A prelude to statistics in Wasserstein metric spaces

Chon Van Le; Uyen Hoang Pham
A icle
A p elude o s a is ics in Wasse s ein me ic spaces
Asian Jou nal o Economics and Banking (AJEB)
P o ided in Coope a ion wi h:
Ho Chi Minh Uni e si y o Banking (HUB), Ho Chi Minh Ci y
Sugges ed Ci a ion: Chon Van Le; Uyen Hoang Pham (2024) : A p elude o s a is ics in Wasse s ein
me ic spaces, Asian Jou nal o Economics and Banking (AJEB), ISSN 2633-7991, Eme ald, Leeds, Vol.
8, Iss. 1, pp. 54-66,
h ps://doi.o g/10.1108/AJEB-10-2023-0099
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A p elude o s a is ics
in Wasse s ein me ic spaces
Chon Van Le
In e na ional Uni e si y, Vie nam Na ional Uni e si y –Ho Chi Minh Ci y,
Ho Chi Minh Ci y, Vie nam, and
Uyen Hoang Pham
Uni e si y o Economics and Law, Vie nam Na ional Uni e si y –Ho Chi Minh Ci y,
Ho Chi Minh Ci y, Vie nam
Abs ac
Pu pose –This pape aims mainly a in oducing applied s a is icians and econome icians o he cu en
esea ch me hodology wi h non-Euclidean da a se s. Speci ically, i p o ides he basis and a ionale o
s a is ics in Wasse s ein space, whe e he me ic on p obabili y measu es is aken as a Wasse s ein me ic
a ising om op imal anspo heo y.
Design/me hodology/app oach –The au ho s spell ou he basis and a ionale o using Wasse s ein
me ics on he da a space o ( andom) p obabili y measu es.
Findings –In elabo a ing he new s a is ical analysis o non-Euclidean da a se s, he pape illus a es he
gene aliza ion o adi ional aspec s o s a is ical in e ence ollowing F eche ’s p og am.
O iginali y/ alue –Besides he elabo a ion o esea ch me hodology o a new da a analysis, he pape
discusses he applica ions o Wasse s ein me ics o he obus ness o inancial isk measu es.
Keywo ds F eche mean se s, His og am da a se s, Op imal anspo , Random p obabili y measu es,
Robus ness o inancial isk measu es, Wasse s ein me ics, Wasse s ein sampling spaces, WGAN
Pape ype Resea ch pape
1. In oduc ion
As we a e wi nessing he cu en ex ension o s a is ical analysis o mo e gene al da a se s in
da a science, i is abou ime o le applied s a is icians and econome icians be awa e o his
use ul and impo an phenomenon. The co ne s one o s a is ical heo y o applica ions is
da a. T adi ionally, da a a e elemen s o Euclidean spaces which a e na u ally equipped wi h
Euclidean dis ances which a e essen ial o analysis. Mode n applica ions call o mo e
gene al da a se s, such as his og ams o non-Euclidean da a. To use s a is ics o make
p edic ions and decisions wi h his new ype o da a, we need o ex end adi ional s a is ical
heo y. The i s basic ing edien o gene alize is me ics on new da a se s. This sho no e
aims simply a elabo a ing a bi on a popula new me ic which applied econome icians can
lea n o apply o hei empi ical applica ions om cu en esea ch li e a u e. This popula
new me ic is called Wasse s ein me ic (dis ance) which is shown o be sui able o a a ie y
o non-Euclidean da a space, such as Wasse s ein space which is a space o p obabili y
dis ibu ions equipped wi h a Wasse s ein me ic.
Simple and elemen a y examples will se e as illus a ing he use ulness and a ionale o
mode n s a is ics wi h non-Euclidean da a. The no e elabo a es heo e ical aspec s in simple
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© Chon Van Le and Uyen Hoang Pham. Published in Asian Jou nal o Economics and Banking.
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Recei ed 8 Oc obe 2023
Re ised 17 Oc obe 2023
19 Oc obe 2023
31 Oc obe 2023
Accep ed 6 No embe 2023
Asian Jou nal o Economics and
Banking
Vol. 8 No. 1, 2024
pp. 54-66
Eme ald Publishing Limi ed
e-ISSN: 2633-7991
p-ISSN: 2615-9821
DOI 10.1108/AJEB-10-2023-0099
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se ings, as well as men ioning some conc e e applica ions. Ou pu pose is simply in oducing
applied s a is icians and econome icians o mode n da a analysis based upon s a is ical
heo y.
