Download Applied Time Series Analysis I. Proceedings of the First by D.F. Findley PDF

By D.F. Findley

ISBN-10: 0122572505

ISBN-13: 9780122572500

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II. The Gn - and en-Transformations. , 1971) and en-transformation (Shanks, 1955), a clear understanding of G-spectral estimation seems unlikely and hence we devote this section to those topics. DEFINITION 1. Let z (0) z (k) z(l) (1 ) and F(t) = Then ~define t f a f(x)dx. (2) the Gn-transformation of F(t) F(t) f(t) F(t+h) f(t+h) ~ F (t+nh) f(t+nh) f(t+(n-l)h) . • • f(t+(2n-l)h) 1 1 •.. • f (t+nh) f (t) f(t+(n-l)h) . f(t+(2n-l)h) Hn+l(F(t+jh); f(t+(i+j)h» Hn+l(l ; f(t+(i+j)h» (3) H. L. Gray et al.

54 THEOREM 7. ) Hn(m+ ) [l;f;+J')' l -- n J ' n+l J ' (m+l) (m) Hn +l [l;f i + j ) and Hn + 2 [1;f i + j ) ~ not ~, then 4 G(m) n+l G (m+l) _ G (m) n n G (m) + n (29) 1 - R (m) n+l where (m+1) r n+l r(m) n+l R(m) n+l (30) In order to calculate the G-transform recursively, it is clear from (29) that a recursive calculation of R~~i also is needed. This is accomplished by the following relations which also were established in [51). [Sns r (m+l) n (m) r n +1 lmHl n (m) -1] , -r n (m+l) s. •. and m s o rl (m) (m) (m+l) ] rn - 1 [ r (m) n = , r r n n (m) 'I 0 (m) 0 0, 1, 2, •..

DEFINITION 3. Let {x(t)} be a continuous or discrete index set covariance stationary stochastic process with autocorrelation function p(T), and let PT(T) be an estimator for p(T) based on a sample record from -T to T. Further let f T 0 e-i2nwT PT(T)dT -TO ( 19) (20) where 0 < NO + (2k-l)h < T and 1001 ~ 1/2. ), if the index set is continuous and ~ H. L. Gray et al. 50 (21) if the index set is discrete. ;;;~ Although it is not necessary for our results to hold, it is convenient in the future to assume real processes.

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