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>  # chex44.r                         J F Monahan, June 2001, May 2010
>  # 
>  # likelihood computation for ARMA(1,1) model
>  #
>  # set parameter values
>  phi <- .9
>  th  <- .5
>  n <- 4
>  # table covariance function -- gamma(0) stored in gam[1]
>  gam <- rep(0,n+1)
>  gam[1] <- (1+th*th-2*phi*th)/(1-phi*phi)
>  gam[2] <- (1-phi*th)*(phi-th)/(1-phi*phi)
>  for(j in 3:(n+1)) gam[j] <- phi*gam[j-1]
>  "covariance function"
[1] "covariance function"
>  gam
[1] 1.8421053 1.1578947 1.0421053 0.9378947 0.8441053
>  # Toeplitz covariance matrix
>  d <- 1 + abs(matrix(1:n,n,n) - t(matrix(1:n,n,n)))
>  A <- matrix(gam[d],n,n)
>  "covariance matrix"
[1] "covariance matrix"
>  A
          [,1]     [,2]     [,3]      [,4]
[1,] 1.8421053 1.157895 1.042105 0.9378947
[2,] 1.1578947 1.842105 1.157895 1.0421053
[3,] 1.0421053 1.157895 1.842105 1.1578947
[4,] 0.9378947 1.042105 1.157895 1.8421053
> 
>  "Cholesky factor"
[1] "Cholesky factor"
>  Lt <- chol(A)
>  Lt
         [,1]      [,2]      [,3]      [,4]
[1,] 1.357242 0.8531234 0.7678111 0.6910300
[2,] 0.000000 1.0555973 0.4763721 0.4287349
[3,] 0.000000 0.0000000 1.0127394 0.4177550
[4,] 0.000000 0.0000000 0.0000000 1.0031201
>  det <-prod(diag(Lt))
>  "determinant"
[1] "determinant"
>  det*det
[1] 2.118421
>  z <- c(-2, 1, 0, 1)
>  one <- rep(1,4)
>  LiBz <- forwardsolve(t(Lt),z)
>  LiBone <- forwardsolve(t(Lt),one)
>  zAiz <- sum( LiBz * LiBz )
>  zAione <- sum( LiBz * LiBone )
>  oneAione <- sum( LiBone * LiBone )
>  muhat <- zAione / oneAione
>  "muhat"
[1] "muhat"
>  muhat
[1] -0.07971014
>  "quadratic form"
[1] "quadratic form"
>  qf <- zAiz - zAione*muhat 
>  qf
[1] 7.857101
>  # clean up
>  rm(list=ls())
>  q()