The Consistency of Quantile Regression in Linear Mixed Models

Abstract

Quantiles are parameters of a distribution, which are of location and of scale character at the same time. The median, as a location parameter, is even robust and outperforms the mean, whenever there are outliers or extreme values in the data. In linear models quantile regression was firstly introduced by Koenker and Bassett [1978]. The method is well investigated and the asymptotic behaviour, like the consistency, was already proven. Doing so one is able to use the equivalence of the linear quantile model to a linear model with asymmetric Laplacian error terms. The asymmetric Laplace distribution has established a direct link to quantile estimation and is investigated in Yu and Zhang [2005]. In linear mixed models the quantile estimation was recently developed by Geraci and Bottai [2007] as well as Geraci and Bottai [2014]. There also the equivalence to an asymmetric Laplacian mixed model is employed. The estimation of the quantile is possible due to the shift to a maximum likelihood approach. An estimating algorithm of numerical kind is implemented in the open software R (see the package lqmm by Geraci [2016]). Due to the complex appearance of the log-likelihood density analytical solutions are pending. However the asymptotic theory was outstanding, up to this thesis, which shows the consistency of the conditional quantile estimator under some additional conditions. In proofing this property the Weiss’ Theorem (cf. Weiss [1971] and Weiss [1973]) for the dependent observations in the maximum likelihood estimation is applied. In the linear mixed model Miller [1977] and Pinheiro [1994] also employed this theorem, when they proved the asymptotic normality of the parameter estimators for the mean estimation. In the quantile estimation the necessity for its application is the calculation of the second derivatives from the loglikelihood density with respect to the unknown parameters. These constitute a form of the Fisher information matrix. Therefore they determine the asymptotic variance of the parameter estimators and are needed for the proof of the assumption in theWeiss’ Theorem. The resulting asymptotic normality of the parameter estimators imply the consistency of a conditional τ-quantile estimator for a given value τ in (0,1). Both proven properties, the asymptotic normality of the parameter estimators and the consistency of the quantile estimator, are supported by model-based simulation studies. The application of quantile regression in linear mixed models is shown to be applicable for count data. In this special case the consistency is also proven here. Furthermore a method called Microsimulation via Quantiles for the estimation of parameters, which are beyond the mean of a population, is proposed. There the natural connection between quantiles and the distribution function is deployed leading into an estimation of the whole distribution. From there any parameter of interest – e.g. be a quantile, a proportion, or others – can be generated by a Monte Carlo simulation or microsimulation.