Maximum likelihood and maximum a posteriori estimators for the Riesz probability distribution

Abstract

We focus on some statistical facets of the Riesz probability distribution that could replace and exceed the Wishart in many application fields. First, the maximum likelihood (ML) estimators of both Riesz parameters are derived using two approaches. The first one yields an equation solved using an algorithm alternating the Cholesky decomposition with intermediate calculations. The second one provides a closed-form solution of the ML-estimator, which is proven to be asymptotically unbiased. Afterward, we assume a Riesz as a prior for the maximum a posteriori estimator (MAP) of the scale parameter, which heads to a Riesz Inverse Gaussian ($\mathcal{R}\mathcal{I}\mathcal{G}$) posterior distribution. The resulting MAP estimator is simplified and solved via an algorithm alternating the Denman-Beavers algorithm and the Cholesky decomposition. We also characterize the Riesz-$\mathcal{R}\mathcal{I}\mathcal{G}$ model uniquely by a conditional distribution and a regression assumption. Finally, some supporting simulations illustrate the efficiency of these estimators. The corresponding computer codes are provided.

Keywords:

• Cholesky decomposition
• maximum a posteriori estimator
• maximum likelihood estimator
• Riesz inverse Gaussian distribution
• Riesz distribution
• Wishart distribution

2010 Mathematics Subject Classifications:

• 62H12
• 62F15
• 62F12
• 60E10
• 60B20

Disclosure statement

No potential conflict of interest was reported by the author(s).