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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 theGIG. The particular form of GIG densities, involving modified Bessel functions,prevents in general from a closed-form expression for ML estimators, which areobtained at the expense of complex numerical approximation methods. On thecontrary, Method of Moments (MM) estimators allow for concise expressions, buttests based on these estimators suffer from a lack of efficiency compared tolikelihood ratio tests. This is why, in recent years, trade-offs between ML andMM estimators have been proposed, resulting in simpler yet not completelyefficient estimators and tests. In the present paper, we do not propose such atrade-off but rather an optimal combination of both methods, our testsinheriting efficiency from an ML-like construction and simplicity from the MMestimators of the nuisance parameters. This goal shall be reached by attackingthe problem from a new angle, namely via the Le Cam methodology. Besidesproviding simple efficient testing methods, the theoretical background of thismethodology further allows us to write out explicitly power expressions for ourtests. A Monte Carlo simulation study shows that, also at small sample sizes,our simpler procedures do at least as good as the complex likelihood ratiotests. We conclude the paper by applying our findings on two real-data sets.

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