Quantile-distribution Functions and Their Use for Classification

We develop a flexible parametric framework for the estimation of quantile functions. The method involves the specification of an analytical quantile distribution function for the data at hand [1]. We focus on quantile functions that are linear with respect to their parameters, such as the flattened generalized logistic distribution [2]: these can adapt to a wide range of distributional shapes and allow for the estimation to be carried out through a computationally efficient least-squares method based on the order statistics. Inferential properties of this estimator, such as its asymptotic distribution, are derived, and these allow for the definition of a test of hypothesis for the equality of two distributions. The properties of the test are evaluated via a simulation study. Our method of quantile function estimation is implemented as a density estimation method in the naïve Bayes classifier. This innovation is compared to standard approaches for the classifier in a simulation study, and is illustrated on a real data set coming from microRNA profiling in human Medulloblastoma. Moreover, the test of hypothesis is shown to be useful as a variable selection method.