Transformations to symmetry based on the probability weighted characteristic function

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Transformations to symmetry based on the probability weighted characteristic function

Simos Meintanis, Gilles Stupfler

To cite this version:

Simos Meintanis, Gilles Stupfler. Transformations to symmetry based on the probability weighted

characteristic function. 2014. �hal-01018575v2�

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PROBABILITY WEIGHTED CHARACTERISTIC FUNCTION

Simos G. Meintanis

a,b

, Gilles Stupfler

c

a

Department ofEonomis,NationalandKapodistrian Universityof Athens,Athens,Greee,

b

Unit forBusiness MathematisandInformatis, North-WestUniversity, Pothefstroom, SouthAfria

and

c

AixMarseilleUniversité, CNRS,EHESS,Centrale Marseille, GREQAMUMR 7316,

13002 Marseille, Frane

Abstrat. We suggest a nonparametri version of the probability weighted empirial harateristi funtion

(PWECF) introdued by Meintanis et al. (2014) and use this PWECF inorder to estimate the parameters

of arbitrary transformations to symmetry. The almost sure onsisteny of the resulting estimators is shown.

Finitesampleresultsfori.i.d. dataarepresentedandaresubsequentlyextendedtotheregressionsetting. Real

dataillustrations arealsoinluded.

Keywords. Charateristifuntion; Empirialharateristi funtion;Probabilityweightedmoments;Symmetry

transformation

AMS2000lassiationnumbers:62G10,62G20

1 Introdution

Transformationsareappliedongivendatasets inordertofailitatestatistialinferene. Thesetransformations

areoftenusedsoastoinduenitemomentsandlighttailsand/orsymmetry.Thisisimportantasitisommon

knowledgethatertainstatistialproeduresareappliableorperformwellonlyundersuhassumptions.Apart

fromthat,symmetryhasdeniteadvantagesfor identiationandonsistenyofloationestimatorswithi.i.d.

data,aswellasintheontextofregressionwhereBikel(1982)andNewey(1988)studytheexisteneofadaptive

andeientregressionestimatorsundersymmetrierrors. ThereaderisreferredtoChapter6ofHorowitz(2009)

for anie reviewof transformations inregression andotherrelated models. Lately the symmetryassumption

hasalsobeeninvokedfortheonsistenyandeienyofthequasimaximumlikelihoodestimator(QMLE) in

GARCHmodels;seeGonzálezRiveraandDrost(1999)andNeweyandSteigerwald(1997). Finally,wemention

thatpowertransformationshavereentlybeenusedbySavhukandShik(2013)inordertoimprovetherate

ofonvergene ofthe lassialParzen-Rosenblatt(Parzen,1962; Rosenblatt,1956)estimatorof theprobability

densityfuntion.

Thepurpose ofthispaperistosuggestaproedurebymeansofwhihasamplefromanunknowndistribution

is redued to a sample from a symmetri distribution. To this endwe employ the notion of the probability

weightedempirialharateristi funtion(PWECF),introdued reentlyinMeintaniset al. (2014). However,

the PWECF used inMeintanis et al. (2014) is dened in anentirely parametri ontext and it is therefore

notappropriatewhenpursuingnonparametriinferene. Inwhatfollows wesuggestanonparametriversionof

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remainder of this work is outlined as follows. In Setion 2 we reall some properties of the PWCF and the

nonparametriPWECFisintrodued. InSetion3weintroduethe newestimationproedure whihisbased

onanappropriatefuntionalofthisPWECF;themethodisrelatedtothoseinYeoandJohnson(2014)andYeo

etal. (2014). ThestrongonsistenyofourestimatorisgiveninSetion4,whileinSetion5thenitesample

properties ofthemethodare investigatedby meansofasimulationstudy. Real dataexamplesare inludedin

Setion6whilesomeauxiliaryresultsandtheirproofs aredeferredtotheAppendix.

2 The nonparametri PWECF

LetX^denoteânârbitrary^random^variable^withânâbsolutelyôntinuousdistributionfuntionF(x) =P(X≤x)^.

