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The choice of the weights remains to be speci ed We know that the variance is minimal if we take into consideration the covariance structure of yj , yj in the de nition of aj Once again, in the case of uncorrelated variables yj , this leads us to 2 choose aj j In the case of scale invariance, we use the estimator de ned earlier with the variables: yj = log2 1 nj dX (j, k)2
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where g(j) are the correction terms aimed at taking into account the fact that E(log( )) is not log(E( )) and at ensuring that E yj = j + b Hence, this estimator simply consists of a weighted linear regression carried out in the diagram yj against j, referred to as the log-scale diagram [VEI 99] In order to easily implement this estimator, it is 2 necessary to further determine g(j) and j and choose aj To begin with, we assume that dx (j, k) are random Gaussian variables, ie, that they result from the wavelet decomposition of a process which is itself jointly Gaussian Moreover, if we idealize the weak correlation property of the wavelet coef cients in exact independence, then 2 we can calculate g(j) and j analytically: g(j) = (nj /2)/ (nj /2) log 2 log2 (nj /2) 1 , nj log 2 nj ; (230a)
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where and respectively designate the Gamma function and its derivative, and where (2, z) = n=0 1/(z + n)2 de nes a function called the generalized Riemann zeta function Let us note that these analytical expressions, which depend on the known nj s alone, can hence be easily evaluated in practice The numerical simulations presented in depth in [VEI 99] indicate that, for Gaussian processes, this analytical calculation happens to be an excellent approximation of reality, satisfying a posteriori the idealization of exact independence Thus, for Gaussian processes, we obtain an 2 estimator which is remarkably simple to carry out, since the quantities g(j) and j can be analytically calculated and do not need to be estimated from data, and which gives excellent statistical performance From these analytical expressions, we obtain: E = , which indicates that the estimator is unbiased and this is also valid for observations of 2 nite duration With the choice aj = j , its variance reads:
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and it attains the Cram r-Rao lower bound, which is calculated under the same hypotheses (Gaussian process and exact independence) [ABR 95, VEI 99, WOR 96] In addition, if we add the form nj = 2 j n (with n as the number of coef cients in the initial process) induced by the construction of the multiresolution analysis, we obtain the following expression for the variance:
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1 (1 2 J ) 1 j1 , n 2 F
(231)
where F = F (j1 , J) = log2 2 (1 (J 2 /2+2) 2 J +2 2J ) and where J = j2 j1 +1 denotes the number of octaves involved in the linear regression This analytical result shows that the variance of the estimator decreases in 1/n, in spite of the possible presence of a long-range dependence in the analyzed process It is noteworthy that, in practice, relation nj = 2 j n is not exactly satis ed because of boundary effects, which are systematically excluded a priori from the measures In the case of non-Gaussian processes, the implementation of the estimator is 2 more subtle, since we cannot use analytical expressions for g(j) and j Nevertheless, in the case of nite variance processes, the variables (1/nj ) k dX (j, k)2 are asymptotically Gaussian and we can show that correcting terms can be introduced [ABR 00b] to the Gaussian case: g(j) 1 + C4 (j)/2 ; nj log 2 1 + C4 (j)/2 , nj