  When a strict subset of covariates are given, we propose conditional quantile
treatment effect to capture the heterogeneity of treatment effects via the
quantile sheet that is the function of the given covariates and quantile. We
focus on deriving the asymptotic normality of probability score-based
estimators under parametric, nonparametric and semiparametric structure. We
make a systematic study on the estimation efficiency to check the importance of
propensity score structure and the essential differences from the unconditional
counterparts. The derived unique properties can answer: what is the general
ranking of these estimators? how does the affiliation of the given covariates
to the set of covariates of the propensity score affect the efficiency? how
does the convergence rate of the estimated propensity score affect the
efficiency? and why would semiparametric estimation be worth of recommendation
in practice? We also give a brief discussion on the extension of the methods to
handle large-dimensional scenarios and on the estimation for the asymptotic
variances. The simulation studies are conducted to examine the performances of
these estimators. A real data example is analyzed for illustration and some new
findings are acquired.
