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STS430 Eustasio D.B. et al.
ℋ ∶ () ≥ △ vs ℋ ∶ () < △ , (5)
0

0

0

where △ > 0 is a fixed threshold. With this formulation the test decision of
0
rejecting the null hypothesis implies that there is statistical evidence that the
deformation model is approximately true. In this case, rejection would
correspond to small observed values of , (). In subsequent sections, we
provide theoretical results that allow the computation of approximate critical
values and p-values for the testing problems (3) and (5) under suitable
assumptions.

3. Bootstraping Wasserstein’s variations
We present now some general results on Wasserstein distances that will
be applied to estimate the asymptotic distribution of the minimal alignment
cost statistic, , (), defined in (4). In this section, we write ℒ() for the law
of any random variable Z. We note the abuse of notation in the following, in

which is used both for the Wasserstein distance on ℝ and on ℝ , but this

should not cause much confusion.
Our first result shows that the laws of empirical transportation costs are
continuous (and even Lipschitz) functions of the underlying distributions.

Theorem 1. Set , , probability measures in (ℝ ), , … , iid random
′

1

,
vectors with common law , , … , , iid with law ′ and write , for the
′
′

1

corresponding empirical measures. Then
,
[ℒ{ ( , )}, ℒ{ ( , )}] ≤ (, ).
′

The deformation assessment criterion introduced in Section 2 is based on
the Wasserstein r-variation of distributions, Vr. It is convenient to note that
( , … , ) can also be expressed as

1

( , … , ) = inf ∫ ( , … , )( , … , ), (6)

1
1
1

∈∏( 1 ,…, )

where ∏( , … , ) denotes the set of probability measures on ℝ with

1
marginals , … , and
1

1
( , … , ) = min ∑‖ − ‖ .
1

∈ℝ
=1
Here we are interested in empirical Wasserstein r-variations, namely, the r-
variations computed from the empirical measures , coming from
independent samples , … , , of iid random variables with distribution .

1,
Note that in this case, problem (6) is a linear optimization problem for which
a minimizer always exists.
As before, we consider the continuity of the law of empirical Wasserstein
r-variations with respect to the underlying probabilities. This is covered in the
next result.

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