  The Wasserstein distance is a distance between two probability distributions
and has recently gained increasing popularity in statistics and machine
learning, owing to its attractive properties. One important approach to
extending this distance is using low-dimensional projections of distributions
to avoid a high computational cost and the curse of dimensionality in empirical
estimation, such as the sliced Wasserstein or max-sliced Wasserstein distances.
Despite their practical success in machine learning tasks, the availability of
statistical inferences for projection-based Wasserstein distances is limited
owing to the lack of distributional limit results. In this paper, we consider
distances defined by integrating or maximizing Wasserstein distances between
low-dimensional projections of two probability distributions. Then we derive
limit distributions regarding these distances when the two distributions are
supported on finite points. We also propose a bootstrap procedure to estimate
quantiles of limit distributions from data. This facilitates asymptotically
exact interval estimation and hypothesis testing for these distances. Our
theoretical results are based on the arguments of Sommerfeld and Munk (2018)
for deriving distributional limits regarding the original Wasserstein distance
on finite spaces and the theory of sensitivity analysis in nonlinear
programming. Finally, we conduct numerical experiments to illustrate the
theoretical results and demonstrate the applicability of our inferential
methods to real data analysis.
