A globally convergent and consistent method for estimating the shape parameter of a generalized Gaussian distribution

We propose a novel methodology for estimating the shape parameter of a generalized Gaussian distribution (GGD). This new method is based on a simple estimating equation derived from a convex shape equation. The estimating equation is completely independent of gamma and polygamma functions. Thus, no lookup table, interpolation, or additional subroutine to evaluate these functions are required for real-time implementations of the proposed method, which is in contrast to all existing methods. Furthermore, we establish that the shape equation has a unique global root on the positive real line and the Newton-Raphson root-finding algorithm converges to the unique global root from any starting point in a semi-infinite interval Theta_min. More importantly, we show that the sample-based shape estimating equation has a unique global root with probability tending to one and the root is consistent for the true shape parameter. Finally, we prove via fixed point arguments that, with probability tending to one, the Newton-Raphson algorithm converges to the unique global root of the sample shape estimating equation from any starting point in Theta_min. Some numerical experiments are also provided to demonstrate the global convergence and the excellent finite sample performance of the proposed method