
parameter estimation
I'm working with the following conditioned probability density
$\displaystyle \Pr(RN)= \binom {L} {LR}\sum_{r=1}^{R}\binom {R} {r} (1)^{Rr}\left(\frac{r}{L}\right)^N
$
where $\displaystyle L,R,N$ are discrete variables, $\displaystyle L,R,N>0$and $\displaystyle R<=L$.
What I need is to find the maximum likelihood value for N that maximize the probability to have a given R. The approach I've tried is the classical ML estimation, so I set
$\displaystyle \frac{\partial\Pr\left( R=R'N\right)}{\partial N}=0$
that gives
$\displaystyle \binom {L} {LR}\sum_{r=1}^{R}\binom {R} {r} (1)^{Rr}\left(\frac{r}{L}\right)^N\ln\frac{r}{L} =0
$
The problem is that now I'm not able to solve this equation for $\displaystyle N$ (Headbang)
Is there any other way to find an estimate of $\displaystyle N$given a particular $\displaystyle M$ from the first conditioned probability? I need only an estimate so I can consider to make some approximations...but I need a closed form, I cannot resort to numerical methods.
Thanks in advance!