Maximum Likelihood Estimators of Generalized Gaussian Distribution With an ℍ-Proper Quaternion Random Variable

The currently known generalized Gaussian distributions (GGD) with an augmented quaternion random variable are based on the 3D GGD distribution. That is, they are based on random variables with three components. In this article, GGD for an augmented quaternion random variable based on GGD for a random variable with four components is given. Then the distribution obtained is simplified for an
$\mathbb {H}$
-proper quaternion random variable. For this simplified distribution the maximum likelihood estimators are presented. The simplification allows the presentation of the ML estimator for the shape parameter of GGD by eliminating the coefficients related to the covariance from the non-linear equation. The performance of the derived ML estimators is examined in the simulations.