Neumann, M. H. (1996) Spectral density estimation via nonlinear wavelet methods for stationary non-gaussian time series. J. Time Ser. Anal., to appear.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to the methodology available for estimating the density and the hazard rate from randomly censored data. The problem of estimating the log spectrum of a stationary Gaussian time series by wavelet thresholding techniques has been addressed by Gao [43] in his thesis. More generally Neumann [68] applied the thresholding procedure in the framework of spectral density estimation for a stationary, possibly non Gaussian time series. It has also been applied by von Sachs and Schneider [85] to the periodogram of a locally stationary process for the estimation of its evolutionary spectrum. A ....

Neumann, M. H. (1994). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian series. Technical report 99, Institute for applied and stochastic analysis, Berlin.

....we assume that (u) 0, i.e. we do not treat the problem of estimating the mean of the time series. Also, here we restrict ourselves to Gaussian processes mainly for reasons of using proof techniques which are somewhat similar to those of [9] But we like to note that using techniques as in [14] will enable us to derive similar results also for the non Gaussian situation. In this model the smoothness of A in u restricts the departure from stationarity and ensures the locally stationary behavior of the process. It also allows to define a unique underlying time varying spectrum of X ....

.... particular, can be performed by applying non linear thresholding techniques which were introduced by Donoho et al. see [6] 7] e.g. First theoretical investigations in the context of spectral density estimation for stationary time series can be found in [9] for Gaussian time series and in [14] for more general stationary processes. Basically, these non linear techniques are important to benefit also on the empirical side of estimation from a particular nice property of wavelets: They deliver sparse representations for curves with inhomogeneously distributed regularity. This can often ....

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M. H. Neumann, Spectral density estimation via nonlinear wavelet methods for stationary non--Gaussian time series, Preprint, WIAS Berlin, 1994.

....specially smoothed variances. ffl The smoothing is done using simple wavelet thresholding ideas. Note however, that the noise in empirical covariance estimates is neither Gaussian nor homoscedastic. By exploiting considerable experience of statisticians with wavelet thresholding of periodograms [Gao, Neu, NvS] we know that results roughly comparable to the Gaussian case are available in certain non Gaussian settings. ffl The thresholding that is used above must be rather specially chosen in order for our proofs to work. It is of order Const log(T ) p n where n is the number of realizations. Note ....

Neumann, M. H. (1996). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. J. Time Ser. Anal. 17, 601--633.

....in our model which is uniform absolute summability of the autocovariances of the data, being a weak assumption for time series data. Moreover it allows us to derive asymptotic normality of the b fi jk in a uniform way for an increasing number of coefficients by the use of a technique found in Neumann (1996), Lemma 3.1, and also in Neumann and von Sachs (1995) and Dahlhaus et al. 1998) Alternatively, appropriate moment conditions and mixing could be used to derive our results. However, A2) is a very convenient condition which works uniformly in order to deal with a wide range of correlated, ....

....is fulfilled asymptotically as by (A3)b) we assume that inf u2[0;1] f(u; 0) is uniformly bounded away from zero. It is, however, no problem to deal with those coefficients which violate that assumption, as is indicated in Section 2. 2 of Neumann and von Sachs (1995) and is covered in detail in Neumann (1996), Section 4. Lemma 2.2(ii) indicates how to (theoretically) choose the threshold in order for the main theorem (in Section 2.4) to hold. Some ideas, though still preliminary, on automatic threshold choice, more adapted to the situation of heteroskedasticity in the variance of the empirical wavelet ....

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Neumann, M. H. (1996). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. J. Time Ser. Anal. 17, 601--633.

.... can serve as adaptive time frequency smoothers of Fourier based (periodogram based) estimators, as shown in [vSS] This approach on segmented periodograms of locally stationary time series, in the above mentioned sense of nonparametric curve estimation, generalizes the work of [Gao] and of [Neu], which treat the stationary situation. Full adaptivity which goes beyond the question of an appropriate local time frequency smoother, is addressed in [NvS] Here, by applying non linear thresholding to the empirical 2 d wavelet coefficients of a fully localized periodogram (a kind of Wigner ....

