# [R] Likelihood optimization numerically

Mohammad Ehsanul Karim wildscop at yahoo.com
Sun Jan 27 07:47:40 CET 2008

```Dear List,

I am not sure how should i optimize a log-likelihood numerically:
Here is a Text book example from Statistical Inference by George Casella, 2nd
Edition Casella and Berger, Roger L. Berger (2002, pp. 355, ex. 7.4 # 7.2.b):

data = x =  c(20.0, 23.9, 20.9, 23.8, 25.0, 24.0, 21.7, 23.8, 22.8, 23.1, 23.1, 23.5, 23.0, 23.0)
n <- length(x)

# likelihood from a 2 parameter Gamma(alpha, beta), both unknown
llk = -n*log(gamma(alpha)) - n*alpha*log(beta) + (alpha - 1)*(sum(log(x))) - (sum(x))/beta

# analytic 1st derivative solution w.r.t alpha, assuming beta known
# by putting MLE of beta = sum(x)/(n*alpha)
# (to simplify as far as possible analytically)
llk.1st = - n*digamma(alpha) -n*(log(sum(x)/(n*alpha))+1) + (sum(log(x)))

It feels like i should use
nls(... ,  trace=T, start=c(alpha=...),nls.control(maxiter=100,tol=.1))
but not sure "how".

Can anyone provide me hint?
Thank you for your time.

Ehsan

> R.Version()
\$platform
[1] "i386-pc-mingw32"
\$arch
[1] "i386"
\$os
[1] "mingw32"
\$system
[1] "i386, mingw32"
\$status
[1] ""
\$major
[1] "2"
\$minor
[1] "6.1"
\$year
[1] "2007"
\$month
[1] "11"
\$day
[1] "26"
\$`svn rev`
[1] "43537"
\$language
[1] "R"
\$version.string
[1] "R version 2.6.1 (2007-11-26)"

____________________________________________________________________________________
Be a better friend, newshound, and

```