Page 296 - Special Topic Session (STS) - Volume 1

STS430 Eustasio D.B. et al.
ith observation of Xj is such that
, = ( ) ,
,

where the s are iid random variables with unknown distribution . Assume
,
that the functions g1,….,gJ belong to a class of deformation functions, which
model how the distributions ,…., are warped one to another.

1
This model is the natural extension of the functional deformation models
studied in the statistical literature for which estimation procedures are
provided in [23] and testing issues are tackled in [12]. Note that at the era of
parallelized inference where a large amount of data is processed in the same
way but at different locations or by different computers, this framework
appears also natural since this parallelization may lead to small changes with
respect to the law of the observations that should be eliminated.
In the framework of warped distributions, a central goal is the estimation
of the warping functions, possibly as a first step towards registration or
alignment of the (estimated) distributions. Of course, without some
constraints on the class , the deformation model is meaningless. We can, for

instance, obtain any distribution on ℝ as a warped version of a fixed
probability having a density if we take the optimal transportation map as the
warping function; see [35]. One has to consider smaller classes of deformation
functions to perform a reasonable registration.
In cases where is a parametric class, estimation of the warping functions
is studied in [2]. However, estimation/registration procedures may lead to
inconsistent conclusions if the chosen deformation class is too small. It is,
therefore, important to be able to assess the fit to the deformation model
given by a particular choice of . This is the main goal of this paper. We note
that within this framework, statistical inference on deformation models for
distributions has been studied first in [21]. Here we provide a different
approach which allows to deal with more general deformation classes.
The pioneering works [16, 26] study the existence of relationships between
distributions F and G by using a discrepancy measure ∆(F,G) between them
which is built using the Wasserstein distance. The authors consider the
assumption ℋ : ∆(F,G) > Δ versus ℋ : ∆(F,G) ≤ Δ for a chosen threshold Δ .
0

0
0
0
Thus when the null hypothesis is rejected, there is statistical evidence that the
two distributions are similar with respect to the chosen criterion. In this same
vein, we define a notion of variation of distributions using the Wasserstein
distance, Wr, in the set Wr(ℝ ) of probability measures with finite rth moments,

where r ≥1. This notion generalizes the concept of variance for random

distributions over ℝ . This quantity can be defined as
1 ⁄

1

( , … . , ) = inf { ∑ ( , )} ,

1

(ℝ )
=1
285 | I S I W S C 2 0 1 9