R. Fox and M. S. Taqqu. Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. The Annals of Statistics, 14:517--532, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....likelihood principle assuming that the process under analysis is Gaussian. The estimator, unlike the previous ones, provides the estimate through a non graphical method. This estimation takes more time to perform but it has the advantage of providing con dence intervals as well. For details see [17, 3]. For the Bellcore trace, the estimated value of the Hurst parameter is 0.82 and its 95 con dence interval is [0:79; 0:84] 3.3 Multifractal framework In this section we introduce two techniques to analyze multifractal processes. Legendre spectrum Considering a continuous time process Y = fY ....

R. Fox and M. S. Taqqu. Large sample properties of parameter estimates for strongly dependent stationary time series. The Annals of Statistics, 14:517-532, 1986.

.... model is 4 Marc Henry,Paolo Zaffaroni given in [100] Exact maximum likelihood is efficient in the Fr echet DarmoisCram er Rao (hereafter FDCR) sense when the t s are normally and identically distributed ( 26] The Whittle approximate likelihood is asymptotically efficient under Gaussianity ([38]) and remains p n consistent and asymptotically normal for possibly non normal identically and independently distributed t s ( 46] A result that is particularly relevant to macroeconomic time series is the apparently greater robustness properties of the Whittle estimate in small samples when ....

Fox, R., and M. S. Taqqu (1986): "Large sample properties of parameter estimates for strongly dependent stationary Gaussian time series," Annals of Statistics, 14, 517--132.

.... are thus less efficient than p n consistent estimates which employ information across the whole Nykvist band [ # ] on the basis of a complete finiteparameter model for f( such as a fractionally integrated autoregressive moving average (FARIMA) model, as shown in the case of Whittle estimates byFoxandTaqqu (1986). However, such estimates are generally inconsistent if the parameterization is misspecified, as is the case in FARIMA models if either the autoregressiveormoving average orders are under specified, or both are over specified. The greater robustness of the semiparametric estimates might then be ....

Fox, R., and M. S. Taqqu (1986): "Large sample properties of parameter estimates for strongly dependent stationary Gaussian time series," Annals of Statistics, 14, 517--132.

....scale. Thus, characterizing the scaling behavior amounts to estimating some power law exponents. Many methods for estimating these exponents have been developed. By far, the most precise estimations are given by parametric methods such as the maximum likelihood estimator (the Whittle estimator) [10], 11] 12] 13] However, such estimators assume that the parametric form of the spectral density or of the auto correlation function is known. Moreover, choosing an inappropriate parametric form generally leads to drastically biased estimations. Since, in most practical situations, one has no ....

R. Fox and M.S. Taqqu, \Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series," The Annals of Statistics, vol. 14, pp. 517-532, 1986.

....2AE, UK p.m.robinson lse.ac.uk 1. Nonstationary Time Series Exact and approximate maximum likelihood estimation for the parameters of stationary time series has been justi#ed under various sets of conditions, including spectral densities with a peak at the origin due to a persistentbehaviour #e.g. Fox and Taqqu #1986##. Thus it is often assumed that the spectral density f### of an observed covariance stationary sequence satis#es, for 0 #G#1, #1# f### # Gj#j ,2d as # 0; where d 2 #, 1 2 ; 1 2 # is the parameter that governs the degree of memory of the series. This is the interval of values of d for ....

Fox, R. and Taqqu, M.S. #1986#. Large-sample properties of parameter estimates for strongly dependent stationary Gaussian times series. Annals of Statistics 14, 517-532.

....the real exchange rate series 12 X t as the residuals from the regression on equation (7) For the US monthly series, we simply used formula (6) INSERT TABLE 2 ABOUT HERE The next steps were performed with the resulting series. 4 Tests and Results Our approach is based on the Fox and Taqqu (1986) frequency domain approximate maximum likelihood. Using a Whittle approximation to the log likelihood function (Brockwell and Davis 1991, p. 529, equation (13.2.26) the function L(x #) 2 n [n 2] # j=1 I n(# j ) f # # (# j ) 2 n [n 2] # j=1 log f # # (# j ) 8) is minimized, ....

