SIMULTANEOUS CONFIDENCE BANDS FOR THE SPECTRAL ESTIMATE OF TWO-CHANNEL AUTOREGRESSIVE PROCESSES

Abstract. In this paper we derive simultaneous confidence bands for the maximum entropy method spectral estimate of two-channel autoregressive (AR) processes by using the asymptotic theory of the estimation of periodic AR processes.     

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.     

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.     

ESTIMATION OF AUTOREGRESSIVE MOVING-AVERAGE ORDER GIVEN AN INFINITE NUMBER OF MODELS AND APPROXIMATION OF SPECTRAL DENSITIES

Abstract. A modification of the minimum Akaike information criterion (AIC) procedure (and of related procedures like the Bayesian information criterion (BIC)) for order estimation in autoregressive moving-average (ARMA) models is introduced. This procedure has the advantage that consistency for the order estimators obtained via this procedure can be established without restricting attention to only a finite number of models. The behaviour of these newly introduced order estimators is also analysed for the case when the data-generating process is not an ARMA process (transfer function/spectral density approximation). Furthermore, the behaviour of the order estimators obtained via minimization of BIC (or of related criteria) is investigated for a non-ARMA data-generating process.     

CONFIDENCE INTERVALS FOR ROBUST ESTIMATES OF THE FIRST ORDER AUTOREGRESSIVE PARAMETER

Abstract. The confidence interval properties of several estimators of the transition parameter, φ, in the first order autoregressive model are examined by a Monte Carlo study. The least squares confidence interval estimator, as well as two forms of a proposed robust confidence interval based on a generalized M-estimator, are examined under two model alternatives to the classical time series approach: the innovations model (the time series is observed 'perfectly') and the additive effects model (the time series is observed plus an added 'effect'). Samples were generated from a number of symmetric distributions, including the Gaussian and a variety of contaminated distributions with mild to heavy contamination. Over a range of outlier models, values of φ (.25 to.9), and sample sizes (20 to 100), it was found that the GM-estimators possess desirable confidence interval robustness properties in terms of validity and efficiency. In general, the least squares confidence interval is not robust against symmetric heavy-tailed contamination in the innovations model or against the additive effects model.     

ASYMPTOTIC EXPANSIONS FOR THE DISTRIBUTION OF AN ESTIMATOR IN THE FIRST-ORDER AUTOREGRESSIVE PROCESS

Abstract. In this paper, two asymptotic expansions for the distribution for an estimator of the parameter in a first-order autoregressive process are derived, according to two situations. Some well known estimators are special cases of the estimator discussed here. The series expansions are carried to terms of order T ^-1.     

A GOODNESS-OF-FIT TEST FOR AUTOREGRESSIVE MOVING-AVERAGE MODELS BASED ON THE STANDARDIZED SAMPLE SPECTRAL DISTRIBUTION OF THE RESIDUALS

Abstract. A stochastic process derived from the standardized sample spectral density of the residuals of a causal and invertible ARMA( p, q ) model is introduced to construct a goodness-of-fit procedure. The test statistics considered have a proper limiting distribution which is free of unknown parameters and which, unlike some well-known goodness-of-fit statistics based on the residuals, does not depend on the sample size.     

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.     

ASYMPTOTIC RESULTS FOR PERIODIC AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. This paper is concerned with the derivation of asymptotic distributions for the sample autocovariance and sample autocorrelation functions of periodic autoregressive moving-average processes, which are useful in modelling periodically stationary time series. In an effort to obtain a parsimonious model representing a periodically stationary time series, the asymptotic properties of the discrete Fourier transform of the estimated periodic autocovariance and autocorrelation functions are presented. Application of the asymptotic results to some specific models indicates their usefulness for model identification analysis.     

REDUCTION OF THE ASYMPTOTIC BIAS OF AUTOREGRESSIVE AND SPECTRAL ESTIMATORS BY TAPERING

Abstract. The asymptotic bias to terms of order T ^-1, where T is the observed series length, is studied for estimators of the coefficients and disturbance variance in an AR( p ) model. Reduction of the asymptotic bias by tapering is established and, if the tapering function is defined appropriately to depend on T , not only is the asymptotic bias reduced, but the asymptotic distribution of the estimators is not altered. In addition, the asymptotic biases of other time series parameter estimators constructed from the sample covariance function, such as several types of spectral estimators, can also be reduced by tapering.     

CONSISTENT ESTIMATION OF THE FOURTH-ORDER CUMULANT SPECTRAL DENSITY

Abstract. In this paper we consider the estimation of the fourth-order cumulant spectral density. Indeed this is the first case where the cumulant depends on lower-order product moments for a mean-zero stationary process. The proposed estimator of the fourth-order cumulant spectral density is constructed by replacing product moments with appropriately weighted estimates of product moments according to the definition of the fourth-order cumulant spectral density. Asymptotic unbiasedness and consistency are shown to hold for these estimators under stationarity and absolute summability of cumulants up to various orders with no restrictions on the frequencies. An expression for the asymptotic variance is also obtained.     

