  Suppose there are two unknown parameters, each parameter is the solution to
an estimating equation, and the estimating equation of one parameter depends on
the other parameter. The parameters can be jointly estimated by "stacking"
their estimating equations and solving for both parameters simultaneously.
Asymptotic confidence intervals are readily available for stacked estimating
equations. We introduce a bootstrap-based hypothesis test for stacked
estimating equations which does not rely on asymptotic approximations. Test
statistics are constructed by splitting the sample in two, estimating the first
parameter on a portion of the sample then plugging the result into the second
estimating equation to solve for the next parameter using the remaining sample.
To reduce simulation variability from a single split, we repeatedly split the
sample and take the sample mean of all the estimates. For parametric models, we
derive the limiting distribution of sample splitting estimator and show they
are equivalent to stacked estimating equations.
