Given a series of $N$ observations $\left(x_1, \ldots, x_N\right)$ that follow a Gamma distribution with a single parameter, $ \text{Gamma}(k, k)$, what is the maximum likelihood estimate of $ k $?. It should be noted that $k$ is both a shape and scale parameter, not a rate parameter.
My analysis boiled down to approximating the solution for the following equation: $$ \ln k + \psi(k) = \frac{1}{N} \sum_{i = 1}^N \ln x_i + \frac{1}{Nk^2}\sum_{i = 1}^N x_i - 1 $$ How should I estimate the solution for this equation?
Note that
$$\ln k + \psi(k) = E(\ln X) \approx \frac{1}{N} \sum_{i = 1}^N \ln x_i $$
Substituting in the equation for the MLE, you have
$$\frac{1}{N} \sum_{i = 1}^N \ln x_i = \frac{1}{N} \sum_{i = 1}^N \ln x_i + \frac{1}{Nk^2}\sum_{i = 1}^N x_i - 1 $$
$$\Rightarrow 1=\frac{1}{Nk^2}\sum_{i = 1}^N x_i \Rightarrow \hat k = \sqrt {\frac{1}{N}\sum_{i = 1}^N x_i }$$