One maximum likelihood estimation is calculated according to equation:
$$\widehat{\ell}=\operatorname*{argmin}_\ell\mathbb{E}f(\ell,k),$$
where $f(\ell,k)$ is a function of two variables:
$$f(\ell,k)=\binom \ell k \binom {n-\ell} {m-k}.$$
One of arguments $k$ is random variable with hypergeometric distribution:
$$P(k)=\frac{\binom {\ell-g_1}{k-g_2}\binom{h_1-\ell+g_1}{h_2-k+g_2}}{\binom {h_1}{h_2}}.$$
This probability has values more then zero on interval $k\in[g_2,g_2+h_2]$.
I want to prove that if $\ell\in[g_1,g_1+h_1]$, then minimum of objective function locates on one of the borders:
$$\min\limits_\ell\mathbb{E}f(\ell,k)=\min\left(\binom {g_1} {g_2}\binom {n-g_1} {m-g_2},\binom {g_1+h_1} {g_2+h_2}\binom {n-g_1-h_1} {m-g_2-h_2}\right),$$
where all variables and constants in binomial cooefs are nonnegative integers.
To prove it I tried to calculate: $$\mathbb{E}f(\ell,k)=\sum\limits_k f(\ell,k)P(k)$$
$$\mathbb{E}f(\ell,k)={\binom {h_1}{h_2}}^{-1} \sum\limits_{k=\max(g_2,g_2+h_2+\ell-g_1-h_1)}^{\min(\ell-g_1,h_2)} \binom \ell k \binom {n-\ell} {m-k}\binom {\ell-g_1}{k-g_2}\binom{h_1-\ell+g_1}{h_2-k+g_2}$$
But when I read "Concrete mathematics" I can not find some formula or trick to use in this sum. It looks like the some modification of Vandermonde's identity, but ...
- how to use this to get closed form of this? It is possible that closed form doesn't exist.
- Another question is how to find this descrete function extremum?
- Am I going right way trying to calculate last expression? May be, there exist another way to prove that minimum is on borders.
Actually, I use log-likelihood instead likelihood (first equation) to estimate $\ell$ beacuse this helps to avoids big integers. I'm not sure about the equation:
$$\operatorname*{argmin}\limits_\ell\mathbb{E}\log{f(\ell,k)}=\operatorname*{argmin}\limits_\ell \mathbb{E}f(\ell,k)$$
To ilustrate this problem in terms of log-likelihood I posted this function realization and sample mean over 100 realizations. We can see that hypothesis i want to prove can be statistically accepted. Existing of proof is my question to mathematical community.
Thank you for any help or comments!