Abstract: An indirect maximum likelihood estimation method is proposed for the parameter estimation of the generalized inverse
Gaussian distribution. For the case where the order parameter λ < 0, variable substitution is performed on the parameters of the generalized inverse Gaussian distribution, and a covariate following the uniform distribution U（0, 1） is introduced. The substituted parameters to be estimated are then assumed to be a parametric function of this covariate. Furthermore, maximum likelihood estimation is applied to the parameters within this function to estimate the parameters of the generalized inverse Gaussian distribution. The effectiveness and practicality of the proposed method are validated through real-world case studies.

Key words: generalized inverse Gaussian distribution, maximum likelihood estimation, variable substitution, covariates