Fox, Robert and Taqqu, Murad (1986) Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series, Annals of Statistics, 14, 517-532.

....(h(H) 0 ] 2 (4.34) for N j1 large with Cb de ned in (4.30) and h in (4.19) The variance of the estimators is of order 1=N j1 . Long memory processes can also be modeled via state space models. Maximum likelihood estimators (MLE) for such models are analyzed in Dahlhaus [4] and Fox and Taqqu [8]. 4.7.5 Illustration of precision The above result on the distribution of the estimators may be validated by numerical simulation. We generate synthetic realizations of fractional Brownian motion with known parameters. Then we use the above algorithm to estimate these parameters and compare 38 ....

Fox, R., and Taqqu, M., (1986), \Large-sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time Series," in Annals of Statistics, 14, pp. 517-532.

....the spectral density of an ARMA(p; q) process is a bounded rational function. One approach used for the estimation of model parameters fi is based on maximizing the likelihood (1.3) or approximations to (1. 3) and has been discussed by Boes, Davis and Gupta (1989) Brockwell and Davis (1987) Fox and Taqqu (1986), Hosking (1981, 1984) Luce no (1993) and Sowell (1992) Some properties of the maximum likelihood estimators are discussed in Fox and Taqqu (1986, 1987) Dahlhaus (1989) and Cheung and Deibold (1994) The study of asymptotic inference for the ARF IMA(p; d; q) process is of considerable current ....

....on maximizing the likelihood (1.3) or approximations to (1. 3) and has been discussed by Boes, Davis and Gupta (1989) Brockwell and Davis (1987) Fox and Taqqu (1986) Hosking (1981, 1984) Luce no (1993) and Sowell (1992) Some properties of the maximum likelihood estimators are discussed in Fox and Taqqu (1986, 1987) Dahlhaus (1989) and Cheung and Deibold (1994) The study of asymptotic inference for the ARF IMA(p; d; q) process is of considerable current research interest. The use of differential geometry to characterize statistical inference has been widely studied in the last two decades and an ....

Fox, R. and Taqqu, M.S. (1986) Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann.

....memory if 1 P k= Gamma1 jae(k)j 1. See Granger (1980) Lawrance and Kottegoda (1977) and Greene and Fietlitz (1977, 1979) for applications of long memory models. Fractionally differenced autoregressive models provide a rich class of long memory models. Hosking (1981) Yajima (1985, 1991) and Fox and Taqqu (1986) have studied various aspects of the asymptotic estimation problem for fractionally differenced models. In this paper, we introduce a multivariate process based on fractionally differenced autoregressive process. In Section 2, we introduce the model and derive the likelihood function. Section 3 is ....

Fox, R. and Taqqu, M. S. (1986). Large sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist., 14 , 517-532.

....long memory times series. Many heuristic methods have been considered to estimate long memory parameter ( see Beran 1994, for a review) From a frequentist point of view, many authors have studied the Gaussian maximum likelihood estimator as well as the Whittle estimator (see Dahlhaus (1989) Fox and Taqqu (1986), Giraitis and Surgalis (1990) In particular the asymptotic properties of these estimators, such as asymptotic normality, are well established now. Bayesian analysis for ARFIMA models have been introduced by Carlin et al. (1985) More recently, Koop et al. (1994, 1997) and Pai and Ravishanker ....

Fox R.,Taqqu M.S. (1986) Large sample properties of parameter estimates for strongly dependent stationary gaussian time series Annals of Stat., 14, 517-532.

....for inference; forecast distributions can also be easily obtained. Methods for estimating univariate and multivariate long range dependent processes based on approximations to the likelihood function that have the same asymptotic properties as those based on the exact likelihood have been explored (Fox and Taqqu, 1986; Haslett and Raftery, 1989; Beran, 1994) however, their finite sample properties may be much worse (Sowell, 1992) Pai and Ravishanker (1998) estimated the parameters of a univariate ARFIMA process in the Bayesian framework using a MetropolisHastings algorithm to generate samples of d from a ....

