Beyond Jensen's Inequality: Speeding Up ML Estimation of Generalized Hyperbolic Distributions

Abstract

The generalized hyperbolic (GH) distribution is a highly flexible probability distribution that finds applications in various fields, yet estimating its parameters is quite challenging. This paper focuses on the maximum likelihood (ML) estimation of the GH distribution. In the literature, several expectation-maximization (EM) type algorithms have been proposed, which iteratively optimize a surrogate objective called the Q-function—a tractable lower bound of the likelihood function. While these generic EM-type algorithms, derived based on Jensen’s inequality, provide a systematic framework, they often restrict flexibility in algorithm design. In this study, we adopt the block minorization-maximization (BMM) framework, a more general iterative surrogate maximization approach that subsumes EM-type algorithms as special cases, to address the ML estimation problem. We propose efficient, problem-specific algorithms that utilize novel surrogate functions, which provide a provably tighter surrogate function on the likelihood than the Q-function while still allowing for closed-form updates. As a result, the proposed algorithm achieves faster and guaranteed convergence. Numerical experiments on synthetic data confirm the superior convergence speed of our algorithms compared to existing ones.