The pape is o ganized as ollows. In Sec ion 2, we elabo a e on Wasse s ein me ics in a
conc e e da a se consis ing o ( andom) his og ams which a e p obabili y measu es, oge he
wi h he no ion o Wasse s ein me ics. In Sec ion 3, we ouch upon he s a ing poin o
gene alize adi ional s a is ics in Euclidean spaces o Wasse s ein spaces. In Sec ion 4,we
men ion an applica ion o Wasse s ein me ics o he obus ness issue o inancial isk
managemen . Sec ion 5 p o ides he conclusions.
2. Wasse s ein me ics on his og am da a se s
We can ake i as sel -e idence ha s a is ics is based on da a. While we do ha e a gene al
heo y o s a is ics o guide us each ime we need s a is ics, he e is some hing hidden in he
p ac ices o s a is ics ha we s a looking a nowadays.
T adi ionally, mos o ou da a a e Euclidean elemen s and in p ac icing s a is ics on Rk,
we ake o g an ed hei Euclidean dis ances k.k
k
, wi hou bo he ing spelling ou ha ou
da a se is a me ic space (which is, in ac , essen ial o all s a is ical in es iga ions, such as
compa ing da a poin s, summa izing obse ed da a sample).
Be o e ou imes, i.e. be o e we ac ually un in o mode n applica ions whe e ou da a could
be non-Euclidean, Mau ice F eche has o seen he u u e (i.e. nowadays) o us. Indeed,
ecognizing ha ou adi ional da a space is he me ic space ðRd;
:kdÞ,F eche (1906) i s
axioma ized he no ion o a me ic on a bi a y spaces, o ha e igo ous me ic spaces, no
only o ma hema ical unc ional calculus, bu speci ically o p obabili y and s a is ics.
A well-known si ua ion o all s a is icians whe e “da a poin s”a e non-Euclidean is his.
Le X
1
,X
2
,...,X
n
be an obse ed (IID) andom sample d awn om a eal- alued andom
a iable (popula ion) Xwhose dis ibu ion unc ion Fis unknown. To imp o e he classical
p ac ices (e.g. es ima ing some popula ion pa ame e s o in e es ), and o ake in o accoun
he ad an ages o compu e science, he me hod o boo s ap was in en ed o imp o e he
accu acy o es ima o s and hei con idence in e als. The me hod consis s o c ea ing new
“da a poin s” ia simula ions.
Speci ically, gi en he obse ed sample X
1
,X
2
,...,X
n
, we ob ain he known empi ical
dis ibu ion unc ion (bu , ex an e, i is a andom dis ibu ion unc ion):
FnðxÞ¼1
nX
n
j¼1
1ð−∞;xðXjÞ
whose co esponding p obabili y measu e (law) is dFnð:Þ¼1
nPn
j¼1 Xjð:Þ(by Lebesgue-
S iel jes Theo em) whe e Xjð:Þ¼1ð:ÞðXjÞis he ( andom) Di ac p obabili y measu e a X
j
on BðRÞ.
Ha ing he known p obabili y measu e dF
n
, we can c ea e simula ed da a om i ia
F−1
nðUÞ, whe e
F−1
nð:Þ:½0;1→R;F−1
nðuÞ¼in x∈R:FnðxÞ≥ug
is he (uni a ia e) quan ile unc ion o F
n
and Uis he andom a iable uni o mly dis ibu ed
on [0,1].
Roughly speaking, a simula ed sample (a new “da a poin ”) is ob ained as a esul o
d awing wi h eplacemen npoin s om he se {X
1
,X
2
,...,X
n
}, say, m imes, esul ing in m
se s B
k
5{b
1,k
,b
2,k
,...,b
n,k
}, k51, 2, ...,m.
Wasse s ein
me ic spaces
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Because o he d awings wi h eplacemen , he elemen s b
j,k
in each B
k
could be equal, i.e.
appea ing mo e han once in i , so ha each new “da a poin ”B
k
is no eally a subse o n
elemen s o Ras in se heo y. Ins ead, each B
k
is a mul ise , i.e. a collec ion o poin s dis inc o
no (mul iplici ies o occu ences a e allowed).
As a ema k, such a collec ion o npoin s b
j,k
,j51, 2, ...,n, can be iewed as a uzzy subse
o R, consis ing o dis inc poin s whose deg ees o membe ship a e equal o he a ios o hei
mul iplici y o occu ence and he size n.
