  The Mann-Whitney-Wilcoxon rank sum test (MWWRST) is a widely used method for
comparing two treatment groups in randomized control trials, particularly when
dealing with highly skewed data. However, when applied to observational study
data, the MWWRST often yields invalid results for causal inference. To address
this limitation, Wu et al. (2014) introduced an approach that incorporates
inverse probability weighting (IPW) into this rank-based statistics to mitigate
confounding effects. Subsequently, Mao (2018), Zhang et al. (2019), and Ai et
al. (2020) extended this IPW estimator to develop doubly robust estimators.
  Nevertheless, each of these approaches has notable limitations. Mao's method
imposes stringent assumptions that may not align with real-world study data.
Zhang et al.'s (2019) estimators rely on bootstrap inference, which suffers
from computational inefficiency and lacks known asymptotic properties.
Meanwhile, Ai et al. (2020) primarily focus on testing the null hypothesis of
equal distributions between two groups, which is a more stringent assumption
that may not be well-suited to the primary practical application of MWWRST.
  In this paper, we aim to address these limitations by leveraging functional
response models (FRM) to develop doubly robust estimators. We demonstrate the
performance of our proposed approach using both simulated and real study data.