Forγ≥0^,^theprobabilityweightedharateristifuntion(PWCF)ofX ^is^dened^by ϕ(t;γ) :=E

h

W(X;γt)e^itXi

= Z ∞

−∞

W(x;γt)e^itxdF(x), t∈R, ^(2.1)

whereW(x;s) := [F(x)(1−F(x))]^|s|^. Îtîs^noteworthy^that^the^PWCFôfX^has^variousûseful^properties^similar

tothatoftheharateristifuntion(CF)ofX^,^see^Meintanisêtâl. ^(2014);ⁱⁿ^partiular,âdistributionfuntion whihissymmetriaroundzeromustyieldareal-valuedPWCF,seepropertyP5there,andthiswillbethebasis

ofourtransformationproedureinSetion3. Thefatthatforγ >0^the^PWCF^is^no^longer^a^Fourier^transform,

however, makesit diultto prove strongdistributional results suhas aone-to-one orrespondene between

PWCFsand probabilitydistributions. Interestinglythough,intheontext ofloation-salefamilies,whihwas

theoriginalframeworkofMeintanisetal. (2014),wemaystateandprovesuharesult:

Proposition 1. AssumethatF1 ^andF2 ^belong^to^someloation-salefamily,namely

∀x∈R, F1

x−µ1

σ1

=F2

x−µ2

σ2

=G(x)

where G îsânâbsolutely ôntinuous distributionfuntion andµ1, µ2 ∈R^, σ1, σ2 >0^. ^Then, ^forâny γ >0^, F1

andF2 ^yield^the^same ^PWCFîfândônlyîfF1=F2^.

Proof ofProposition 1. Letϕµ,σ ^be^the^PWCF^related^toFµ,σ(x) :=G(σx+µ)^. ^Sine ϕµ,σ(t;γ) =

Z ∞

−∞

[Fµ,σ(x)(1−Fµ,σ(x))]^γ|t|e^itxdFµ,σ(x),

wegetbythehangeofvariablesx=σy+µ^: ϕµ,σ(t;γ) =

Z ∞

−∞

[G(y)(1−G(y))]^γ|t|e^i(σt)y+µdG(y) =e^itµϕ0,1(σt;γ/σ).

AssumenowthatF1^andF2 ^yield^the^same^PWCF,^withσ16=σ2^. ^Then

e^itµ¹ϕ0,1(σ1t;γ/σ1) =e^itµ²ϕ0,1(σ2t;γ/σ2), t∈R, ^(2.2)

whihuptoreparametrization isequivalentto

ϕ0,1(T; Γ) =e^itMϕ0,1(ΣT; Γ/Σ), T ∈R,

for someM ∈R^,Σ6= 1 ^andΓ>0^. ^Without^loss ^ofgenerality, we assumeinwhatfollows thatΣ>1^;ⁱⁿ^this

ase,astraightforwardproofbyindutionshowsthatforanypositiveintegerm^:

|ϕ0,1(T; Γ)|=|ϕ0,1(Σ^mT; Γ/Σ^m)|, T ∈R.

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Observenowthatϕ0,1(0; Γ) = 1^and^for^anyT >0^, ϕ0,1(Σ^mT; Γ/Σ^m) =

Z∞

−∞

[G(y)(1−G(y))]^Γ|T|e^i(Σ^m^T)yg(y)dy

= 1

T Z∞

−∞

[G(z/T)(1−G(z/T))]^Γ|T|g(z/T)e^iΣ^m^zdz

where gîs^the probabilitydensity funtionrelated to G^. ^The^right-hand^side îs,ûp ^to âônstant,^the ^Fourier

transformoftheintegrablefuntion

z7→[G(z/T)(1−G(z/T))]^Γ|T|g(z/T),

evaluated at the point Σ^m^. ^Sine Σ^m → ∞^, ^the Riemann-Lebesgue lemma states that this expression must onvergeto0asm→ ∞^. Âsâônlusion,

ϕ0,1(0; Γ) = 1 ^and ϕ0,1(T; Γ) = 0, T >0.

Thisis aontradition sine T 7→ϕ0,1(T; Γ) îsôntinuous,^see ^property ^P7ⁱⁿ^Meintanis êt âl. ^(2014). ^Hene σ1=σ2^,ând^thuseîtµ¹=eîtµ² ^forâll t∈R^by^(2.2),^whihêntailsµ1=µ2^. ^The^proofîsômplete.

Remark 1. Theloationsale ontext may atually be dropped underadditional moment hypotheses, suh

as the existene of the moment-generating funtionof F1 ^and F2 ⁱⁿ ^a neighborhood of 0, by using analyti ontinuation. Inany ase,ifthePWCFisunique,itanbeusedto assesssymmetryaround zero: It isindeed

learthatfor anytândγ^,^the^PWCFôf−X îsêqual^toϕ(−t;γ)^,ând^thatϕ(−t;γ) =ϕ(t;γ)^,^wherez ^denotes

theomplexonjugateof z^. ^Nowîf ^the^PWCF ôfX îs real-valued,this entailsϕ(−t;γ) =ϕ(t;γ) ând^thus X

and−X ^have^the^same^PWCF,^whene^the^fat^that^thedistributionfuntionofX ^is^symmetri^around^zero.

