  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.