Bu , in he se ing o s a is ics, i is mo e ep esen a i e i we iew he new “da a poin s”B
k
as a his og am (a andom p obabili y measu e on BðRÞ), so ha ou new da a se is a space o
( andom) p obabili y measu es deno ed as PðRÞwhe e each “da a poin ”is no an elemen o
he Euclidean space R, bu is a p obabili y measu e on he me ic space ðR;j:jÞ.
Da a se s which a e ( andom) p obabili y measu es on a me ic space ðX;
ρ
Þabound in
applica ions. As such, we need a sui able me ic be ween p obabili y measu es.
Rema k. Bu we know well ha a la ge pa o p obabili y heo y was abou p ecisely he
me iza ion o weak con e gence o p obabili y measu es on me ic spaces, i.e. p oducing
me ics on he space o p obabili y measu es, see, e.g. Billingsley (1995),Pa hasa a hy (1967).
Can we jus pick some known me ic among, say, Le y, P okho o , To al Va ia ion me ics o
use? Well, i depends on wha we wan ou chosen me ic o “beha e!”So a , me ics on
p obabili y measu es a e in en ed o s udy asymp o ic sampling dis ibu ions, such as in he
Cen al Limi Theo em. They we e no in en ed o handle da a analysis, in which we need, o
example, o use a sui able me ic o compa e p obabili y measu es (as da a poin s in ou new
da a se o an applica ion). Fo example, i we obse e h ee da a “poin s”as h ee p obabili y
densi ies ,g,hwhich a e uni o mly dis ibu ed on [3, 2], [2, 1] (Be n on e al., 2019;
Bha and P ashan h, 2019), espec i ely, (and deno ing Fas he dis ibu ion unc ion wi h
densi y and dF i s associa ed p obabili y measu e) hen
TVðdF;dGÞ¼1
2Z∞
−∞
j ðxÞgðxÞjdx ¼1¼TVðdF;dHÞ
i.e. he o al a ia ion me ic canno cap u e he loca ions o hese his og am da a.
This is simila o he ecogni ion ha Hausdo dis ance on subse s o a space canno be
used when da a a e cu ed in he space, al hough cu es a e subse s. The eason is clea :
Hausdo dis ance does no cap u e he s uc u e o cu es which is needed in da a analysis
when cu es a e da a “poin s”.
So, wha a e o he me ics (on space o p obabili y measu es) which can be used o da a
analysis/s a is ics wi h da a se s as his og ams?
We need o compa e his og ams (as da a poin s) in applica ions when each his og am
ep esen s he obse ed in o ma ion abou an “objec ”, o he e u n o a s ock in inancial
econome ics. Then i is ob ious ha we mus ake in o accoun o hei loca ions! On he
o he hand, i da a poin s a e elemen s o an Euclidean space, e.g. x;y∈ðR;j:jÞ, he sui able
me ic Wwe wish o ha e should be a na u al ex ension o he Euclidean me ic j.jon R, in he
sense ha W(
x
,
y
)5jxyj, i.e. when we iden i y a numbe x∈Rwi h he Di ac p obabili y
measu e
x
.
We a e going o “men ion”a sui able and popula me ic W(., .) on his og am da a. I seems
impo an o applied s a is icians and econome icians o ha e a good unde s anding o ha
me ic o eel com o able o use i in eal-wo ld applica ions, a he han jus ake i o
g an ed!.
The ollowing elabo a ion is o his pu pose.
As a as his o y is conce ned, i is ai o s a wi h Mau ice F eche , he pionee o mode n
s a is ics.
In 1937, Le y (1937) de ined se e al me ics on p obabili y measu es on BðRÞ. One is
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LðF;GÞ¼in
ε
>0:Gðx
ε
Þ
ε
≤FðxÞ≤Gðxþ
ε
Þþ
ε
;
∇
x∈Rg
which me ized he con e gence in dis ibu ion (o weak con e gence o p obabili y
measu es, i.e. Fn→
wFi F
n
(x)→F(x), as n→∞, o any x∈C(F), he con inui y se o F(.))
i.e. Fn→
wF5LðFn;FÞ→0.
Pu suing Le y’s wo k, in 1957, F eche (1957) obse ed ha Le y’s dis ance L(F,G) o he
dis ibu ion unc ions o wo andom a iables Xand Yin ol ed Fand Galone. He sugges ed
ha a “global”dis ance W
H
(X,Y) should in ol e he join dis ibu ion unc ion H(x,y) o he
andom ec o (X,Y), say, W
H
(F,G) whe e Hð:; :Þ:R2→½0;1is he join dis ibu ion wi h
ma ginals F,G, i.e. H(x,∞)5F(x), H(∞,y)5G(y).
