Page 300 - Special Topic Session (STS) - Volume 1

STS430 Eustasio D.B. et al.
Theorem 2. With the above notation,

1

′
′
[ℒ { ( 1 ,1 , … , , )} , ℒ { ( ′ 1 ,1 , … , , }] ≤ ∑ ( , ).

=1
A useful consequence of the above results is that empirical Wasserstein
distances or r-variations can be bootstrapped under rather general conditions.
To be more precise, in Theorem 1 we take = , the empirical measure on
′

∗
∗
, … , , and consider a bootstrap sample , … , of iid (conditionally given
1
1

, … , ) observations with common law . We will assume that the
1

∗
resampling size satisfies → ∞ , = () and write for the

empirical measure on , … , and ℒ () for the conditional law of Z given
∗
∗
∗
1

, … , . Theorem 1 now reads

1
∗
∗
[ℒ { ( , )}, ℒ( { , )}] ≤ ( , ).

⁄
Hence, if ( , ) = (1 ) for some sequence > 0 such that

/ → 0 as → ∞ , then using the fact that {ℒ(), ℒ()} =

{ℒ(), ℒ()} for > 0, we see that

∗
∗
[ℒ { ( , )}, ℒ{ ( , )}] ≤ ( , ) → 0

in probability.
Assume that, in addition, ( , ) ⇝ () for a smooth distribution

() . If ̂ () denotes that αth quantile of the conditional distribution

∗
∗
ℒ { ( , )}, then

lim Pr { ( , ) ≤ ̂ ()} = ; (7)

⟶∞
see, e.g., Lemma 1 in [24]. We conclude in this case that the quantiles of
( , ) can be consistently estimated by the bootstrap quantiles, ̂ (),

which, in turn, can be approximated through Monte Carlo simulation.
As an example, if d=1 and r=2, under integrability and smoothness
assumptions on ν, we have
⁄
()
√ ( , ) ⇝ [∫ 0 1 { 2 −1 ()} ] 1 2 ,
2

2
as → ∞, where f and −1 are the density and the quantile function of ,
respectively; see [18]). Therefore, Eq. (7) holds. Bootstrap results have also
been provided in [20].
For the deformation model (1), statistical inference is based on , (),
introduced in (4). Now consider ′ , , (), the corresponding version obtained
′
from samples with underlying distributions . Then, a version of Theorem 2 is

valid for these minimal alignment costs, provided that the deformation classes
are uniformly Lipschitz, namely, under the assumption that, for all ∈ {1, … , },
= sup ‖ ()− ()‖ (8)

≠, ∈ ‖−‖
is finite. Theorem3. If = max ( , … , ) < ∞, with as in (8), then

1
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