WhileMeintanis etal. (2014)estimatedthe PWCF ina parametriway,it is interestingto onsider thease

whereF îsômpletelyûnknown. În^thisôntext,îtîsâ^naturalîdea ^to^deneânêstimatorôf^the^PWCFⁱⁿân

entirelynonparametriway. TothisendnotiethatthePWCFin(2.1)maybewrittenas

ϕ(t;γ) = Z 1

0

[x(1−x)]^γ|t|e^itQ(x)dx, ^(2.3)

whereQ(x) = inf{t∈R|F(t)≥x}^denotes^the^quantile^funtion^ofX^.

Inviewof(2.3)wesuggestthefollowingnonparametriestimatorofthePWCF:

b

ϕn(t;γ) = Z 1

0

[x(1−x)]^γ|t|e^it^Q^bⁿ^(x)dx, ^(2.4)

withQbn(x)^denoting ^the^empirial ^quantile ^funtion. ^We ^shall^allϕbn(t;γ) ^theprobabilityweighted empirial harateristifuntion(PWECF),andforthepurposeofestimationwewilluse

∀k∈ {1, ..., n}, ∀x∈ k−1

n ,k n

, Qbn(x) =Xk:n,

where X1:n ≤ · · · ≤ Xn:n ^denote ^the ^order ^statistis orresponding to independent opies X1, . . . , Xn ^of ^the

randomvariableX^.

3 L2type proedures for symmetry transformation

The problem we shall onsider is to estimate the parameters of a given transformation whih, if applied on

the original nonsymmetrially distributed observations X1, . . . , Xn^, ^yields transformed observations that are

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approximatelysymmetriallydistributedwithloationzero. Tothisend, writeϑ= (δ, λ)∈Θ⊂R×Λ^for ^the

transformationparametervetor,whereδ^denotes^loationândλ^denotes^the^shape^parameter^whihîsâssumed

tolieinasubsetΛôf^the^real^line. ^Fôrϑ= (δ, λ)∈Θ^,^we^letQZ(·;ϑ)^be^the^quantile^funtionôf^thetransformed randomvariableZ(ϑ) =ψ(X;λ)−δ^,^whereψîsâ^speitransformationfamily,andwedene

S(t;γ;ϑ) = Z 1

0

[x(1−x)]^γ|t|sin(tQZ(x;ϑ))dx,

the imaginarypart of the PWCFof Z(ϑ)^. Ît îs ^thusâ ônsequene ôf^Remark ¹^that îf^the transformedran- dom variable Z ^has â ^symmetri distribution around zero then S(t;γ;ϑ) = 0 ^for âll t ∈ R^, ôr equivalently

R∞

−∞S²(t;γ;ϑ) = 0^.

Thisobservation isthe basiidea we needtobuild ourestimator: we introdueZk(ϑ) = ψ(Xk;λ)−δ^, ^we ^let QbZ(x;ϑ)^be^the^empirial^quantile^funtion^related^toZ1(ϑ), . . . , Zn(ϑ)^and^we^dene

Sbn(t;γ;ϑ) = Z 1

0

[x(1−x)]^γ|t|sin(tQbZ(x;ϑ))dx,

theimaginarypartofthePWECFofZ1(ϑ), . . . , Zn(ϑ)^. ^ThenSbn(t;γ;ϑ)îs^theêmpirialôunterpartôfS(t;γ;ϑ)^.

We suggest to estimate the true value ϑ0 = (δ0, λ0) ^(see ^Setion⁴ ^for â ^disussion ôf ^the ûniqueness ôf ^this

parameter)byϑbn^,^where

ϑbn= arg min

ϑ∈Θ

∆n(γ;ϑ), ^with ∆n(γ;θ) = Z ∞

−∞

Sb_n²(t;γ;ϑ)dt. ^(3.1)

Remark2. ThePWCFϕ(t;γ)ând^PWECFϕbn(t;γ)ôfâ^random^variableX âre^suh^that|ϕ(t;γ)| ≤(1/4)^γ|t|

and|ϕbn(t;γ)| ≤(1/4)^γ|t| ^for êvery(t, γ)∈R×R⁺^. Âsâônsequene,^for ânyϑ^,^the întegral∆n(ϑ)îs^positive

andnite.

