Mao Zai TIAN, Man Lai TANG, Ping Shing CHAN
The classic hierarchical linear model formulation provides a considerable flexibility formodelling the random effiects structure and a powerful tool for analyzing nested data that arise invarious areas such as biology,economics and education.However,it assumes the within-group errorsto be independently and identically distributed(i.i.d.)and models at all levels to be linear.Mostimportantly,traditional hierarchical models(just like other ordinary mean regression methods)cannotcharacterize the entire conditional distribution of a dependent variable given a set of covariates and failto yield robust estimators.In this article,we relax the aforementioned and normality assumptions,anddevelop a so-called Hierarchical Semiparametric Quantile Regression Models in which the within-grouperrors could be heteroscedastic and models at some levels are allowed to be nonparametric.We presentthe ideas with a 2-level model.The level-1 model is specified as a nonparametric model whereas level-2model is set as a parametric model.Under the proposed semiparametric setting the vector of partialderivatives of the nonparametric function in level-1 becomes the response variable vector in level 2.Theproposed method allows us to model the fixed effiects in the innermost level(i.e.,level 2)as a functionof the covariates instead of a constant effiect.We outline some mild regularity conditions required forconvergence and asymptotic normality for our estimators.We illustrate our methodology with a realhierarchical data set from a laboratory study and some simulation studies.
