using the result that the eigenvectors and eigenvalues of \u03a3 occur in pairs ie v

Using the result that the eigenvectors and

• 592

This preview shows page 501 - 504 out of 592 pages.

using the result that the eigenvectors and eigenvalues of Σ occur in pairs, i.e., ( v 0 c , v 0 s ) 0 and ( v 0 s , - v 0 c ) 0 , where v c - iv s denotes the eigenvector of f xx . Show that 1 2 ( Z - μ ) 0 Σ - 1 ( Z - μ )) = ( X - M ) * f - 1 ( X - M ) so p ( X ) = p ( Z ) and we can identify the density of the complex multivariate normal variable X with that of the real multivariate normal Z . 7.2 Prove b f in ( 7.6 ) maximizes the log likelihood ( 7.5 ) by minimizing the negative of the log likelihood L ln | f | + L tr { b ff - 1 } in the form L X i ( λ i - ln λ i - 1 ) + Lp + L ln | b f | , where the λ i values correspond to the eigenvalues in a simultaneous diago- nalization of the matrices f and ˆ f ; i.e., there exists a matrix P such that P * fP = I and P * b fP = diag ( λ 1 , . . . , λ p ) = Λ. Note, λ i - ln λ i - 1 0 with equality if and only if λ i = 1, implying Λ = I maximizes the log likelihood and f = b f is the maximizing value. Section 7.3
490 7 Statistical Methods in the Frequency Domain 7.3 Verify ( 7.18 ) and ( 7.19 ) for the mean-squared prediction error MSE in ( 7.11 ). Use the orthogonality principle, which implies MSE = E ( y t - X r = -∞ β 0 r x t - r ) y t and gives a set of equations involving the autocovariance functions. Then, use the spectral representations and Fourier transform results to get the final result. Next, consider the predicted series b y t = X r = -∞ β 0 r x t - r , where β r satisfies ( 7.13 ). Show the ordinary coherence between y t and b y t is exactly the multiple coherence ( 7.20 ). 7.4 Consider the complex regression model ( 7.28 ) in the form Y = X B + V , where Y = ( Y 1 , Y 2 , . . . Y L ) 0 denotes the observed DFTs after they have been re- indexed and X = ( X 1 , X 2 , . . . , X L ) 0 is a matrix containing the reindexed in- put vectors. The model is a complex regression model with Y = Y c - iY s , X = X c - iX s , B = B c - iB s , and V = V c - iV s denoting the representation in terms of the usual cosine and sine transforms. Show the partitioned real re- gression model involving the 2 L × 1 vector of cosine and sine transforms, say, Y c Y s = X c - X s X s X c B c B s + V c V s , is isomorphic to the complex regression regression model in the sense that the real and imaginary parts of the complex model appear as components of the vectors in the real regression model. Use the usual regression theory to verify ( 7.27 ) holds. For example, writing the real regression model as y = xb + v, the isomorphism would imply L ( b f yy - b f * xy b f - 1 xx b f xy ) = Y * Y - Y * X ( X * X ) - 1 X * Y = y 0 y - y 0 x ( x 0 x ) - 1 x 0 y. Section 7.4
Problems 491 7.5 Consider estimating the function ψ t = X r = -∞ a 0 r β t - r by a linear filter estimator of the form b ψ t = X r = -∞ a 0 r b β t - r , where b β t is defined by ( 7.42 ). Show a sufficient condition for b ψ t to be an unbiased estimator; i.e., E b ψ t = ψ t , is H ( ω ) Z ( ω ) = I for all ω . Similarly, show any other unbiased estimator satisfying the above condition has minimum variance (see Shumway and Dean, 1968), so the esti- mator given is a best linear unbiased (BLUE) estimator.