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.     

SPECTRAL DISCRIMINATION FOR TWO GROUPS OF TIME SERIES

Abstract. The problem of discriminating in the frequency domain between two groups of Gaussian stationary time series is examined. A test aimed at detecting differences between the windowed spectra of the two groups is used. The effect of windows on the resulting quadratic discriminant function is considered. Two examples taken from seismology and neurology are given.     

SPECTRAL ANALYSIS FOR AMPLITUDE-MODULATED TIME SERIES

Abstract. Let { X ( n )} be a non-observed strictly stationary process, { a ( n )} a sequence independent of { X ( n )} and Y ( n ) = a ( n ) X ( n ) the observed process. This work deals with the estimation of the spectral density function fx ( Λ ) of the process of interest, { X ( n )}, using observations of the modulated process { Y ( n )}. We obtain estimators of fx ( Λ ) for three types of modulating functions:deterministic, random independent and random correlated.     

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     

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.     

Tests for Harmonic Components in the Spectra of Categorical Time Series

A test for categorical time series is developed which is based on Fisher's test for continuous-parameter time series. Instead of using a test based on the Fourier periodog ram for spectral analysis, we utilize the Walsh–Fourier periodogram for testing purposes. We briefly explain the theory behind Walsh–Fourier analysis and some of its recent applications. Asymptotic results for the distribution of the new test statistic for Walsh–Fourier spectra are presented and compared with a simulated distribution. We also perform power studies in order to assess the detection capability of the test. In the presence of multiple peaks in the spectrum, this test tends to lose power. Therefore, we also explore several alternatives to the test for Walsh–Fourier spectra and apply all of the alternative methods to a realization of geomagnetic reversals     

RATE OF CONVERGENCE FOR LOGSPLINE SPECTRAL DENSITY ESTIMATION

Abstract. The logarithm of the spectral density function for a stationary process is approximated by polynomial splines. The approximation is chosen to maximize the expected log-likelihood based on the asymptotic properties of the periodogram. Estimates of this approximation are shown to possess the usual nonparametric rate of convergence when the number of knots suitably increases to infinity.     

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.     

LOCALLY ADAPTIVE LAG-WINDOW SPECTRAL ESTIMATION

Abstract. We propose a procedure for the locally optimal window width in nonparametric spectral estimation, minimizing the asymptotic mean square error at a fixed frequency Λ of a lag-window estimator. Our approach is based on an iterative plug-in scheme. Besides the estimation of a spectral density at a fixed frequency, e.g. at frequency Λ = 0, our procedure allows to perform nonparametric spectral estimation with variable window width which adapts to the smoothness of the true underlying density.     

BISPECTRAL ANALYSIS OF RANDOMLY SAMPLED DATA

Abstract. A continuous time series is often observed or sampled at discrete intervals. Most literature has dealt with the case when the sampling intervals are equally spaced. For irregularly sampled data, most existing literature is concerned with second-order moments or anti-aliasing spectral estimations. We study the estimation of higher-order spectral density functions with the emphasis on the bispectral estimate when the continuous time series is sampled by a random point process. Estimates under the Poisson sampling scheme are studied in detail. Asymptotic bias and covariances are obtained. In particular, it is shown explicitly how the information of the sampling process comes into play in obtaining a consistent estimate of the bispectral density function of a continuous time series. In contrast to the second-order spectral density function estimation where the Poisson sampling scheme results in a constant correction term, a consistent bispectral density function estimate results in a nonlinear correction term even in the Poisson sampling scheme. A simple simulation example is presented for illustration.     

CONTRIBUTIONS TO EVOLUTIONARY SPECTRAL THEORY

Abstract. The purpose of this paper is to discuss several fundamental issues in the theory of time-dependent spectra for univariate and multivariate non-stationary processes. The general framework is provided by Priestley's evolutionary spectral theory which is based on a family of stochastic integral representations. A particular spectral density function can be obtained from the Wold—Cramér decomposition, as illustrated by several examples. It is shown why the coherence is time invariant in the evolutionary theory and how the theory can be generalized so that the coherence becomes time dependent. Statistical estimation of the spectrum is also considered. An improved upper bound for the bias due to non-stationarity is obtained which does not rely on the characteristic width of the process. The results obtained in the paper are illustrated using time series simulated from an evolving bivariate autoregressive moving-average process of order (1, 1) with a highly time-varying coherence.     

LOGSPLINE ESTIMATION OF A POSSIBLY MIXED SPECTRAL DISTRIBUTION

Abstract. Cubic splines and indicator functions are used to estimate the spectral density function and line spectrum, respectively, for a stationary time series. A fully automatic procedure involving maximum likelihood, stepwise addition and deletion of basis functions, and the Bayes information criterion (BIC) is used to select the final model.     

