The inversion of correlation matrix for MA(1) process

K e y w o r d s I n v e r s i o n of the correlation matrix of AR(1) process, Partitioning, Inversion of a submatrix. 1. I N T R O D U C T I O N Cons ide r a t ime-ser ies rea l i za t ion {Yt: t = 1 , . . . , n} of l eng th n f rom a s t a t i o n a r y Gauss i an process. Suppose t h a t Ptt' refers to t he cor re la t ion be tween Yt and Yt' for t, t ' = 1 , . . . , n. Obviously , Ptt = 1. If P t t ' = P (say), for all t ~ t', t , t ' = 1 , . . . ,n , t h e n the process is known to be exchangeab le or equ icor re l a t ion ( E Q C ) process. Fur the r , suppose t h a t CE = (1 p)In + PJn deno tes t he n × n cor re la t ion m a t r i x of th is E Q C process, where Jn is an n × n m a t r i x of ls . T h e n its inverse CE 1 f rom [1] is g iven by CE 1 = (a -b)In + bJn, where a = {1 + (n 2)p}/[(1 -p){1 + (n 1)p}] and b = p / [ ( 1 p){1 + (n 1)p}]. If {y t : t = 1 , . . . , n } follow t h e AR(1 ) process Yt = ¢Yt-1 + at, where 1 < ¢ < 1 is t he i .~ . N ( 0 , qa2), t h e n the cor re la t ion m a t r i x CA (¢ l t t ' l ) of th is p a r a m e t e r of t he process, and at = This research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. Authors thank the referee for the valuable comments. 0893-9659/03/$ see front matter (~) 2003 Elsevier Science Ltd. All rights reserved. Typeset by AA/~-2~X PII: S0893-9659(02)00199-4 318 B.C. SUTRADHAR AND P. KUMAR process has its inverse CA 1 [2] CA 1 = 1 1 ~ 1 --¢ 0 0 0 0 --¢ 1 + ¢ 2 --¢ 0 . . . 0 0 0 ¢ 1+¢2 _ ¢ . . . 0 0 : : : : : : : 0 0 0 0 . . . 1 + ¢2 _ ¢ 0 0 0 0 . . . . ¢ 1 (1.1) If {ye:t = 1 , . . . ,n}, however, follow the MA(1) process Y, = at Oat-l, where 1 < 8 < 1 is the pa ramete r of the process, and at i.i~. N(0, a2), then it is not so easy to invert the correlation matr ix CM of this process [2], where the n x n CM matr ix is given by 1 81 0 0 ''" 0 0 81 1 81 0 . . . 0 0 0 81 1 81 . . . 0 0 CM = , (1.2) 0 0 0 0 . . . 1 81 0 0 0 0 ' ' ' 81 1 where 81 = 0 / ( 1 + 82). As a remedy, some approximations were suggested to obtain the inverse of the CM matr ix [3,4]. More specifically, let c w denote the (t, t ' ) th element of the approximate inverse mat r ix of CM. Following [3,4], these elements are given by ( I ) ] e V [ { i " v / ~ } lt-t'' (1.3) (1 + 0~) ~ 201 Later on, Shaman [5] considered three new approximations to derive the inverse of the CM matrix. Under certain modifications, these three approaches appear to agree with the approximat ion given in (1.3). In the next section, we provide the exact expression for the inverse CM 1 of the correlation matr ix CM. A numerical illustration is given in Section 3 to compare the exact inverse CM 1 with the approximate inverse of CM. In Section 4, we discuss an immediate application of this inversion process to the longitudinal da ta analysis. 2. D E R I V A T I O N O F CM 1 THEOREM 2.1. For t, t' = 1 , . . . , n, the (t, it) th element of the inverse matr ix o r e M (1.2) is given by 1 + 8 2 / ~ ;O'*t" -~2(n+2) ' t ' -2~ -0 '+" PROOF. The technique of the derivation depends on the fact tha t the inverse of the correlation mat r ix of the AR(1) process, i.e., CA 1, contains an ( n 2) x ( n 2) symmetr ic mat r ix which has the same s t ructure as the correlation mat r ix of the MA(1) process. More specifically, part i t ion the CA 1 mat r ix in (1.1) as cA 1 ~_-~ G ~ , (2.1) where G = [0, 0 , . . . , ¢ / ( 1 + ¢2)]T is the (n 1) x 1 vector, Q = 1/(1 + ¢2) is a scalar quantity, G T is the t ranspose of G, and P is the leading (n 1) x (n 1) symmetr ic mat r ix which may further be part i t ioned as [A P = DM '