TESTING FOR WHITE NOISE AGAINST MULTIMODAL SPECTRAL ALTERNATIVES

Abstract. The test for white noise proposed by Reschenhofer ( Biometrika 76 (1989) 629–32) is adaptive in that the form of its test statistic depends on the data. It proceeds (i) by using the real part of the finite Fourier transform of the data for the selection of the form of the test statistic and (ii) by applying the selected test statistic to the imaginary part.
In this paper it is shown that independently of using information contained in the real part to construct a test statistic for the imaginary part, information contained in the imaginary part can be used to construct a test statistic for the real part. Because of the independence of these tests, they can easily be combined.
Then a further test is proposed which again is adaptive but where both the real and imaginary parts contribute simultaneously to a single test statistic.     

Testing for Stationarity in Multivariate Locally Stationary Processes

In this article, we propose a nonparametric procedure for validating the assumption of stationarity in multivariate locally stationary time series models. We develop a bootstrap‐assisted test based on a Kolmogorov–Smirnov‐type statistic, which tracks the deviation of the time‐varying spectral density from its best stationary approximation. In contrast to all other nonparametric approaches, which have been proposed in the literature so far, the test statistic does not depend on any regularization parameters like smoothing bandwidths or a window length, which is usually required in a segmentation of the data. We additionally show how our new procedure can be used to identify the components where non‐stationarities occur and indicate possible extensions of this innovative approach. We conclude with an extensive simulation study, which shows finite‐sample properties of the new method and contains a comparison with existing approaches.     

Bootstrapping the Local Periodogram of Locally Stationary Processes

Abstract. Locally stationary processes are non‐stationary stochastic processes the second‐order structure of which varies smoothly over time. In this paper, we develop a method to bootstrap the local periodogram of a locally stationary process. Our method generates pseudo local periodogram ordinates by combining a parametric time and non‐parametric frequency domain bootstrap approach. We first fit locally a time varying autoregressive model so as to capture the essential characteristics of the underlying process. A locally calculated non‐parametric correction in the frequency domain is then used so as to improve upon the locally parametric autoregressive fit. As an application, we investigate theoretically the asymptotic properties of the bootstrap method proposed applied to the class of local spectral means, local ratio statistics and local spectral density estimators. Some simulations demonstrate the ability of our method to give accurate estimates of the quantities of interest in finite sample situations and an application to a real‐life data‐set is presented.     

Bernstein polynomial estimation of a spectral density

Abstract. We consider an application of Bernstein polynomials for estimating a spectral density of a stationary process. The resulting estimator can be interpreted as a convex combination of the (Daniell) kernel spectral density estimators at m points, the coefficients of which are probabilities of the binomial distribution bin(m ? 1, |λ|/π), λ ∈ Π ≡ [?π, π] being the frequency where the spectral density estimation is made. Several asymptotic properties are investigated under conditions of the degree m. We also discuss methods of data‐driven choice of the degree m. For a comparison with the ordinary kernel method, a Monte Carlo simulation illustrates our methodology and examines its performance in small sample.     

Testing equality of stationary autocovariances

Abstract.  This article studies tests for assessing whether two stationary and independent time series have the same dynamics – specifically, whether the autocovariances of both series coincide at all lags. Frequency domain statistics previously proposed for this purpose are reviewed. A time domain statistic is then developed and investigated. The performance of these statistics are compared. Multivariate versions of the results are constructed. The methods are applied in the analysis of temperatures and precipitations from Atlanta and Athens, Georgia. Our interest here is driven by the need to identify a good climatological reference series for a given station.     

Nonparametric Tests of Change-Points with Tapered Data

In this work we build two families of nonparametric tests using tapered data for the off-line detection of change-points in the spectral characteristics of a stationary Gaussian process. This is done using the Kolmogorov–Smirnov statistics based on integrated tapered periodograms. Convergence is obtained under the null hypothesis by means of a double indexed (frequency– time) process together with some extensions of Dirichlet and Fejer kernels. Consistency is proved using these statistics under the alternative. Then, using numerical simulations, we observe that the use of tapered data significantly improves the properties of the test, especially in the case of small samples.     

