  In some high-dimensional and semiparametric inference problems involving two
populations, the parameter of interest can be characterized by two-sample
U-statistics involving some nuisance parameters. In this work we first extend
the framework of one-step estimation with cross-fitting to two-sample
U-statistics, showing that using an orthogonalized influence function can
effectively remove the first order bias, resulting in asymptotically normal
estimates of the parameter of interest. As an example, we apply this method and
theory to the problem of testing two-sample conditional distributions, also
known as strong ignorability. When combined with a conformal-based rank-sum
test, we discover that the nuisance parameters can be divided into two
categories, where in one category the nuisance estimation accuracy does not
affect the testing validity, whereas in the other the nuisance estimation
accuracy must satisfy the usual requirement for the test to be valid. We
believe these findings provide further insights into and enhance the conformal
inference toolbox.
