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T.W. Anderson 305The characteristic roots are λ j = cos 2πj/n and sin 2πj/n, j ∈{1,...,n}.If n is even, the roots occur in pairs. The distribution of the circular serialcorrelation is the distribution ofn∑ / ∑nλ j zj2 zj 2 , (27.1)j=1where z 1 ,...,z n are independent standard Normal variables. Anderson studiedthe distribution of the circular serial correlation, its moments, and otherproperties.j=127.3 Periodic trendsDuring World War II, R.L. Anderson and I were members of the PrincetonStatistical Research Group. We noticed that the jth characteristic vector ofA had the form cos 2πjh/n and/or sin 2πjh/n, h ∈{1,...,n}. Thesefunctionsare periodic and hence are suitable to represent seasonal variation. Weconsidered the modely i = β 1 x 1i + ···+ β k x ki + u i ,where x hi = cos 2πhi/n and/or sin 2πhi/n. Then the distribution of∑ (yi − ∑ β h x hi )(y i−1 − ∑ β h x h,i−1 )r = ∑ (yi − ∑ β h x hi ) 2is the distribution of (27.1), where the sums are over the z’s corresponding tothe cos and sin terms that did not occur in the trends. The distributions ofthe serial correlations have the same form as before.Anderson and Anderson (1950) found distributions of r for several cyclicaltrends as well as moments and approximate distributions.27.4 Uniformly most powerful testsAs described in Anderson (1948), many problems of serial correlation areincluded in the general model[K exp − α {}](y − µ) ⊤ Ψ (y − µ)+λ (y − µ) ⊤ Θ (y − µ) ,2where K is a constant, α>0, Ψ a given positive definite matrix, Θ a givensymmetric matrix, λ aparametersuchthatΨ − λΘ is positive definite, and