In this dissertation we st udy quantile regression in varying coefficien t models using one particular nonparametric techni que called P-splines. Quantile regression off ers an alternative for mean regression analys is. It has been widely used in statistical modelin g. In traditional mean regression, the condit ional mean of the response for each fixed value of the predictors, is investigated.Similar ly, as a special case of quantile regression, ¨median regression focuses on a central location o f the response based on a fixed setof pr edictors. When the distribution is highly ske wed, median regression is more reasonable to¨ be used than mean regression. In a more gener al setting, quantile regression is an important to ol to describe the characteristics of a¨ conditional distribution function. It describes th e conditional quantile functions of the respo nse variable given fixed values of the predic tors. Quantile regression is able to give moreinfo rmation about the conditional distribution of ¨the response variable.
To¨ allow for analyzing complex data situations,¨ several flexible regression models have been¨ introduced. Among these are the varying coefficien t models, that differ with a classical linear ¨regression model by the fact that the r egression coefficients are no longer constant ¨but functions that vary with the value taken ¨by another variable, such as for example time. In ¨this dissertation we study quantile regressi on in varying coefficient models for longitudinaldata. The quantile function is modeled a s a function of the covariates and the main t ask is to estimate the unknown regression coeffici ent functions. We approximate each coeff icient function by means of P-splines.The keystatistical tools used in this dissertation are presented inChapter 1.
In Chapter 2, we investigate the theoretical prope rties of the estimators, such as rate of convergence and asymptotic distributional results. The estimation methodology requests solving ¨an optimization problem that also involves asmoothing parameter. For a special case the ¨optimization problem can be transformed into ¨a linear programming problem for which then a Fri sch-Newton interior point algorithm is used,¨ leading to a computationally fast and efficie nt procedure. Several data-driven choices of the s moothing parameters are briefly discussed, an d their performances are illustrated in a simulati on study. Some real data analysis demonstrate s the use of the developed method.
Population conditional quantile funct ions cannot cross for different quantile orde rs. Unfortunately estimated regression quantile cu rves often violate this non-crossingness ¨property, which can be very annoying for interpre tations and further analysis. Under the flexi ble varying coefficient modelling, we developmethods for quantile regression that ensure that¨ the estimated quantilecurves do not cross. Th is is done in Chapter 3. Additionally, we allow fo r some heteroscedasticity in the error modell ing, and also estimate the associated variabi lity function. Chapter 4 investigates a more¨ general setting of heteroscedasticity, and di scusses estimation methods in a more flexiblevariability setting of the error term. For b oth chapters (Chapters 3 and 4), we inve stigate the finite-sample performances of the disc ussed methods via simulation studies. Some ap plications to real data illustrate the use of themethods in practical settings.
The computational issues for all ch apters (Chapters 2, 3 and 4) are (mostly) bas ed on linear programming optimization problems. De aling with the optimization problems implies¨ the need of translating the quantile objectivefunction into a primal-dual linear prog ramming problem. We then apply a method which ¨is the so-called Frisch-Newton interior- point algorithm to find an optimal solution.¨For the computation of the algorithm, we use¨ the function rq.fit.sfn which is available in¨ the quantreg R-package. As an altern ative, in Chapter 2, we also propose to use aMatlab-based modelling system called CVX. So me detailed information about a selection of¨ encountered linear programming problems arepresent ed in the Appendix.
Finally, in Chapter 5 we dra w some conclusions and discuss some possible¨ topics for further research.