Quantile regression for interval censored data using an Enriched Laplace distribution

Consider a linear quantile regression model in which the response is subject to interval censoring. Quantile regression is an alternative to the common mean regression, and has the advantage that it allows to give a broader view of the conditional distribution and it is less sensitive to outliers. We propose novel estimators of the quantile regression coefficients for any quantile level 0<τ<1 by converting the quantile regression problem into a maximum likelihood problem. However, unlike the case of uncensored data, the quantile regression problem cannot be regarded as a maximum likelihood problem with a Laplace distribution for the error term. Instead, the error distribution will be approximated by a generalization of the Laplace distribution employing Laguerre polynomial expansions. This enriched Laplace distributuion will by construction have τ−th quantile equal to zero and can approximate the true underlying error distribution arbitrarily well. We show the consistency of the proposed estimator and investigate the finite sample performance of the method by means of extensive simulations. The estimator is also applied on data regarding the starting salary of newly graduated students.