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1) methods related to maximum likelihood, which, by nature, aim at estimating all the parameters of the model; 2) the estimation of the memory exponent d in ( ) = f ( )| | 2d where is the spectral density; 3) the estimation of the autosimilarity parameter in fractional Brownian motion For fractional ARMA processes whose transfer function has the form:
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where dj are integer and where 1 < d < 1 (referred to as ARFIMA), parametric 2 2 methods give satisfactory results, provided that the order of the model is known a priori (see, for example, [GON 87]) When the order of the model is not known, semi-parametric methods, as in point 2), are fruitfully used We can schematically describe these methods as follows Let us use the exible framework of models with spectral density: (ei ) = |1 ei | 2d g( ), where g can be written g( ) = exp( k=0 k cos k ) and is a very regular function The unique singularity of is located at = 0 Then, the natural tool to estimate the spectral density is the periodogram:
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The log of the periodogram estimates log :
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log (ei ) = 2d log |1 ei | +
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After a truncation of the sum at the order q 1, the parameters d, 0 , 1 , , q 1 are estimated by a regression of log I( ) against ( 2 log |1 ei |, cos k ) The dif culty resides in two points: the choice of q; the study of the properties of the estimators In [MOU 00], recent procedures based on penalization techniques give automatic methods for the choice of q, that can be tuned to the function g The asymptotic
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properties ensure convergences, for example towards Gaussian, of the estimator of d Many estimators that rely on this principle were proposed, some of them being compared in [BARD 01, MOU 01] When the singularities of the transfer function are not located at z = 1, another model is used Let us assume that the transfer function is written: F (z) = (1 e
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with | j | < 1, 0 = 0 The spectral density takes the form ( ) = |1 ei( + 0 ) |2d g( ) with a very regular g If 0 is known, various authors [OPP 00] think that the estimate of d is made according to ideas developed for = 1 When 0 is not known, it has to be estimated The idea is to use the frequency location where the periodogram takes its max Although more sophisticated, the procedures elaborated by Yajima [YAJ 96], Hidalgo [HID 99] and Giraitis et al [GIR 01], provide convergences in probability and in law of the estimators towards normal law 834 Simulated example Several methods were proposed to simulate trajectories of fractional processes Granger and Joyeux [GRAN 80] use an autoregressive approximation of order 100 obtained by truncating the AR( ) representation combined with an initialization procedure based on the Cholesky decomposition Geweke and Porter-Hudak [GEW 85] or Hosking [HOS 81] elaborate on the autocovariance and use a Levinson-Durbin-Whittle algorithm to generate an autoregressive approximation In both cases, quality is not quanti ed although it can be Gray et al [GRAY 89] approximate (X(n)) by a long MA obtained by truncating the MA ( ) representation The second method seems inadequate in our case because there is no expression for the autocovariance function of fractional ARMA processes These methods can easily be established thanks to differential equations (85) However, when (X(n)) is long-ranged, these methods require very long trajectories of white noise because of the slow decay of an ; for example, Gray et al [GRAY 89] use moving averages of order around 290,000 Another idea consists of approximating F by a rational fraction B and simulating A the ARMA process with representation A(L)Y (n) = B(L) (n) This is the chosen approach for the example below The algorithm used to calculate the polynomials A and B is developed by Baratchart et al [BARA 91, BARA 98] In principle, this algorithm, as with the theoretical results, is only valid when F has no singular point on the unit disc However, it provides satisfactory results, from the perspective detailed below The studied lter F reads: F (z) = z exp(2i 0231)
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