On Fri, 8 Dec 2006, [EMAIL PROTECTED] wrote:
Dear R users, I??m a graduate students and in my master thesis I must
obtain the values of the parameters x_i which maximize this Multinomial
log??likelihood function log(n!)-sum_{i=1]^4 log(n_i!)+sum_ {i=1}^4 n_i
log(x_i)
under the following constraints:
a) sum_i x_i=1, x_i>=0,
b) x_1<=x_2+x_3+x_4
c)x_2<=x_3+x_4
I have been using the
??ConstrOptim?? R-function with the instructions I report below, and I
have tried to implement them with different values of ??n??. BUT I have
encountered 2 problems:
The problem is that the first constraint is not an inequality but an
equality. Writing it as two inequalities results in the feasible region
for the optimization being a very narrow slice of four-dimensional space,
which makes the optimization difficult.
There are at least two ways to fix the problem. The first is to note that
the loglikelihood is monotone in each x, so that sum_i x_i <=1 is
sufficient when maximizing. The second is to reparametrize in terms of
three parameters.
Minimization is more challenging, because the loglikelihood does not have
a minimum. It is negative infinite when any x_i is zero and the
corresponding n is non-zero.
-thomas
My R instructions
n=c(10,20,3,5)
n1=n[1]
n2=n
[2]
n3=n[3]
n4=n[4]
logfr=function(x) { ##function to maximize
x1= x
[1]
x2= x[2]
x3= x[3]
x4= x[4]
log(factorial(sum(n)))-sum(log
(factorial(n)))+sum(n*log(x))
}
grr.log <- function(x) { ## Gradient
of 'log fr'
x1=x[1]
x2=x[2]
x3=x[3]
x4=x[4]
return(n/x)
}
par.start=
c(.19999999,.15,.4,.25)
constr.coeff = rbind(diag(1,4,4),c(-1,1,1,1),c
(0,-1,1,1),c(-1,-1,-1,-1), c(1,1,1,1))
constr.tn= c(0,0,0,0,0,0,-1,.
9999999)
min= constrOptim(par.start, logfr, grr.log, ui=constr.coeff,
ci=constr.tn)
max=constrOptim(par.start, logfr, grr.log, ui=constr.
coeff, ci=constr.tn, control=list(fnscale=-1))
______________________________________________
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PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Thomas Lumley Assoc. Professor, Biostatistics
[EMAIL PROTECTED] University of Washington, Seattle
______________________________________________
[email protected] mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
