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Sample text
1 SPECTRUM PARAMETER ESTIMATION IN TIME SERIES ANALYSIS 45 In fact, the spectrum f(X) in [45] is strictly positive (f(X) > m > 0) and belongs to the class ^ of functions g(X\ — π<λ<π, with Fourier transform gt, such that QO Σ tgf < oo. 17) It is seen that the latter condition on / implies l o g / e ^ and l/f e &. 2. 18) var[L r (X r ) - L r (X T )] = 0(1). 19) Here the tilde above the function of λ indicates the Fourier coefficient of the corresponding function (so that, for example, fk = b(k), while hk is the /cth Fourier coefficient of h(X) = l/4n2f(X)).
26 when oc1 = \. 4, matrix Γ(ο) = Π7ι ? · · · ? Ίρ,ν1) is simply diagonal with nonzero entries only along the main diagonal: Γ(0) = diag{l,. . , 1,1/2σ4}. 28) of y l 5 . . 6 forj = 1,. . , p. Let us consider also some examples of spectra vanishing at λ = 0. As mentioned previously, strict positivity of the spectrum in Eq. 8) is only sufficient for the proof given above, but not necessary for its validity. 6, allow one to extend Eq. 8) to this class. 35) is estimated by (σ2)* = ¿*(0)/2, under the Gaussian assumption, one gets ΤΙ)[(σ 2 )*] -> 3σ4 as T->oo.
35) is estimated by (σ2)* = ¿*(0)/2, under the Gaussian assumption, one gets ΤΙ)[(σ 2 )*] -> 3σ4 as T->oo. )->2(l - ßx) as Τ^ oo, where the estimate ßX of ßx in Eq. 38). Here Γ[β,σ 2 ] - diag{l/(l - βχ)2, 1/2σ4} and hence e{ß*) = (1 + ßJ/2. Finally, if the root of Eq. 39), then l i m ^ ^ TD(af) = 2 Σ®=0 d/x{/(l — a 2 ) 2 , where d0 = d8 = 1, d1 = d7 = 2, d2 = . . ). Consequently, e(otf) = ( l - a f ) 3 / 2 £j=o djOc{ so that the asymptotic efficiency of af approaches one-half or zero when ax -► 0 or 1, respectively.