A new look at the inverse Gaussian distribution

TL;DR

This paper introduces a reparameterized inverse Gaussian distribution with a mode-based approach, simplifying its application in various statistical methods like nonparametric smoothing, robust modeling, and clustering, supported by EM algorithm implementations.

Contribution

It proposes a new mode-based parameterization of the inverse Gaussian distribution, facilitating its use in nonparametric, robust, and mixture modeling contexts.

Findings

The reparametrized IG (rIG) simplifies statistical modeling.

A boundary bias-free estimator is developed using rIG kernels.

The contaminated IG model effectively detects outliers.

Abstract

The inverse Gaussian (IG) is one of the most famous and considered distributions with positive support. We propose a convenient mode-based parameterization yielding the reparametrized IG (rIG) distribution; it allows/simplifies the use of the IG distribution in various statistical fields, and we give some examples in nonparametric statistics, robust statistics, and model-based clustering. In nonparametric statistics, we define a smoother based on rIG kernels. By construction, the estimator is well-defined and free of boundary bias. We adopt likelihood cross-validation to select the smoothing parameter. In robust statistics, we propose the contaminated IG distribution, a heavy-tailed generalization of the rIG distribution to accommodate mild outliers; they can be automatically detected by the model via maximum a posteriori probabilities. To obtain maximum likelihood estimates of the parameters, we illustrate an expectation-maximization (EM) algorithm. Finally, for model-based clustering and semiparametric density estimation, we present finite mixtures of rIG distributions. We use the EM algorithm to obtain ML estimates of the parameters of the mixture model. Applications to economic and insurance data are finally illustrated to exemplify and enhance the use of the proposed models.