ITEM METADATA RECORD
Title: P-splines quantile regression in varying coefficient models
Other Titles: P-splines kwantielregressie in variërende coëfficiënten
Authors: Andriyana, Yudhie
Issue Date: 14-Jan-2015
Abstract: 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.


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