Representative Points of the Inverse Gaussian Distribution and Their Applications.

The inverse Gaussian (IG) distribution, as an important class of skewed continuous distributions, is widely applied in fields such as lifetime testing, financial modeling, and volatility analysis. This paper makes two primary contributions to the statistical inference of the IG distribution. First, a systematic investigation is presented, for the first time, into three types of representative points (RPs)-Monte Carlo (MC-RPs), quasi-Monte Carlo (QMC-RPs), and mean square error RPs (MSE-RPs)-as a tool for the efficient discrete approximation of the IG distribution, thereby addressing the common scenario where practical data is discrete or requires discretization. The performance of these RPs is thoroughly examined in applications such as low-order moment estimation, density function approximation, and resampling. Simulation results demonstrate that the MSE-RPs consistently outperform the other two types in terms of approximation accuracy and robustness. Second, the Harrell-Davis (HD) and three Sfakianakis-Verginis (SV1, SV2, SV3) quantile estimators are introduced to enhance the representativeness of samples from the IG distribution, thereby significantly improving the accuracy of parameter estimation. Moreover, case studies based on real-world data confirm the effectiveness and practical utility of this quantile estimator methodology.