An autoregressive-moving average model in which all roots of the autoregressive polynomial are reciprocals of roots of the moving average polynomial and vice versa is called an all-pass time series model. All-pass models generate uncorrelated (white noise) time series, but these series are not independent in the non-Gaussian case. An approximate likelihood for a causal all-pass model is given and used to establish asymptotic normality for maximum likelihood estimators under general conditions. Behavior of the estimators for finite samples is studied via simulation. A two-step procedure using all-pass models to identify and estimate noninvertible autoregressive-moving average models is developed and used in the deconvolution of a simulated water gun seismogram.

We consider maximum likelihood estimation for both causal and noncausal autoregressive time series processes with non-Gaussian αstable noise. A nondegenerate limiting distribution is given for maximum likelihood estimators of the parameters of the autoregressive model equation and the parameters of the stable noise distribution. The estimators for the autoregressive parameters are n 1/α -consistent and converge in distribution to the maximizer of a random function. The form of this limiting distribution is intractable, but the shape of the distribution for these estimators can be examined using the bootstrap procedure. The bootstrap is asymptotically valid under general conditions. The estimators for the parameters of the stable noise distribution have the traditional n 1/2 rate of convergence and are asymptotically normal. The behavior of the estimators for finite samples is studied via simulation, and we use maximum likelihood estimation to fit a noncausal autoregressive model to the natural logarithms of volumes of Wal-Mart stock traded daily on the New York Stock Exchange. . This reprint differs from the original in pagination and typographic detail. 1 2 B. ANDREWS, M. CALDER AND R. A. DAVIS Rachev [28]), signal processing (Nikias and Shao [29]) and teletraffic engineering (Resnick [32]). The focus of this paper is maximum likelihood (ML) estimation for the parameters of autoregressive (AR) time series processes with non-Gaussian stable noise. Specific applications for heavy-tailed AR models include fitting network interarrival times (Resnick [32]), sea surface temperatures (Gallagher [20]) and stock market log-returns (Ling [24]). Causality (all roots of the AR polynomial are outside the unit circle in the complex plane) is a common assumption in the time series literature since causal and noncausal models are indistinguishable in the case of Gaussian noise. However, noncausal AR models are identifiable in the case of non-Gaussian noise, and these models are frequently used in deconvolution problems (Blass and Halsey [3], Chien, Yang and Chi [10], Donoho [16] and Scargle [36]) and have also appeared for modeling stock market trading volume data (Breidt, Davis and Trindade [5]). We, therefore, consider parameter estimation for both causal and noncausal AR models. We assume the parameters of the AR model equation and the parameters of the stable noise distribution are unknown, and we maximize the likelihood function with respect to all parameters. Since most stable density functions do not have a closed-form expression, the likelihood function is evaluated by inversion of the stable characteristic function. We show that ML estimators of the AR parameters are n 1/α -consistent (n represents sample size) and converge in distribution to the maximizer of a random function. The form of this limiting distribution is intractable, but the shape of the distribution for these estimators can be examined using the bootstrap procedure. We show the bootstrap procedure is asymptotically valid provided the bootstrap sample s...

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An autoregressive-moving average model in which all roots of the autoregressive polynomial are reciprocals of roots of the moving average polynomial and vice versa is called an all-pass time series model. All-pass models are useful for identifying and modeling noncausal and noninvertible autoregressive-moving average processes. We establish asymptotic normality and consistency for rank-based estimators of all-pass model parameters. The estimators are obtained by minimizing the rank-based residual dispersion function given by Jaeckel [Ann. Math. Statist. 43 (1972) 1449--1458]. These estimators can have the same asymptotic efficiency as maximum likelihood estimators and are robust. The behavior of the estimators for finite samples is studied via simulation and rank estimation is used in the deconvolution of a simulated water gun seismogram.Comment: Published at http://dx.doi.org/10.1214/009053606000001316 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

We consider a rank-based technique for estimating GARCH model parameters, some of which are scale transformations of conventional GARCH parameters. The estimators are obtained by minimizing a rank-based residual dispersion function similar to the one given in Jaeckel (1972).They are useful for GARCH order selection and preliminary estimation. We give a limiting distribution for the rank estimators which holds when the true parameter vector is in the interior of its parameter space, and when some GARCH parameters are zero. The limiting theory is used to show that the rank estimators are robust, can have the same asymptotic efficiency as maximum likelihood estimators, and are relatively efficient compared to traditional Gaussian and Laplace quasi-maximum likelihood estimators. The behavior of the estimators for finite samples is studied via simulation, and we use rank estimation to fit a GARCH model to exchange rate log-returns.