Improving linear quantile regression for replicated data

Abstract: This paper deals with improvement of linear quantile regression, when there are a few distinct values of the covariates but many replicates. On can improve asymptotic efficiency of the estimated regression coefficients by using suitable weights in quantile regression, or simply by using weighted least squares regression on the conditional sample quantiles. The asymptotic variances of the unweighted and weighted estimators coincide only in some restrictive special cases, e.g., when the density of the conditional response has identical values at the quantile of interest over the support of the covariate. The dominance of the weighted estimators is demonstrated in a simulation study, and through the analysis of a data set on tropical cyclones.