^{-1} \sum _ {t=1}^ T J_ T(t/T,\lambda)\). It is proved that the proposed estimators converge to the corresponding target quantities with a rate that matches the optimal rate in some smoothness classes of the parameters.
Mathematics Subject Classification ID: 62G07
Mathematics Subject Classification ID: 62M15
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 65T60
zbMATH DE Number: 2102370
zbMATH Keywords: locally stationary processes
zbMATH Keywords: nonlinear thresholding
zbMATH Keywords: nonparametric curve estimation
zbMATH Keywords: preperiodogram
zbMATH Keywords: time series
zbMATH Keywords: wavelet estimators
reviewed by: Jiří Anděl
MaRDI profile type: Publication
cites work: Q3865207
cites work: Q4112938
cites work: An exponential model for the spectrum of a scalar time series
cites work: Wavelets on the interval and fast wavelet transforms
cites work: On the Kullback-Leibler information divergence of locally stationary processes
cites work: Fitting time series models to nonstationary processes
cites work: A likelihood approximation for locally stationary processes
cites work: Nonlinear wavelet estimation of time-varying autoregressive processes
cites work: Orthonormal bases of compactly supported wavelets
cites work: Ideal spatial adaptation by wavelet shrinkage
cites work: Minimax estimation via wavelet shrinkage
cites work: Q3851439
cites work: Performance of wavelet methods for functions with many discontinuities
cites work: Formulae for mean integrated squared error of nonlinear wavelet-based density estimators
cites work: Ondelettes sur l'intervalle. (Wavelets on the interval)
cites work: SPECTRAL DENSITY ESTIMATION VIA NONLINEAR WAVELET METHODS FOR STATIONARY NON-GAUSSIAN TIME SERIES
cites work: Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra
cites work: Large deviations for estimates of spectrum of stationary series
cites work: A general lemma on probabilities of large deviations
cites work: Wavelet smoothing of evolutionary spectra by nonlinear thresholding
cites work: Q4289674
cites work: Estimates of semiinvariants and centered moments of stochastic processes with mixing. I
cites work: Q3363781
cites work: A functional large deviations principle for quadratic forms of Gaussian stationary processes
full work available at URL: s0304-4149(00)00060-0
OpenAlex ID: W2148847609
Wikidata QID: Q127176372
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description: scientific article; zbMATH DE number 2102370