  The research in this paper gives a systematic investigation on the asymptotic
behaviours of four inverse probability weighting (IPW)-based estimators for
conditional average treatment effect, with nonparametrically,
semiparametrically, parametrically estimated and true propensity score,
respectively. To this end, we first pay a particular attention to
semiparametric dimension reduction structure such that we can well study the
semiparametric-based estimator that can well alleviate the curse of
dimensionality and greatly avoid model misspecification. We also derive some
further properties of existing estimator with nonparametrically estimated
propensity score. According to their asymptotic variance functions, the studies
reveal the general ranking of their asymptotic efficiencies; in which scenarios
the asymptotic equivalence can hold; the critical roles of the affiliation of
the given covariates in the set of arguments of propensity score, the bandwidth
and kernel selections. The results show an essential difference from the
IPW-based (unconditional) average treatment effect(ATE). The numerical studies
indicate that for high-dimensional paradigms, the semiparametric-based
estimator performs well in general {whereas nonparametric-based estimator, even
sometimes, parametric-based estimator, is more affected by dimensionality. Some
numerical studies are carried out to examine their performances. A real data
example is analysed for illustration.
