The true maximum-likelihood estimators for the generalized Gaussian distribution with p = 3, 4, 5

The generalized Gaussian distribution with location parameter μ, scale parameter σ, and shape parameter p contains the Laplace, normal, and uniform distributions as particular cases for p = 1, 2, +∞, respectively. Derivations of the true maximum-likelihood estimators of μ and σ for these special cases are popular exercises in many university courses. Here, we show how the true maximum-likelihood estimators of μ and σ can be derived for p = 3, 4, 5. The derivations involve solving of quadratic, cubic, and quartic equations.