  Semiparametric models are often considered for analyzing longitudinal data
for a good balance between flexibility and parsimony. In this paper, we study a
class of marginal partially linear quantile models with possibly varying
coefficients. The functional coefficients are estimated by basis function
approximations. The estimation procedure is easy to implement, and it requires
no specification of the error distributions. The asymptotic properties of the
proposed estimators are established for the varying coefficients as well as for
the constant coefficients. We develop rank score tests for hypotheses on the
coefficients, including the hypotheses on the constancy of a subset of the
varying coefficients. Hypothesis testing of this type is theoretically
challenging, as the dimensions of the parameter spaces under both the null and
the alternative hypotheses are growing with the sample size. We assess the
finite sample performance of the proposed method by Monte Carlo simulation
studies, and demonstrate its value by the analysis of an AIDS data set, where
the modeling of quantiles provides more comprehensive information than the
usual least squares approach.