EXPERIENCES WITH THE BRILLINGER SPECTRAL ESTIMATOR APPLIED TO SIMULATED IRREGULARLY OBSERVED PROCESSES

Abstract. Shannon interpolation is used to assign values from a readily simulated discrete time process to the times of a point process, simulated by Ogata's thinning technique. The result is a set of unequally spaced samples from a hypothetical continuous time process with spectrum equal to that of the discrete time process for frequencies |ω| ≤π/Δ and identically equal to zero for |ω| > π/Δ, where Δ is the discrete time step. The spectra are theoretically known both for the sampled process and for the sampling point process. We calculate Brillinger spectral estimates for examples of a process with autoregressive spectrum, sampled at the times of a Hawkes Self Exciting Point Process. The success of the Brillinger estimator is demonstrated but it is shown to have an inherently high variance. An approximate confidence interval is discussed.     

A Frequency‐Domain Test to Check Equality in Spectral Densities of Multiple Time Series With Unequal Lengths

In this paper, a new frequency‐domain test is proposed to check the equality of spectral densities of two or more stationary time series. The proposed test is able to deal with multiple independent time series of different lengths naturally, based on some regression models of log periodograms. The asymptotic null distribution of the proposed test is obtained. The consistency is shown under any fixed alternative and a sequence of local alternatives. A simulation study is conducted to examine the finite sample performance of the test. By jointly modeling all log periodograms, the test is empirically robust when multiple time series are mutually dependent to some extent. It also works well for non‐Gaussian time series. The proposed test is applied to compare several vibrational signals for damage detection of a mechanical system.     

A UNIFIED APPROACH TO CONFIDENCE BOUNDS FOR THE AUTOREGRESSIVE SPECTRAL ESTIMATOR

Abstract. Confidence bounds for the spectral density of a stationary time series are derived. A unified method begins with the autoregressive spectral estimate and produces both confidence intervals at single frequencies chosen a priori and a simultaneous confidence band for multiple a posteriori comparisons. The crux is optimization of a quadratic form subject to the constraint imposed by the F -statistic. An approximate method that may produce tighter bounds is described. The former methods are demonstrated on the Waldmeier annual mean sunspot numbers.     

SPECTRAL ANALYSIS OF A STATIONARY BIVARIATE POINT PROCESS WITH APPLICATIONS TO NEUROPHYSIOLOGICAL PROBLEMS

Abstract. In this paper we discuss the spectral analysis of a stationary bivariate point process applied to the study of a complex physiological system. An estimate of the cross-spectral density can be obtained by smoothing the modified cross-periodogram statistic. The smoothed estimate is calculated by dividing the whole length of the data into a number of disjoint subrecords. Estimates of the coherence function and the cross-intensity function follow directly from the estimate of the cross-spectral density. It is shown that the asymptotic properties of the estimate of the cross-intensity function can be improved by inserting a convergence factor in it. Examples of the estimates are illustrated by using two data sets from neurophysiological experiments and their importance is emphasized by examining the behaviour of the complex physiological system under study.     

Tests for Comparing Time‐Invariant and Time‐Varying Spectra Based on the Pearson Statistic

Two tests are proposed in this paper for comparing spectra of two univariate time series. One is a Pearson‐like statistic based only on periodograms of the compared time series and applicable for testing the equality of two time‐invariant spectra of two independent or dependent time series, with an asymptotic chi‐squared distribution under the null hypothesis. The other is based on the maximum of the Pearson‐like statistics. Not only does this test, again, depend only on periodograms but also approximately equals the maximum of a chi‐squared distribution of the same degrees of freedom under the null. It can be used to test the equality of spectra of two locally stationary time series regardless of whether they are dependent or independent. Multiple simulation examples show that both statistics achieve good performance. The proposed approach is illustrated by an application to longitudinal vibration data from a container ship.     

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.     

MULTIVARIATE WALSH-FOURIER ANALYSIS

Abstract. In this paper we establish a statistical methodology for the spectral analysis of stationary multivariate time series via the Walsh-Fourier transform. Theoretical results pertaining to the definition and estimation of the Walsh-Fourier spectral matrix and functions of that matrix including cross-spectra, coherency and phase are given. An example of the statistical techniques developed in this paper is given; in particular, the methodologies are applied to neonatal sleep data collected from a study of the effect of maternal substance use during pregnancy.     

FORECASTING TIME SERIES:A COMPARATIVE ANALYSIS OF ALTERNATIVE CLASSES OF TIME SERIES MODELS

Abstract. Performance of the state dependent model developed by Priestley is evaluated relative to that of bilinear and standard linear models using two well-known time series. The results indicate the use of broader classes of time series models beyond the conventional ARMA class is likely to lead to significant reductions in forecasting error. However, there are difficult problems relating to the identification of the order of the model, estimation of the parameters, and determination of the correct nonlinear model.