Parametric quantile regression based on the inverse Gaussian distribution

The inverse Gaussian (IG) distribution is one of the most important distributions to analyse positive asymmetric data. However, the IG distribution does not have a closed form for its parametric quantile function, which hinders its utilization for quantile regression purposes. In this paper, we develop a particular fully parametric quantile regression model based on the IG distribution for modelling positive response variables in different quantiles. In particular, we propose a new and straightforward reparameterization of the IG distribution based on an approximation to its quantiles. In contrast to the original parameterization, the proposed parameterization leads to regression coefficients directly associated (approximately) with the quantiles of the response variable. We discuss the estimation of model parameters by maximum likelihood. A Monte Carlo experiment is conducted to evaluate the performance of these estimators in finite samples, examining the influence of outliers, and confirming that the proposed regression model seems to be a new robust alternative for modelling asymmetric positive real data. The effectiveness of our approach is illustrated with a dataset related to the price of cars in India. The results of our model with this dataset demonstrate the practical potential of the proposed parametric quantile regression model for fitting positive response variables.