Efficiency combined with simplicity: new testing procedures for Generalized Inverse Gaussian models

TL;DR

This paper introduces new, simple, and efficient testing procedures for the Generalized Inverse Gaussian family that combine the advantages of ML and MM estimators using Le Cam's methodology, with proven effectiveness through simulations and real data.

Contribution

It proposes an innovative combination of ML and MM estimators via Le Cam's approach to create simple yet efficient tests for GIG models, avoiding complex numerical methods.

Findings

Tests perform at least as well as likelihood ratio tests in simulations.

Procedures are simpler and computationally less intensive.

Effective on real-world data sets.

Abstract

The standard efficient testing procedures in the Generalized Inverse Gaussian (GIG) family (also known as Halphen Type A family) are likelihood ratio tests, hence rely on Maximum Likelihood (ML) estimation of the three parameters of the GIG. The particular form of GIG densities, involving modified Bessel functions, prevents in general from a closed-form expression for ML estimators, which are obtained at the expense of complex numerical approximation methods. On the contrary, Method of Moments (MM) estimators allow for concise expressions, but tests based on these estimators suffer from a lack of efficiency compared to likelihood ratio tests. This is why, in recent years, trade-offs between ML and MM estimators have been proposed, resulting in simpler yet not completely efficient estimators and tests. In the present paper, we do not propose such a trade-off but rather an optimal combination of both methods, our tests inheriting efficiency from an ML-like construction and simplicity from the MM estimators of the nuisance parameters. This goal shall be reached by attacking the problem from a new angle, namely via the Le Cam methodology. Besides providing simple efficient testing methods, the theoretical background of this methodology further allows us to write out explicitly power expressions for our tests. A Monte Carlo simulation study shows that, also at small sample sizes, our simpler procedures do at least as good as the complex likelihood ratio tests. We conclude the paper by applying our findings on two real-data sets.