Ano he de ini ion o Le y’s dis ance on dis ibu ion unc ions on Ris o he o m
WðF;GÞ¼in WHðF;GÞ:H∈CðF;GÞg
whe e C(F,G) is he se o join dis ibu ions wi h ma ginals F,G(la e in 1959, Abe Skla
speci ied i as copulas).
Bu o W(F,G) o be a bona ide “me ic”(in pa icula , W(F,G)5F5G), he abo e
in imum mus be a ained a some special H*.
Le ’s see whe he i is he case o no o he example gi en in F eche (1957)
WHðF;GÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
EHðXYÞ2
q¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ZR2
ðxyÞ2dHðx;yÞ
s
Rema k. In 1969, Vasse sh ein (Wasse s ein) (1969) p oposed exac ly
W1ð
μ
;
ν
Þ¼in ZR2
xy
dλðx;yÞ:λ∈Πð
μ
;
ν
Þ

whe e Π(
μ
,
ν
) is he se o p obabili y measu es on BðR2Þwi h p ojec ions (ma ginals)
μ
,
ν
.
Up on : in ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
EHðX−YÞ2
q:H∈CðF;GÞg
is a ained a H*(x,y)5F(x)∧G(y) because,
o Uuni o mly dis ibu ed on [0, 1], X¼
DF−1ðUÞ,Y¼
DG−1ðUÞ, he join dis ibu ion unc ion o
(F
1
(U), G
1
(U)) is H* and
WH*ðF;GÞ¼WH*F−1ðUÞ;G−1ðUÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Z1
0F−1ðuÞG−1ðuÞ2
du
s
which is he minimum o ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
EHðX−YÞ2
q:H∈CðF;GÞ

.
I su ices o show ha W1ðF;GÞ¼in RR2x−ydHðx;yÞ:H∈CðF;GÞg is a ained a
H*(x,y)5F(x)∧G(y). The same esul holds o W
p
,p≥1, whe e
WpðF;GÞ¼ in ZR2
xyjpdHðx;yÞ:H∈CðF;GÞ

1
p
He e a e he de ails, see Vallende (1973), ha he in imum o RR2jx−yjdHðx;yÞo e H∈C(F,
G) is indeed a ained (a H(x,y)5F(x)∧G(y)).
Le X;Y:ðΩ;A;PÞ→ðR;BðRÞÞ be andom a iables wi h dis ibu ions F,G,
espec i ely.
Wasse s ein
me ic spaces
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Since
jXYj¼ðXYÞ1ðX≥YÞþðYXÞ1ðX<YÞ
we le
α
5max(XY, 0) and β5max(YX), 0), so ha
EjXYj¼E
α
þEβ
Since
α
≥0, we ha e
Eð
α
jY¼yÞ¼Z∞
0
Pð
α
>zjY¼yÞdz
Now, E(
α
)5EE(
α
jY), so ha
E
α
¼Z∞
−∞
dGðyÞZ∞
0
PðXY≥zjY¼yÞ¼
Z∞
−∞
dGðyÞZ∞
0
PðX≥yþzjY¼yÞdz ¼
Z∞
−∞
dGðyÞZ∞
y
PðX≥xjY¼yÞdx ¼
ZZðx;yÞ;x>y
PðX≥y;Y<yÞdx ¼Z∞
−∞
PðX≥y;Y<yÞdy
Simila ly,
Eβ¼Z∞
−∞
PðY≥y;X<yÞdy
Thus,
EjXYj¼Z∞
−∞
PðX≥y;Y<yÞdy þZ∞
−∞
PðY≥y;X<yÞdy ¼
Z∞
−∞
½PðX<y;Y≥yÞþPðY<y;X≥yÞdy
Now, look a he e en (X<y,Y≥y).
Le A5(X<y) and B5(Y<y), hen (X<y,Y≥y)5A∩B
c
. Bu A5(B
c
∩A)∪(A∩B),
so ha
PðX<y;Y≥yÞ¼PðAÞPðA BÞ¼PðX<yÞPðX<y;Y<yÞ
Thus,
EjXYj¼Z∞
−∞
½PðX<yÞþPðY<yÞ2PðX<y;Y<yÞdy
No e ha P(X<y,Y<y) is he alue o he join dis ibu ion H(y,y) o he ec o (X,Y), and i is
well known ha H(x,y)≤F(x)∧G(y) which is a join dis ibu ion wi h ma ginals F,G( om
F eche ’s wo k (1956) on co ela ion analysis wi h gi en ma ginals o om copula heo y),
so ha
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EjXYj≥Z∞
−∞
½FðyÞþGðyÞ2min FððyÞ;GðyÞdy ¼Z∞
−∞
jFðyÞGðyÞjdy
by no ing ha jxyj5xþy2(x∧y).
