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ISBN 3-900051-07-0

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>  # augrain1.r                  
>  #  Example 9.5 -- maximize likelihood/concentrated likelihood
>  #
>  #  data
>   both <- matrix(scan("augrain.dat"),45,2,byrow=TRUE)
>   y <- both[,2]
>   y
 [1] 4.63 4.18 6.69 1.77 3.66 1.37 3.65 1.95 1.88 5.86 6.45 2.53 5.07 2.94 1.37
[16] 4.05 0.92 2.20 4.95 4.15 3.82 2.85 3.36 4.68 5.34 3.99 7.67 3.84 4.74 3.94
[31] 4.20 4.71 2.33 4.28 4.20 5.89 5.43 3.95 2.98 1.19 2.85 3.88 6.97 2.89 2.60
>  #                  define log-likelihood function
>     unlike <- function (theta) { # loglikelihood for 2 param gamma
+              n <- length(y) 
+              sumy <- sum(y)
+              sumly <- sum(log(y))
+       unlike <-  n*theta[1]*log(theta[2]) + n*lgamma(theta[1]) - 
+                     (theta[1]-1)*sumly + sumy/theta[2] }
>  #
>  #                  concentrated/profile log-likelihood function
>     lcalph <- function (alpha) { # profile llike 
+              n <- length(y) 
+              sumy <- sum(y)
+              sumly <- sum(log(y))
+          lcalph <- - n*alpha*log(sumy/(n*alpha)) - n*lgamma(alpha) + 
+                     (alpha-1)*sumly - n*alpha }
>  #
>  #                  define gradient function
>     grunlike <- function (theta) { # gradient of unlike
+              n <- length(y) 
+              sumy <- sum(y)
+              sumly <- sum(log(y))
+              dldalph <- n*log(theta[2]) + n*digamma(theta[1]) - sumly
+              dldbeta <- n*theta[1]/theta[2] - sumy/(theta[2]**2)
+           grunlike <- c(dldalph,dldbeta) }
>  #  
>  #  starting values from method of moments
>     n <- length(y)
>     sy <- sum(y)
>     vy <- var(y)
>     b0 <- n*vy/sy
>     a0 <- sy*sy/(n*n*vy)
>     v <- unlike(c(a0,b0))
>     print("method of moments estimates for starting values")
[1] "method of moments estimates for starting values"
>     print(c(a0,b0,v))
[1]  5.8357604  0.6582023 84.7092968
>  #  maximize profile like
>     that1 <- optimize(f=lcalph, c(a0/5,a0*5), maximum=TRUE )
>     print("maximize concentrated loglikelihood")
[1] "maximize concentrated loglikelihood"
>     that1
$maximum
[1] 5.1414

$objective
[1] -84.50925

>     print("value at maximum")
[1] "value at maximum"
>     ta <-that1$maximum
>     ta
[1] 5.1414
>     f <- lcalph(ta)
>     f
[1] -84.50925
>  #  check gradient
>     tb <- sy/(n*ta)
>     print(c(ta,tb))
[1] 5.1414002 0.7470944
>     grad <- grunlike(c(ta,tb))
>     grad
[1] 2.676173e-06 5.684342e-14
>  #  check original loglikelihood function
>     f <- unlike( c(ta,tb) )
>     f
[1] 84.50925
>  #                  done
>     rm(list=ls())
>   q()