Modelling and classification with quantile-based distributions

In this work, we explore and demonstrate the potential for modeling and classification using quantile-based distributions, which are random variables defined by their quantile function. In the first part we formalize a least squares estimation framework for the class of linear quantile functions, leading to unbiased and asymptotically normal estimators. Among the distributions with a linear quantile function, we focus on the flattened generalized logistic distribution (fgld), which offers a wide range of distributional shapes. A novel naïve-Bayes classifier is proposed that utilizes the fgld estimated via least squares, and through simulations and applications, we demonstrate its competitiveness against state-of-the-art alternatives. In the second part we consider the Bayesian estimation of quantile-based distributions. We introduce a factor model with independent latent variables, which are distributed according to the fgld. Similar to the independent factor analysis model, this approach accommodates flexible factor distributions while using fewer parameters. The model is presented within a Bayesian framework, an MCMC algorithm for its estimation is developed, and its effectiveness is illustrated with data coming from the European Social Survey. The third part focuses on depth functions, which extend the concept of quantiles to multivariate data by imposing a center-outward ordering in the multivariate space. We investigate the recently introduced integrated rank-weighted (IRW) depth function, which is based on the distribution of random spherical projections of the multivariate data. This depth function proves to be computationally efficient and to increase its flexibility we propose different methods to explicitly model the projected univariate distributions. Its usefulness is shown in classification tasks: the maximum depth classifier based on the IRW depth is proven to be asymptotically optimal under certain conditions, and classifiers based on the IRW depth are shown to perform well in simulated and real data experiments.