ON THE ASYMPTOTIC DISTRIBUTION OF THE GENERALIZED PARTIAL AUTOCORRELATION FUNCTION IN AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. It has been conjectured and illustrated that the estimate of the generalized partial autocorrelation function (GPAC), which has been used for the identification of autoregressive moving-average (ARMA) models, has a thick-tailed asymptotic distribution. The purpose of this paper is to investigate the asymptotic behaviour of the GPAC in detail. It will be shown that the GPAC can be represented as a ratio of two functions, known as the θ function and the Λ function, each of which itself has a useful pattern for ARMA model identification. We shall show the consistencies of the extended Yule-Walker estimates of the three functions and present their asymptotic distributions.     

ESTIMATION FOR NONNEGATIVE AUTOREGRESSIVE PROCESSES WITH AN UNKNOWN LOCATION PARAMETER

Abstract. Consider an AR(1) process given by X ^t=γ+ø X _t+ Z _t≥ 1. where 0 ≤γ, 0 ≤ø 1 are unknown parameters and the innovations Z _t, ≥ 1, are independently and identically distributed positive random variables. We propose estimates of (γø) which are obtained as the solution to a linear programming problem and establish their strong consistency. When the Z _ts have the exponential distribution. our estimate becomes the conditional maximum likelihood estimate given X ₀. Under the assumption of regular variation of the innovation distribution at its left and right endpoints (assumed to be 0 and ∝ respectively), we establish asymptotic limit laws for the estimates. Consistent estimators for a class of moving-average processes with heavy-tailed innovation distribution are also presented.     

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.     

ADEQUACY OF ASYMPTOTIC THEORY FOR GOODNESS-OF-FIT CRITERIA FOR SPECTRAL DISTRIBUTIONS

Abstract. Any of the Cramér-von Mises, Anderson-Darling, and Kolmogorov-Smirnov statistics can be used to test the null hypothesis that the standardized spectral distribution of a stationary stochastic process is a specified one. The asymptotic distributions of the criteria have been characterized (Anderson, 1993). They are the same as for probability distributions if the observations are independent (all autocorrelations zero), but are different when there is dependence. In this paper simulation with 10000 replications has been used to determine the distributions of the criteria for samples of size 6, 10, 30 and 100 when the observations are independent. These empirical distributions have been compared with the asymptotic distributions in order to ascertain the sample sizes necessary for using the asymptotic tables. For practical purposes they are 30 for the Cramér-von Mises and Kolmogorov statistics and over 100 for Anderson-Darling.     

RECURSIVE GENERALIZED M ESTIMATES FOR AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. Outliers in time series seriously affect conventional parameter estimates. In this paper a robust recursive estimation procedure for the parameters of auto-regressve moving-average models with additive outliers is proposed. Using 'cleaned' residuals from an initial robust fit of an autoregression of high order as input, bounded influence regression is applied recursively. The proposal follows certain ideas of Hannan and Rissanen, who suggested a three-stage procedure for order and parameter estimation in a conventional setting.
A Monte Carlo study is performed to investigate the robustness properties of the proposed class of estimates and to compare them with various other suggestions, including least squares, M estimates, residual autocovariance and truncated residual autocovariance estimates. The results show that the recursive generalized M estimates compare favourably with them. Finally, possible modifications to master even vigourous situations are suggested.     

ESTIMATION OF AUTOREGRESSIVE PARAMETERS AND ORDER SELECTION FOR ARMA MODELS

Abstract. A quick algorithm for obtaining estimates of autoregressive parameters for autoregressive moving-average model is presented. The algorithm is recursive in the orders, and can be used for model selection by providing a criterion and a two-way table of certain partial covariances. Consistency and asymptotic normality of the estimates are shown.     

SPECTRAL DENSITY ESTIMATION FROM NONLINEARLY OBSERVED DATA

Abstract. The estimation of the spectral density function of a stationary Gaussian process at the input of an instantaneous nonlinearity is considered when the nonlinearity is known and a finite set of observations of the output process is given. A class of spectral estimates is considered and their quadratic-mean consistency is established; precise asymptotic expressions for their bias and covariance are derived and their asymptotic normality is obtained.     

ON THE LAGRANGE MULTIPLIER TEST FOR AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. This paper examines the score or Lagrange multiplier statistic for testing the adequacy of a fitted autoregressive moving-average model and gives a simple closed-form expression for this test statistic. Some singularities arising as the order of the alternative model is increased are examined.     

MAXIMUM LIKELIHOOD ESTIMATION FOR AUTOREGRESSIVE PROCESSES DISTURBED BY A MOVING AVERAGE

Abstract. Maximum likelihood estimation for stationary autoregressive processes when the signal is subject to a moving-average sampling error is discussed. A modified maximum likelihood estimator is proposed. An algorithm for computing derivatives of the modified likelihood is suggested. Maximum likelihood estimators of the parameter vector are shown to be strongly consistent and to have a multivariate normal limiting distribution. A Monte Carlo simulation shows that the modified maximum likelihood estimator performs better than other available estimators. US current labour force data are analysed as an example.