Fox, R. and Taqqu, M. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series, Ann. Statist., 14, 517-532.

....we employ the approximate Whittle estimator that is obtained by maximizing an approximation of the likelihood function in the frequency domain. In this procedure, the parameter vector 0 = ff 0 ; d; fi 0 ) is estimated by minimizing the implied white noise variance (Whittle (1951) Fox and Taqqu (1986)) oe 2 T ( 2 T T Gamma1 X u=1 I T ( u ) g( u ; 8) where the periodogram is defined at the Fourier frequencies u = 2 u=T; u = 1; T Gamma 1 as I T ( u ) j(2 T ) Gamma 1 2 T X j=1 (x j Gamma x)e ij u j 2 : 9) and g( 2 f( oe 2 . ....

Fox, R. and Taqqu, M.S. (1986). `Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series', The Annals of Statistics, Vol. 14,pp. 517 -- 532.

....domain maximum likelihood methods are employed in this analysis, the Whittle estimator and its approximation (Whittle (1951) They will be briefly described in this section. Let = ff 0 ; d; fi 0 ) 0 denote the parameter vector which is to be estimated. Following the presentation of Fox and Taqqu (1986), the key element of both methods is the approximation of the inverted covariance matrix Sigma Gamma1 ( of the stochastic process by an expression in the frequency domain A T ( where each element [A T ( jk is given by 1 2 Z Gamma 1 g( e i(j Gammak) d and g( 2 ....

Fox, R. and Taqqu, M.S.. "Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series.", The Annals of Statistics, 14 (1986): 517 -- 532.

....to be able to estimate the spectral exponent fl from an observed sample path. Many methods to achieve this have been proposed. They range from least squares estimation of the slope of log axes plots of sample periodograms [26, 17] through to approximate and exact maximum likelihood estimation [7, 16, 26, 28] and direct measuring of fractal dimension of observed sample paths [17, 12, 14] Aside from the maximum likelihood (ML) based schemes, these methods assume that the sample path observation fx k g is not corrupted by any other noise sources. Various convergence results (which will be surveyed) are ....

....measured sample path as the previous figure is used, but now it is corrupted by white Gaussian noise of variance 0:001. The true value is still fl = 1:8 but now the estimate is found as b fl = 1:29. where [T N ( 1 2 Z Gamma OE( e j (r Gammas) d : For fl 2 (0; 1) Fox and Taqqu [16] have shown the approximate ML estimator to be strongly consistent, asymptotically Gaussian and efficient. Leu and Papamorcou [26] have extended these results (by employing stronger assumptions on OE( that disallow estimation with white noise corrupted measurements) to also hold for fl 2 (1; ....

R. Fox and M. S. Taqqu, Large sample properties of parameter estimates for strongly dependent stationary time series, The Annals of Statistics, 14 (1986), pp. 517--532.

....used to alleviate much of this burden. Sowell (1992b) studies the properties of this estimator for the model in equation (1) and its autoregressive and moving average extensions, assuming = 0. To avoid some of the computation associated with exact MLE, Cheung and Diebold (1994) suggest using the Fox and Taqqu (1986) frequencydomain approximation to the likelihood function. Hauser (1992) reports small sample properties of a frequency domain estimator similar to the one suggested by Cheung and Diebold (1994) Hauser s estimator, which minimizes the Whittle likelihood function, appears to perform better in ....

....function. Hauser (1992) reports small sample properties of a frequency domain estimator similar to the one suggested by Cheung and Diebold (1994) Hauser s estimator, which minimizes the Whittle likelihood function, appears to perform better in small samples than the estimator suggested by Fox and Taqqu (1986). The Whittle likelihood function is defined as L W (d; X T ) m X j=1 log(f( j ; d) m X j=1 I T ( j ; X T ) f( j ; d) 4) where m = T Gamma 1) 2, j = 2 j=T , I T ( j ; X T ) is the periodogram of the sample X T and oe 2 f( j ; d) is the spectral density function 2 of the ....