The e o e,
W1ðdF;dGÞ¼in WHðF;GÞ:H∈CðF;GÞg ≥Z∞
−∞
jFðyÞGðyÞjdy
Bu
Z∞
−∞
jFðyÞGðyÞjdy ¼Z1
0
jF−1ðuÞG−1ðuÞjdu
(by an “analy ic”p oo below) so ha he in imum o RR2jx−yjdHðx;yÞo e H∈C(F,G)is
R1
0jF−1ðuÞ−G−1ðuÞjdu which u ns ou o be a minimum since
Z1
0
jF−1ðuÞG−1ðuÞjdu ¼EjF−1ðUÞG−1ðUÞj ¼ EH*jXYj
whe e H*(x,y)5F(x)∧G(y) is he join dis ibu ion unc ion o
(F
1
(U), G
1
(U)). Q.E.D.
Rema ks.
(1) Le X¼
DF½−1ðUÞand Y¼
DG½−1ðUÞ, we ha e dH*¼du◦ðF−1;G−1Þ−1, so ha
H*ðx;yÞ¼dH*ðð−∞;x3ð−∞;yÞ ¼ dunu:F−1ðuÞ≤x;G−1ðuÞ≤yo¼
du u:u≤FðxÞ;u≤GðyÞg ¼ du u:u≤FðxÞ∧GðyÞg ¼ FðxÞ∧GðyÞ
(2) P oo o
Z∞
−∞
jFðyÞGðyÞjdy ¼Z1
0
jF−1ðuÞG−1ðuÞjdu
is as ollows. The ollowing is jus i ied by Fubini’s heo em, namely i R
A3B
j (x,y)jd(x,
y)<∞, hen
ZA3B
j ðx;yÞjdðx;yÞ¼ZAZB
ðx;yÞdy

dx ¼ZBZA
ðx;yÞdx

dy
Now, o u∈(0, 1), we ha e
jF½−1ðuÞG½−1ðuÞj ¼ hF½−1ðuÞG½−1ðuÞi1 u:F½−1ðuÞ>G½−1ðuÞgðuÞþ
hG½−1ðuÞF½−1ðuÞi1 u:F½−1ðuÞ≤G½−1ðuÞgðuÞ
So le A¼nu∈ð0;1Þ:F½−1ðuÞ>G½−1ðuÞo
Wasse s ein
me ic spaces
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Ac¼nu∈ð0;1Þ:F½−1ðuÞ≤G½−1ðuÞo
We ha e
Z1
0
jF½−1ðuÞG½−1ðuÞjdu ¼
ZA
jF½−1ðuÞG½−1jðuÞjdu þZAc
jF½−1ðuÞG½−1ðuÞjdu
whe e we can w i e
ZA
jF½−1ðuÞG½−1jðuÞjdu ¼ZAZF½−1ðuÞ
G½−1ðuÞ
dx
"#
du
Now, obse e ha , by de ini ion o he quan ile unc ions, we ha e
G
[1]
(u)≤x5u≤G(x) (and o cou se, x<F
[1]
(u)5u>F(x)), so ha
ZAZF½−1ðuÞ
G½−1ðuÞ
dx
"#
du ¼ZRZGðxÞ
FðxÞ
1AðuÞ1 FðxÞ≤GðxÞgðxÞdu
"#
dx
Simila ly,
ZAc
jF½−1ðuÞG½−1ðuÞjdu ¼ZRZFðxÞ
GðxÞ
1AcðuÞ1 FðxÞ>GðxÞgðxÞdu
"#
dx
Hence,
ZRZGðxÞ
FðxÞ
1AðuÞ1 FðxÞ≤GðxÞgðxÞdu
"#
dx þZRZFðxÞ
GðxÞ
1AcðuÞ1 FðxÞ>GðxÞgðxÞdu
"#
dx ¼
ZR
jFðxÞGðxÞjdx
Q.E.D.
