True ML Estimator for the Location Parameter of the Generalized Gaussian Distribution with p = 4

Estimation of the location parameter of the generalized Gaussian distribution with shape parameter p=4 is studied and an explicit solution for the maximum likelihood estimator is derived. The Cramér Rao lower bound is derived and the mean square error of the new estimator is compared to it. The new maximum likelihood estimator attains the Cramér Rao lower bound for a moderate number of samples. Simulation results show the explicit maximum likelihood estimator has superior performance compared to the mean estimator and slightly better performance than the moment/Newton-step estimator. The new maximum likelihood estimator has similar computational complexity to the moment/Newton-step estimator.