Study of a family of asymmetric densities and flexible quantile regression

Abstract

Probability distributions are fundamental tools for data analysis and statistical inference. Although often a symmetric distribution (e. g. normal distribution) is used for statistical modelling, it is unsuitable for multiple other applications. In this dissertation, we study a broad family of asymmetric distributions of a continuous random variable that can describe skewed data properly.

We establish expressions for important characteristics of the distributions and study estimation of the parameters via two estimation techniques. The distributional properties of both estimators are investigated and their asymptotic behavior (for large sample sizes) is compared. The results are applied to some specific examples of asymmetric densities.

When covariates come into play we consider a conditional setting, via flexible modeling allowing for semiparametric conditional quantile regression in this broad family of asymmetric densities. This approach in a regression setting shares basic philosophy with generalized (non)linear models but focuses on conditional quantile estimation instead of conditional mean estimation. The practical use of the proposed methods is illustrated in real data applications. A simulation study shows that the semiparametric quantile estimation method can lead to considerably smaller prediction errors than when using a nonparametric quantile estimation method. In addition, we develop two formal testing tools for the test of symmetry in the (semi)parametric framework. A small simulation study has been conducted to investigate the performances of both tests. Real data applications are added for illustrating the proposed testing procedures. Finally, an R-package QBAsyDist (available at the CRAN website) is developed and contains all proposed methods.