....for heteroskedastic correlated noise) achieve the near minimax L 2 rate for the risk of estimation, as given below in Theorem 3.1. First theoretical investigations in the context of spectral density estimation for stationary time series can be found in [Gao] for Gaussian time series and in [Neu] for more general stationary processes. Generalizations to time varying spectra of locally stationary processes are covered by our work, which is [vSS] and [NvS] for estimation and [vSNeu] for testing the hypothesis of stationarity. A related approach of estimating time varying autoregressive ....

Neumann, M. H. (1996) Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. J. Time Ser. Anal. 17, 601--633.

....which were introduced by Donoho et al. see, e.g, Do] DJKP] and which non linearly threshold the empirical coefficients b d jk . First theoretical investigations in the context of spectral density estimation for stationary time series can be found in [Gao] for Gaussian time series and in [Neu] for more general stationary processes. The general idea of non linear thresholding is to set to zero, by the now common rules of soft or hard thresholders, those empirical wavelet coefficients which do not exceed a suitable chosen threshold = T , where T denotes the observed sample size. For ....

Neumann, M. H. (1996) Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. J. Time Ser. Anal. to appear.

....A subordinate task would then be to detect and estimate the precise location of these break points and here wavelet methods again prove useful. Finally, for this little review, we mention that wavelet shrinkage can be used for the estimation of spectral densities of stationary time series (Neumann, 1996; Gao, 1997) and of time varying spectra using localised periodogram based estimators (von Sachs and Schneider, 1996; Neumann and von Sachs, 1997) Deterministic time series. A completely different use of wavelets in statistical time series analysis was motivated by how wavelets originally ....

Neumann, M. 1996. Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. J. Time Ser. Anal., 17, 601--633.

.... been used in the estimation of densities (Hall and Patil (1993) Donoho (1993) Donoho et al. (1996) in nonparametric regression (Nason (1995) Hall and Patil (1993) Donoho and Johnstone (1995) Donoho et al. (1995) and in the estimation of spectral densities (Gao (1993, 1997) Moulin (1994) Neumann (1996)) Brillinger (1994, 1996) used wavelets to detect change of levels in hydrologic series and develops mean level function estimations based on wavelets. Nason and Silverman (1994) developed a software for the computation of the discrete wavelet transform. Bruce and Gao (1996) presented some ....

Neumann, M.H. (1996). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. Journal of Time Series Analysis, 17, 601-633.

....compression, turbulence, statistics, numerical analysis, etc. Good mathematical references are Chui (1992) and Daubechies (1992) References for uses of wavelets in statistics are Donoho and Johnstone (1990) Donoho (1993) Nason (1994) For uses in time series analysis see Brillinger (1994a, b) Neumann (1996), von Sachs and Schneider (1996) von Sachs, Nason and Kroisandt (1996) Gao (1997) Neumann and von Sachs (1997) and Chiann and Morettin (1998) In section 2 we give the basic ideas on locally stationary processes and the concept of evolutionary spectra. Section 3 presents two dimensional ....

Neumann, M.H. (1996). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. Journal of Time Series Analysis, 17, 601-633.

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Neumann, M. H. (1994). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. J. Time Ser. Anal., to appear.

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Neumann, M.H. (1996b). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. J. Time Ser. Anal.

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Neumann, M. H. (1994). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series, Manuscript.

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Neumann, M. H. (1996). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series, J. Time Ser. Anal. 17, 601--633.

.... in non Gaussian, and also some dependent cases, through an asymptotic risk equivalence to the Gaussian case, in a wide variety of settings, as noted in Neumann (1995) 28] Details are given for regression by Neumann and Spokoiny (1995) 29] and for spectral density estimation in Neumann (1994) [27]. More serious departures from our assumptions include heteroscedasticity and a nonequally spaced design. While wavelet methods can be adapted for these cases, study of these using exact risk tools is beyond the scope of this paper. 1.2 Wavelet bases The Fourier basis can effectively compress ....

Neumann, M. H. (1994). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. Preprint No. 99, Institut fur Angewandte Analysis und Stochastik, Berlin.

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Neumann, M. H. (1996) Spectral density estimation via nonlinear wavelet methods for stationary non-gaussian time series. J. Time Ser. Anal., to appear.

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Neumann, M.H.: Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. Journal of Time Series Analysis 17(1996), 601-633.

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Neumann, M.H. (1996). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. J. Time Series Anal., 17, 601-633.

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Neumann, M.H. (1996). Spectral density estimation via nonlinear wavelet methods for stationary nonGaussian time series. J. Time Series Anal., 17, 601-633.

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