VARIANCE ESTIMATION FOR QUADRATIC STATISTICS

Abstract. A frequency domain smoothing procedure is presented which provides variance estimates for linear functions of the periodogram. The method provides consistent estimates of the true asymptotic variance, has good small-sample efficiency properties in the Gaussian case and reflects non-Gaussian structure. The relationship of the estimates to frequency domain bootstrap algorithms is discussed.     

SPECTRAL ANALYSIS WITH TAPERED DATA

Abstract. A new method based on an upper bound for spectral windows is presented for investigating the cumulants of time series statistics. Using this method two classical results are proved for tapered data. In particular, the asymptotic normality for a class of spectral estimates including estimates for the spectral function and the covariance function is proved under integrability conditions on the spectra using the method of cumulants.     

THE COMPARISON OF LEAST SQUARES AND THIRD-ORDER PERIODOGRAM PROCEDURES IN THE ESTIMATION OF BIFREQUENCY

Abstract. Given a stretch of time series values, the third-order periodogram is investigated as a criterion for use in the estimation of bifrequencies, that is of frequency triples (ω₁ω₂, ω₃) with ω₃= 2ω -ω₁ω₂. The least squares estimates of such frequencies are compared with estimates derived by maximizing the modulus of the third-order periodogram. It is found that neither estimation procedure is uniformly better than the other.     

A Wavelet-Based Test for Stationarity

We develop a test for stationarity of a time series against the alternative of a time-varying covariance structure. Using localized versions of the periodogram, we obtain empirical versions of a reasonable notion of a time-varying spectral density. Coefficients with respect to a Haar wavelet series expansion of such a time-varying periodogram are an indicator of whether there is some deviation from covariance stationarity. We propose a test based on the limit distribution of these empirical coefficients.     

Local Cross-validation for Spectrum Bandwidth Choice

We investigate an automatic method of determining a local bandwidth for non-parametric kernel spectral density estimates at a single frequency. This procedure is a modification of a cross-validation technique for global bandwidth choices, avoiding the computation of any pilot estimate based on initial bandwidths or on approximate parametric models. Only local conditions on the spectral density around the frequency of interest are assumed. We illustrate with a Monte Carlo study the performance in finite samples of the bandwidth estimates proposed.     

CONSISTENCY FOR NON-LINEAR FUNCTIONS OF THE PERIODOGRAM OF TAPERED DATA

Abstract. In this paper we investigate the merits of using a data taper in non-linear functional of the periodogram of a stationary time series. To this end, we show consistency for a general class of statistics of the form , where A(ω) is a function of bounded variation and where φ is allowed to be a non-linear function of the periodogram I_T(ω) of the tapered data. The key step in deriving our asymptotic results is an Edgeworth expansion for the finite Fourier transform of the tapered data, which do not have to follow a particular distribution (i.e. we allow for non-Gaussianity). Important applications are the estimation of , choosing φ to be a suitable transform of a given function g (see Taniguchi, On estimation of the integrals of certain functions of spectral density. J. Appl. Prob. 17 (1980). 73–83), the peak-insensitive spectrum estimator of von Sachs (Peak-insensitive nonparametric spectrum estimation. J. Time Ser. Anal. 15 (1994), 429–52), where φ is chosen to be a bounded (robustifying) σ function, and the parametric approach of Chiu (Peak-insensitive parametric spectrum estimation. Stoch. Proc. Appl. 35 (1990). 121–40) on robust estimation of the parameters of the continuous spectrum of the time series.     

REGRESSION OF SPECTRAL ESTIMATORS WITH FRACTIONALLY INTEGRATED TIME SERIES

Abstract. Assuming a normal distribution we supplement the proof of periodogram regression suggested by Geweke and Porter-Hudak ( J. Time Ser. Anal. 4 (1983) 221–38) in order to estimate and test the difference parameter of fractionally integrated autoregressive moving-average models. The procedure proposed by Kashyap and Eom ( J. Time Ser. Anal. 9 (1988) 35–41) arises as a special case and is found to be correct if the true parameter value is negative. Regression of the smoothed periodogram yields estimators for the difference parameter with much faster vanishing variance; no asymptotic distribution can be derived, however. In computer experiments we find that the smoothed periodogram regression may be superior to pure periodogram regression when we have to discriminate between autoregression and fractional integration