P. Whittle, "Estimation and information in stationary time series," Arkiv Matematick, vol. B-2, no. 23, pp. 423--434, 1953.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 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, ....

P. Whittle. Estimation and information in stationary time series. Arkiv for Matematik, 2:423-434, 1953.

....10.8.1. and that e I n;X ( can be considered as an estimator of g( fi 0 ) For Gaussian (X t ) t2Z the estimator fi n is closely related to least squares and maximum likelihood estimators and it is a standard estimator for ARMA processes with finite variance. The idea goes back to Whittle (1953), see also Dzhaparidze (1986) Fox and Taqqu (1986) and Dahlhaus (1989) It is wellknown that in the classical case fi n is consistent and asymptotically normal (cf. Brockwell and Davis (1991) We showed in Mikosch et al. 1994) that fi n is also for ARMA processes with infinite variance a ....

Whittle, P. (1953). Estimation and information in stationary time series.

....due to the covariance matrix being high dimensional (for long time series) and often numerically unstable for certain values of d such that inverting the matrix might be a problem. There are several ways to compute an approximate likelihood function like ( Whittle s approximate MLE , see [18]) We take an alternative approach as given in Section 5.6 of [16] Consider (1) Assuming that the long memory time series has a causal linear representation one could write x t as x t = 1 X l=1 b(l)x t Gammal ffl t (11) where f ffl t g is a sequence of i.i.d. innovations and asymptotic ....

P. Whittle, "Estimation and information in stationary time series," Ark. Mat., vol. 2, pp. 423--434, 1953.

....are defined by a vector of parameters by using the methods of maximum likelihood; a special case of this then becomes the method of interest to us where = oe 2 ; fl] and OE( oe 2 j j Gammafl . Two approaches have been analysed in the literature. Firstly, the methods of Whittle [43] can be used to approximate the log likelihood function by using the periodogram so that an approximate Maximum Likelihood estimate b is given by b = arg min aeZ Gamma log OE( I N ( OE( d oe : Secondly, at (significantly) more computational expense the exact ....

P. Whittle, Estimation and information in stationary time series, Arkiv For Matematik, 2 (1953), pp. 423--434.

.... to difficult non convex optimisation problems, so that a compromise is often made with AR models in the Gaussian case by approximating the ML cost function with a quadratic one this amounts to discarding a matrix determinant term that can be shown to tend to zero with increasing data length N [24]. This same approach of using the least squares estimate will be employed here, so that the estimate b is chosen as b = arg min 2R p ( 1 N N Gamma1 X k=0 (y k Gamma OE T k ) 2 ) which is well known to have closed form solution b = R p (N) Gamma1 1 N N Gamma1 X k=0 ....

P. Whittle, Estimation and information in stationary time series, Arkiv For Matematik, 2 (1953), pp. 423--434.

....Chapter 5 of [45] It is formulated by maximising the log likelihood function with respect to the parameter vector. However the procedure is very computationally intensive, especially for long datasets and it can be unstable when H 1. Instead Whittle s approximation to the MLE method is employed [58]. One disadvantage of this method is that it is only optimal for Gaussian self similar processes, which is often not a valid assumption for real teletraffic data. 3.4.5 Comparing the estimation techniques In order to evaluate the performance of the estimation techniques we applied them to a ....

P. Whittle, "Estimation and information in stationary time series," Ark. Mat., vol. 2, 423-434 1953.

....of the form (1.9) Note that from the definition of the periodogram it follows that I N ( I N (2 Gamma ) therefore (1.9) can be rewritten in terms of the periodogram ordinates at the first N=2 Fourier frequencies only. For details on Whittle approximation the reader is referred to Whittle (1953) and, in the context of long memory processes, Beran (1994b) 1.3 Long memory processes: definition and examples. Let X be a stationary process with spectral density f . Assume that, for some number d in the interval (0,1) f( is asymptotically equivalent to Gammad times a constant, as ....

Whittle, P. (1953). Estimation and information in stationary time series, Ark. Mat.

....f 2 by maximizing L (T ) M (f) with respect to f 2 F 0 . This approach has been proposed by Hawkes and Adamopoulos (1973) and further discussed by Brillinger (1975) and Tuan (1981) for parametric families, for which the procedure is a point process version of a procedure suggested by Whittle (1953) for the analysis of time series. Subsequently, we will use the following continuous version of L (T ) M L (T ) f) Gamma Z R n log f( I (T ) f( o w( d (3.5) with w : R R satisfying Assumption (A3) and w( 0 for all 0. Let f (T ) denote a sequence of functions ....

Whittle, P. (1953), Estimation and information in stationary time series, Ark. Mat. 2, 423-434.

....density [10] More sophisticated methods have to be applied to obtain useful estimates of H. Several periodogram based estimators can be found in the literature. In this paper we will focus on an MLE as presented in [1, 13] which is based on Whittle s approximate MLE for Gaussian processes [12]. For Gaussian sequences this estimator is asymptotically normal and efficient [4, 3] The spectral density of the self similar process is denoted by f( where the parameter vector of the process = 1 ; M ) is structured as follows. 1 = oe 2 ffl is a scale parameter, where ....

P. Whittle. Estimation and information in stationary time series. Arkiv for Matematik, 2(23):423--434, 1953.

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P. Whittle, "Estimation and information in stationary time series," Arkiv Matematick, vol. B-2, no. 23, pp. 423--434, 1953.

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P. Whittle, "Estimation and information in stationary time series", Arkiv Matematick, vol. B-2, pp. 423--434, 1953.

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P. Whittle, "Estimation and information in stationary time series", Arkiv Matematick, vol. B-2, pp. 423--434, 1953.

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P. Whittle (1953). Estimation and Information in Stationary Time Series Arkiv for Matematik, 2:423--434.

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P. Whittle (1953). Estimation and Information in Stationary Time Series Arkiv for Matematik, 2:423--434.

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P. WHITTLE, Estimation and information in stationary time series, Arkiv For Matematik, 2 (1953), pp. 423--434.

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Whittle, P. (1953). Estimation and information in stationary time series. Ark. Mat. Astr. Fys. 2, 423--434.

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P. Whittle, "Estimation and Information in Stationary Time Series", Ark. Mat. 2, 423-434, 1953.

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