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.     

ON THE SPECTRAL DENSITY MATRIX OF A PERIODIC ARMA PROCESS

Abstract. After reviewing the spectral representation theorems for periodic stationary process, we derive a parametric formula for the spectral density of a periodic ARMA process via a new approach. The equivalence with the existing approach is shown.     

Second-order properties of locally stationary processes

Abstract.  In this article, we investigate an optimal property of the maximum likelihood estimator of Gaussian locally stationary processes by the second-order approximation. In the case where the model is correctly specified, it is shown that appropriate modifications of the maximum likelihood estimator for Gaussian locally stationary processes is second-order asymptotically efficient. We also discuss second-order robustness properties.     

Elimination of Third-series Effect and Defining Partial Measures of Causality

Using the one-way effect extraction method, this paper presents a set of partial causal measures which represents quantitatively the interdependence between a pair of vector-valued processes in the presence of a third process. Those measures are defined for stationary as well as for a class of non-stationary time series. In contrast to conventional conditioning methods, the partial concept defined in the paper would be mostly devoid of feedback distortion by the third process. The paper also discusses statistical inference on the proposed measures.     

BOOTSTRAPPING STATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. In this paper we develop an asymptotic theory for application of the bootstrap to stationary stochastic processes of autoregressive moving-average (ARMA) type, with known order ( p, q ). We give a proof of the asymptotic validity of the bootstrap proposal applied to M estimators for the unknown parameter vector of the process. For this purpose we derive an asymptotic expansion for M estimators in ARMA models and construct an estimate for the unknown distribution function of the residuals which in principle are not observable. A small simulation study is also included.     

ESTIMATING THE ARCH PARAMETERS BY SOLVING LINEAR EQUATIONS

Abstract. This paper discusses the asymptotics of two-stage least squares estimator of the parameters of ARCH models. The estimator is easy to obtain since it involves solving two sets of linear equations. At the same time, the estimator has the same asymptotic efficiency as that of the widely used quasi-maximum likelihood estimator. Simulation results show that, even for small sample size, the performance of our estimator compared to the quasi-maximum likelihood estimator is better.     

A FORMULA FOR THE INVERSE AUTOCORRELATION FUNCTION OF AN AUTOREGRESSIVE PROCESS

Abstract. The determination of the inverse autocorrelation function of a weakly stationary autoregressive process using the autocorrelation function is considered. Usually this is carried out either by using frequency domain methods or by solving first the parameters of the process and then using them. In this paper we give a simple formula by which the inverse autocorrelation function can be determined directly from the autocorrelation function.     

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.     

Note on the Asymptotic Efficiency of Sample Covariances in Gaussian Vector Stationary Processes

In this note certain results obtained by Porat ( J. Time Ser. Anal. 8 (1987), 205–20) and Kakizawa and Taniguchi ( J. Time Ser. Anal. 15 (1994), 303–11) concerning the asymptotic efficiency of sample autocovariances of a zero-mean Gaussian stationary process are extended to the case of m -vector processes. It is shown that, for Gaussian vector AR( p ) processes, the sample autocovariance matrix at lag k is asymptotically efficient if 0 ≤ k ≤ p . Further, none of the sample autocovariance matrices is asymptotically efficient for Gaussian vector MA( q ) processes.     

SUBORDINATION OF STATIONARY PROCESSES

Abstract. Let X ={ X ( t ), t ε T ∁ R} be a ( L ₂ -) stationary process and suppose that N ={ N ( t ), t ≥ 0} is an infinitely divisible process, independent of X. Then ={ ( t ) = X ( N ( t )), t ≥ 0} is again a stationary process. In this paper, we relate the spectral properties of the original process X and the derived or subordinated process .     

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.