Now, he dis ance W
1
(F,G)o W1ðdF;dGÞ¼R1
0jF−1ðuÞ−G−1ðuÞjdu does ake in o
accoun he loca ions o he his og am da a “poin s”. Indeed, o he his og ams ,g,hin he
p e ious example (wi h associa ed dis ibu ions F,G,H, espec i ely), we ha e W
1
(F,G)51
and W
1
(F,H)55, showing ha he his og am is close o g han h.
On he o he hand, W
1
is a na u al ex ension om Euclidean da a poin s o his og am da a
poin s. Indeed, o x;y∈R, we iden i y hem as
xðAÞ¼dFxðAÞ¼1AðxÞ; yðBÞ¼¼dGyðBÞ¼1BðyÞ
so ha
Fxð Þ¼ xðð−∞; Þ ¼ 1½x;∞Þð Þ
Since we conside eal- alued andom a iable, i.e. wi h alues in R¼ð−∞;∞Þ, hei
quan ile unc ions, e.g. F−1
xð:Þ:ð0;1Þ→R:
AJEB
8,1
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F−1
xðuÞ¼in ∈R:Fxð Þ≥ug¼x1ð0;1ÞðuÞ
and hence
W1ð x; yÞ¼Z1
0
jF−1
xðuÞF−1
yðuÞjdu ¼Z1
0
jxyj1ð0;1ÞðuÞdu ¼jxyj
Rema ks.
(1) In 1969, Wasse s ein Vasse sh ein (1969) conside ed W
2
o in es iga e he
uniqueness o he s a iona y dis ibu ion o a Ma ko p ocess. And in 1970,
Dob ushin (1970) used Wasse s ein me ic o in es iga e s ochas ic p ocesses by
condi ional dis ibu ions.
(2) In 1972, Mallows (1972) conside ed he same W
2
me ic, wi hou e e ing o i s
exis ence yea s ago!
(3) Sho ack and Wellne (1986) used Wasse s ein me ics o in es iga e he con e gence
o empi ical p ocesses in hei book in 1986.
(4) Fo gene al Wasse s ein me ics in Op imal T anspo Theo y, see Villani (2003)
3. Typical posi ions in F eche ’s p og am
F om a his o ical pe spec i e, he pionee ing wo k o F eche (1948) can be iewed as he i s
a emp o gene alize p obabili y backg ound o s a is ics, such as gene al andom elemen s
in a bi a y me ic spaces, hei ypical posi ions (e.g. mean), gene al pa ame e s, gene al
s a is ics and hei con e gences ( o asymp o ics, e.g. consis ency o es ima o s).
Nowadays, we a e wi nessing e o s o heo e ical s a is icians o speci y F eche ’s ision
while applied s a is icians in a ious ields, such as economics and machine lea ning (ML),
s a ed by implemen ing i in eal-wo ld applica ions, see, e.g. Be n on e al. (2019),Bha and
P ashan h (2019),Bigo (2020),Cha ie (2013) and Kiesel e al. (2016).
We will elabo a e on hese cu en e o s in he con ex o Wasse s ein me ic spaces as
da a se s. Fo an in i a ion o he heo e ical aspec s o s a is ics in Wasse s ein space, see
Pana e os and Zemel (2020).
Rema k. As B eiman (2001) spelled ou he use ul ma iage be ween s a is ics and ML, see,
e.g. Mo ize (2020),Shale -Shwa z and Ben-Da id (2014), Wasse s ein me ics a e used also
in ML, e.g. in WGAN.
As a s a ing poin , le ’s discuss he no ion o “ ypical posi ions”o a andom elemen X
wi h alues in an a bi a y me ic space ðX;
ρ
Þ.
Acco ding o F eche (1948), gene alized ypical posi ions such as median and mean could
be de ined ia app op ia e cha ac e iza ions o classical no ions on Euclidean spaces. Fo
simplici y, conside ðR;j:jÞ.
Le Xbe a eal- alued andom a iable wi h dis ibu ion unc ion F(and law dF). In
classical p obabili y heo y, he median m(X)o Xis a alue on Rwhich is “equip obable”
(always exis ed). The mean o Xis he quan i y EX ¼RRxdFðxÞwhich exis s when his
in eg al is ini e.
To gene alize hese ypical posi ions o a bi a y me ic spaces, we need o
“cha ac e ize” hem.
Fi s , a cha ac e iza ion o m(X) is ob ained when s a is icians use LAD (Leas Absolu e
De ia ion) EjXajas e o , say, in quan ile eg ession.
Wasse s ein
me ic spaces
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