Fox, Robert and Murad S. Taqqu (1986), Large-Sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time Series, The Annals of Statistics 14, 517--532.

....0 C b (2; 2) log 2 (h(H) 0 ] 2 (2.26) for N j 1 large with C b defined in (2.22) and h in (2.11) The variance of the estimators is of order 1=N j 1 . Long memory processes can also be modeled via state space models. Maximum likelihood estimators (MLE) for such models are analyzed in [3, 7]. 2.7 Illustration of precision The above result on the distribution of the estimators may be validated by numerical simulation. We generate synthetic realizations of fractional Brownian motion with known parameters. Then we use the above algorithm to estimate these parameters and compare the ....

R. Fox and M. Taqqu, Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series, Ann. Statist., 14, pp. 517-532, 1986.

.... and include heuristics such as analysis of the rescaled adjusted range statistics (R=S statistic, in short; e.g. see [38, 57, 83] and variancetime analysis of the aggregated processes [12, 83] examples of frequencydomain techniques are the periodogram analysis [31, 37, 83] and Whittle s method [87, 9, 16, 28]. For a wavelet domain approach, see [1, 25, 2, 3] Leland et al. 47] introduced the self similarity and LRD concepts in the modeling of data network trac. Starting with the extensive analyzes of trac measurements from Ethernet local area networks (LANs) over a four year period reported in [46, ....

R. Fox and M. S. Taqqu. Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. The Annals of Statistics, 14:517-532, 1986.

....f is sufficiently smooth, for example, if f is a polynomial. The condition (1.3) thus extends the univariate condition P t2Z jr f (t)j 1 to quadratic forms. However, in contrast to the CLT for univariate sums, it does not cover certain additional cases, first discovered by Fox and Taqqu [8], where the CLT for quadratic form also holds. Specifically, it excludes the possible compensation of the long range dependence of (X t ) by a fast decay of the weights b(t) These cases, which do not have a simple formulation in the time domain, are best characterized in the spectral domain. ....

R. Fox and M. S. Taqqu. Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. The Annals of Statistics, 14:517--532, 1986.

....stable innovations. The results are presented numerically, in tables and through boxplots. The goal of these studies is to provide a benchmark that will be useful to practitioners. The standard Whittle method for estimating the parameter H of longrange dependence has been widely used (see Fox and Taqqu (1986), Beran (1994) It is a parametric method in that it assumes that the spectral density of the series is known with the exception of a few parameters, which are to be estimated. This assumption allows for very precise estimation when the series being examined fits the assumed model exactly. If, on ....

....where p and q are known, then j also includes the unknown coefficients in the autoregressive and moving average parts. For such series the Whittle estimate H of H converges to its true value at the rate of N 1=2 and the asymptotic distribution of p N( H Gamma H) is Gaussian. For details see Fox and Taqqu (1986). 3.2 The Aggregated Whittle Estimator The aggregated Whittle estimator provides an additional technique for obtaining a robust estimate of H. It can be used if the time series is long enough. The idea is to aggregate the data, which creates a new, shorter series X (m) i : 1 m mi X ....

Fox, R. & Taqqu, M. S. (1986), `Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series', The Annals of Statistics 14, 517--532.

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R. Fox and M. S. Taqqu. Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. The Annals of Statistics, 14:517--532, 1986.

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R. Fox and M.S. Taqqu, "Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series," The Annals of Statistics, vol. 14, pp. 517--532, 1986.

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R. Fox and M. S. Taqqu, "Large-sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time," The Annals of Statistics, Vol.14, No.2, pp.517--532, 1986.

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FOX, R. & M.S. TAQQU (1986): \Large-sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time Series", Annals of Statistics, ##, 517-532.

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Fox, R., and Taqqu, M.S. (1986). Large-Sample properties of Parameter estimates for Strongly Dependent Stationary Gaussian Time Series. The Annals of Statistics, 14:517-- 532.

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Fox, R. and Taqqu, M. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series, Annals of Statistics, 14, 517--532.

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