Gamma distribution

Maximum likelihood estimation

There are no explicit solutions for ${\hat{\alpha }}_{MLE}$ and ${\hat{\beta }}_{MLE}$. One should use numerical methods to find the estimations.

Method of moments estimation

Let $\alpha$ and $\beta$ unknown for $\text{Gamma}(\alpha ,\beta )$. We need two moments to estimate both these

$$\begin{array}{rl}{m}_1& =\bar{x}=\hat{\alpha }\hat{\beta }\\ m_2& =s^2+{\bar{x}}^2=\hat{\alpha }{\hat{\beta }}^2+(\hat{\alpha }\hat{\beta }{)}^2\\ \bar{x}& =\hat{\alpha }\hat{\beta }\\ s^2& =\hat{\alpha }{\hat{\beta }}^2\\ \frac{s^2}{\bar{x}}& =\frac{\hat{\alpha }{\hat{\beta }}^2}{\hat{\alpha }\hat{\beta }}⟹{\hat{\beta }}_{MME}=\frac{s^2}{\bar{x}}\\ \hat{\alpha }\frac{s^2}{\bar{x}}& =\bar{x}⟹{\hat{\alpha }}_{MME}=\frac{{\bar{x}}^2}{s^2}\end{array}$$