I'm having trouble with an exercise about maximum likelihood estimators.
Specifically, the exercise gives me values of a protein which was found in 50 adults. We assumed that the data follow a gamma distribution:
$X \sim \Gamma(r,\lambda)= \frac {\lambda^{r}}{\Gamma(r)}x^{r-1}e^{-\lambda x} $ if $x\ge0$.
It asks me to find the maximum likelihood estimators of parameters $\lambda$ and $r$.
How can I find those parameters given that from the data I have $E(X),Var(X)$?
We know that $\Gamma(r,\lambda)= \frac {1}{\Gamma(r)}\lambda^{r}x^{r-1}e^{-\lambda x} $ if $x\ge0$. In this case the likelihood function $L$ is $$\prod_i \Gamma(r,\lambda)_{x_i}=\frac{1}{\Gamma(r)^{n}}\lambda^{nr}x_1^{r-1}x_2^{r-1}...x_n^{r-1}e^{-\lambda T}$$ where $T=x_1+...+x_n$;
By apllying the logaritmic function to $L$ we semplificate the problem so
$$logL=(r-1)\sum_ilogx_i-\lambda T +(nr)log\lambda -nlog(\Gamma(r))$$
and now we must find the point of max of $logL$, so $\frac{\partial L}{\partial\lambda}= -T+\frac{nr}{\lambda}=0$ which have as solution $\hat\lambda = \frac{nr}{T}$.
With the same method you can obtain the extimation for $r$. (Find $\frac {\partial L}{\partial r}$ and put it equal to $0$).