P3-2: Uncertainty Quantification in Quantile Mixed Models

PhD student: Peter Kramlinger
Supervisor: Prof. Dr. Tatyana Krivobokova
Group: Institute for Mathematical Stochastics

Quantile regression provides a convenient framework for analysing regression effects on conditional quantiles of a response distribution and therefore is one special case of distributional regression. In particular, quantile regression allows to study empirical phenomena such as heteroscedasticity or skewness arising due to the modulating effect of covariates. While the classical framework on linear quantile regression models with a purely parametric predictor is now well understood, semiparametric and nonparametric extensions are not yet as well established. Here certain numerical and theoretical difficulties arise due to combination of different types of a fitting criterion and penalties.

In this project, we focus on semiparametric quantile mixed models, which combine nonparametric fixed effects and complex random effects, including spatial ones. This kind of mixed models allow to handle various scaling issues that arise in statistical modelling. Thereby, both frequentist and Bayesian frameworks for estimation and inference will be considered. In particular, assuming a nonparametric effect to follow a Cauchy process and random effects to follow a Laplace distribution leads to the corresponding Bayesian formulation of semiparametric quantile mixed models.

In addition to the derivation of point estimates and appropriate estimation schemes, uncertainty quantification is of major interest in any statistical model. One possibility is the construction of pointwise or simultaneous confidence bands for the fixed and/or random effects part of a semiparametric model to be able to test the significance of these effects. Such uncertainty assessment in semiparametric quantile mixed models is challenging, first of all due to the distribution-free formulation.

Furthermore, we aim to study the asymptotic properties of confidence intervals both in the frequentist and Bayesian model formulations. For this task, we will consider different asymptotic scenarios arising from the consideration of classical mixed models (relating to a hypothetical population of individuals of infinite size) or small area statistics (where the number of spatial regions/individuals is fixed). These scenarios have important implications for the asymptotic properties that will be studied.

In summary, this project combines various aspects of statistical modelling for scaling problems (random effects in mixed models, spatial effects in small area statistics, semiparametric effects) in a quantile regression framework. The aim is to provide theoretical results that enable uncertainty quantification in a general class of quantile mixed models under various asymptotic scenarios. This yields a deeper general understanding of such models and is particularly important in applied problems, due to the possibility for the construction of corresponding tests for the significance of certain model effects.