  With the increasing availability of datasets, developing data fusion methods
to leverage the strengths of different datasets to draw causal effects is of
great practical importance to many scientific fields. In this paper, we
consider estimating the quantile treatment effects using small validation data
with fully-observed confounders and large auxiliary data with unmeasured
confounders. We propose a Fused Quantile Treatment effects Estimator (FQTE) by
integrating the information from two datasets based on doubly robust estimating
functions. We allow for the misspecification of the models on the dataset with
unmeasured confounders. Under mild conditions, we show that the proposed FQTE
is asymptotically normal and more efficient than the initial QTE estimator
using the validation data solely. By establishing the asymptotic linear forms
of related estimators, convenient methods for covariance estimation are
provided. Simulation studies demonstrate the empirical validity and improved
efficiency of our fused estimators. We illustrate the proposed method with an
application.
