ROBUST REGRESSION AND INTERPOLATION FOR TIME SERIES

Abstract. In this paper we shall consider the interpolation problem under the condition that the spectral density of a stationary process concerned is vaguely known (i.e., Huber's ε -contaminated model). Then we can get a minimax robust interpolator for the class of spectral densities S ={ g:g(x)=(1-ε)f(x)+εh(x)ε Ar D_o, 0<ε<1}, where f(x) is a known spectral density and D ₀ is a certain class of spectral densities. Also we shall consider the time series regression problem under the condition that the residual spectral density is vaguely known. Then we can get a minimax robust regression coefficient estimate for the class of the residual spectral densities S .     

NON-PARAMETRIC APPROACH IN TIME SERIES ANALYSIS

Abstract. Suppose that { X _t } is a Gaussian stationary process with spectral density f ( Λ ). In this paper we consider the testing problem , where K (Λ) is an appropriate function and c is a given constant. This test setting is unexpectedly wide and can be applied to many problems in time series. For this problem we propose a test based on K { f _n ( Λ )} dΛ where f _n ( Λ ) is a non-parametric spectral estimator of f ( Λ ), and we evaluate the asymptotic power under a sequence of non-parametric contiguous alternatives. We compare the asymptotic power of our test with the other and show some good properties of our test. It is also shown that our testing problem can be applied to testing for independence. Finally some numerical studies are given for a sequence of exponential spectral alternatives. They confirm the theoretical results and the goodness of our test.     

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.     

ON VECTOR AUTOCORRELATIONS AND GENERALIZED SECOND-ORDER FUNCTIONS FOR TIME SERIES

Abstract. Some structural properties of certain vector generalizations of second-order functions of a stationary stochastic process based on determinantial functions of autocovariances are discussed. In particular, a generalized autocovariance function which retains all properties of the ordinary autocovariance function is considered and the linear dependence structure of certain scalar stochastic processes associated with this function is investigated. Properties of the normalized function are discussed and a duality property is found, according to which this function also generalizes in a natural way the ordinary partial autocorrelation function of stochastic processes.     

TESTING EQUALITY OF VARIANCES FOR PAIRED TIME SERIES

Abstract. We develop a simple test for testing equality of variances for paired stationary Gaussian time series. The test statistic is a modified z statistic. It is based on the periodograms of the two series and consistent estimation of the difference between the two spectral densities. Simulations illustrate the validity of the asymptotic results for finite samples. An application to EEG data is discussed.     

SPECTRAL DENSITY ESTIMATION VIA NONLINEAR WAVELET METHODS FOR STATIONARY NON-GAUSSIAN TIME SERIES

Abstract. In the present paper we consider nonlinear wavelet estimators of the spectral density f of a zero mean, not necessarily Gaussian, stochastic process, which is stationary in the wide sense. It is known in the case of Gaussian regression that these estimators outperform traditional linear methods if the degree of smoothness of the regression function varies considerably over the interval of interest. Such methods are based on a nonlinear treatment of empirical coefficients that arise from an orthonormal series expansion according to a wavelet basis.
The main goal of this paper is to transfer these methods to spectral density estimation. This is done by showing the asymptotic normality of certain empirical coefficients based on the tapered periodogram. Using these results we can show the risk equivalence to the Gaussian case for monotone estimators based on such empirical coefficients. The resulting estimator of f keeps all interesting properties such as high spatial adaptivity that are already known for wavelet estimators in the case of Gaussian regression.
It turns out that appropriately tuned versions of this estimator attain the optimal uniform rate of convergence of their L ₂ risk in a wide variety of Besov smoothness classes, including classes where linear estimators (kernel, spline) are not able to attain this rate. Some simulations indicate the usefulness of the new method in cases of high spatial inhomogeneity.     

SUFFICIENT STATISTICS FOR STATIONARY DISCRETE-TIME GAUSSIAN RANDOM PROCESSES

Abstract. It is known that the distribution of N samples of a stationary Gaussian autoregressive process admits a sufficient statistic whose dimension is independent of N . We show that this property depends not on the absence of spectral zeros in autoregressive models, but rather on the fact that the class of models has a fixed set of spectral zeros.     

TIME SERIES RESIDUALS WITH APPLICATION TO PROBABILITY DENSITY ESTIMATION

Abstract. A linear stationary and invertible process y _t models the second-order properties of T observations on a discrete time series, up to finitely many unknown parameters θ. Two estimators of the residuals or innovations ɛ _t of y _t are presented, based on a θ estimator which is root- T consistent with respect to a wide class of ɛ _t distributions, such as a Gaussian estimator. One sets unobserved y _t equal to their mean, the other treats y _t as a circulant and may be best computed via two passes of the fast Fourier transform. The convergence of both estimators to ɛ _t is investigated. We apply the estimated ɛ _t to estimate the probability density function of ɛ _t . Kernel density estimators are shown to converge uniformly in probability to the true density. A new sub-class of linear time series models is motivated.     

A PARAMETER‐DRIVEN LOGIT REGRESSION MODEL FOR BINARY TIME SERIES

We consider a parameter‐driven regression model for binary time series, where serial dependence is introduced by an autocorrelated latent process incorporated into the logit link function. Unlike in the case of parameter‐driven Poisson log‐linear or negative binomial logit regression model studied in the literature for time series of counts, generalized linear model (GLM) estimation of the regression coefficient vector, which suppresses the latent process and maximizes the corresponding pseudo‐likelihood, cannot produce a consistent estimator. As a remedial measure, in this article, we propose a modified GLM estimation procedure and show that the resulting estimator is consistent and asymptotically normal. Moreover, we develop two procedures for estimating the asymptotic covariance matrix of the estimator and establish their consistency property. Simulation studies are conducted to evaluate the finite‐sample performance of the proposed procedures. An empirical example is also presented.     

ASYMPTOTIC EFFICIENCY OF THE SAMPLE COVARIANCES IN A GAUSSIAN STATIONARY PROCESS

Abstract. This paper deals with the asymptotic efficiency of the sample autocovariances of a Gaussian stationary process. The asymptotic variance of the sample autocovariances and the Cramer–Rao bound are expressed as the integrals of the spectral density and its derivative. We say that the sample autocovariances are asymptotically efficient if the asymptotic variance and the Cramer–Rao bound are identical. In terms of the spectral density we give a necessary and sufficient condition that they are asymptotically efficient. This condition is easy to check for various spectra.     

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.     

A new frequency domain approach of testing for covariance stationarity and for periodic stationarity in multivariate linear processes

In modelling seasonal time series data, periodically (non‐)stationary processes have become quite popular over the last years and it is well known that these models may be represented as higher‐dimensional stationary models. In this article, it is shown that the spectral density matrix of this higher‐dimensional process exhibits a certain structure if and only if the observed process is covariance stationary. By exploiting this relationship, a new L₂‐type test statistic is proposed for testing whether a multivariate periodically stationary linear process is even covariance stationary. Moreover, it is shown that this test may also be used to test for periodic stationarity. The asymptotic normal distribution of the test statistic under the null is derived and the test is shown to have an omnibus property. The article concludes with a simulation study, where the small sample performance of the test procedure is improved by using a suitable bootstrap scheme.