>  # finder.r                   October 2008, 2009
>  #
>  #  Example 8.3 -- maximize likelihood/root of likelihood equation
>  #
>   x <- c(1,1,1,1,1,1,2,2,2,3)           # data
>  #                  define log-likelihood function
>     llike <- function (theta) { # loglikelihood for log-series model
+              n <- length(x) 
+              sumx <- sum(x)
+              llike <-  sumx*log(theta) - n*log(-log(1-theta)) }
>  # 
>  # note that llike only used data x -- not general
>  #
>  #                  derivative log-likelihood function
>     dllike <- function (theta,u) { # derivative of log-series llike 
+              n <- length(u) 
+              sumu <- sum(u)
+             dllike <- sumu/theta + n/((1-theta)*log(1-theta)) }
>  #
>  # note generality of dllike -- data as argument to function
>  #
>  #                  another way to define & use function
>     dllikex <- function(theta) dllike(theta,x)
>  #
>  #                  still another approach
>     llikex <- function (x) { # loglikelihood for log-series model
+              n <- length(x) 
+              sumx <- sum(x)
+              function(theta) sumx*log(theta) - n*log(-log(1-theta)) }
>  # 
>  #  First approach ** maximize llike
>  #
>  #  with one parameter, use optimize
>  #                    function  limits    
>     that1 <- optimize(f=llike, c(.02,.98), maximum=TRUE )
>     print("maximize loglikelihood")
[1] "maximize loglikelihood"
>     that1
$maximum
[1] 0.5335877

$objective
[1] -6.71288

>     print("value at maximum")
[1] "value at maximum"
>     t <-that1$maximum
>     t
[1] 0.5335877
>     f <- llike(t)
>     f
[1] -6.71288
>  #
>  #  Second approach ** find root of likelihood equation
>  #
>  #  with one parameter, use uniroot
>  #                   function   limits               pass data
>     that2 <- uniroot(f=dllike, lower=.02, upper=.98, u=x )
>     print("root of likelihood equation")
[1] "root of likelihood equation"
>     that2
$root
[1] 0.5335918

$f.root
[1] -8.666671e-05

$iter
[1] 7

$estim.prec
[1] 6.103516e-05

>     print("value at root")
[1] "value at root"
>     t <- that2$root
>     t
[1] 0.5335918
>  #
>  #  Third approach ** find root of likelihood equation
>  #
>  #  again with uniroot but with specific function
>  #                   function   limits              v  no x passed
>     that3 <- uniroot(f=dllikex, lower=.02, upper=.98 )
>     print("root of likelihood equation")
[1] "root of likelihood equation"
>     that3
$root
[1] 0.5335918

$f.root
[1] -8.666671e-05

$iter
[1] 7

$estim.prec
[1] 6.103516e-05

>     print("value at root")
[1] "value at root"
>     t <- that3$root
>     t
[1] 0.5335918
>     f <- llike(t)
>     f
[1] -6.71288
>  #  
>  #  Fourth approach ** maximize llike
>  #
>  #  with one parameter, use optimize
>  #                    function  limits 
>     xm1 <- c(0,0,0,0,0,0,1,1,1,2)
>     that4 <- optimize(f=llikex(xm1+1), c(.02,.98), maximum=TRUE )
>     print("maximize loglikelihood")
[1] "maximize loglikelihood"
>     that4
$maximum
[1] 0.5335877

$objective
[1] -6.71288

>     print("value at maximum")
[1] "value at maximum"
>     t <-that1$maximum
>     t
[1] 0.5335877
>     ff <- llikex(xm1+1)       # creates function
>     ff
function (theta) 
sumx * log(theta) - n * log(-log(1 - theta))
<environment: 0x18308868>
>     ff(t)                     # evaluate function at t
[1] -6.71288
>  #                  done
>     rm(list=ls())
>   q()