Remark3. Notiethatwhilewewriteϑbn^,^theêstimatorîmpliitly^dependsôn^the^valueôfγând^therefore^we

haveessentiallyafamilyofestimators{ϑbn(γ),0< γ <∞}^indexed^byγ^.

Remark 4. Possible hoies forthe transformationfamilyψ ^are ^the^Box-Coxtransformation (1964),afamily introduedbyBurbidgeetal.(1988)aswellasthereentlyintroduedmethodofYeoandJohnson(2000). Note

thatwhilethepopularBox-Coxtransformation,

ψ(x;λ) =







 x^λ−1

λ ^ifλ6= 0, logx ^ifλ= 0,

applies onlyto positiverandom variables(if λîs ^notâ ^nonzeroînteger), îts^modiations ^suggested^by ^Manly

(1976),JohnandDraper(1980)andBikelandDoksum(1981)weredesignedtoallownegativevaluesaswell.

AfavorablefeatureofthespeidenitionofthenonparametriPWECFin(2.4)isthatitleadstoariterion

in (3.1) whih is onvenient from the omputational point of view. To see this notie that from (2.4) it is

straightforwardtoomputetheimaginarypartofthePWECFofZ1(ϑ), . . . , Zn(ϑ)^as Sbn(t;γ;ϑ) =

Xn

k=1

υk,n(t;γ) sin(tZk:n(ϑ)) ^with υk,n(t;γ) = Z k/n

(k−1)/n

[x(1−x)]^γ|t|dx.

Thentheriterionstatistiin(3.1)followsbydiretalulationas

∆n(γ;ϑ) =1 2

Xn

j,k=1

I_jk⁻(γ;ϑ)−I_jk⁺(γ;ϑ)

whereI_jk⁻(γ;ϑ) :=I(j, k;γ;Zj:n(ϑ)−Zk:n(ϑ))^andI_jk⁺(γ;ϑ) :=I(j, k;γ;Zj:n(ϑ) +Zk:n(ϑ))^with I(j, k;γ;x) =

Z ∞

−∞

υj(t;γ)υk(t;γ) cos(tx)dt.

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Here,weassumethatγ >0^and^that^the^following^hold:

(A1)^The^supportDôf^thedistributionofX îsânôpenîntervalândF îsôntinuousând^stritlyînreasing

onD^.

(A2)^Thetransformationfamilyψîs^suh^that(x, λ)7→ψ(x;λ)îsôntinuousônD ×Λ^. (A3)^Forâllλ∈Λ^,x7→ψ(x;λ)îs^stritlyînreasing.

Assumption(A2)îs âlsoûsedⁱⁿ^Yeoând^Johnson^(2001), ^while(A3) ^means^that ^the^familyôftransformations preservesordering: iftwoobservations X1 ândX2 âre ^suh^that X1 < X2^, ^then^the transformedobservations

ψ(X1;λ)ândψ(X2;λ)âre^suh^thatψ(X1;λ)< ψ(X2;λ)^. În^partiular,ⁱⁿ^this^setting, îtîsstraightforwardto showthat

QZ(x;ϑ) =ψ(Q(x);λ)−δ ^and QbZ(x;ϑ) =ψ(Q(x);b λ)−δ. ^(4.1)

Undertheseassumptions,wemaystateastrongonsistenyresultforourestimator:

Theorem 1. Assume that(A1)^, (A2)ând(A3) ^hold. ^LetΘ^beâômpat^subsetôf R² ôntained ⁱⁿR×Λ^. Îf,

overΘ^,^there êxistsâ ûnique^global^minimumϑ0 ôf^the^funtion

ϑ7→

Z ∞

−∞

S²(t;γ;ϑ)dt

thenϑbn→ϑ0 ^almost^surely.

Proof ofTheorem1. ByLemma2intheAppendix,

Hn(ϑ) :=

Z ∞

−∞

Sbn²(t;γ;ϑ)dt→H(ϑ) :=

Z∞

−∞

S²(t;γ;ϑ)dt

almostsurely,uniformlyinϑ∈Θ^.^Reall^that S(t;γ;ϑ) =

Z 1

0

[x(1−x)]^γ|t|sin(tQZ(x;ϑ))dx.

Beausefor anyx^the^funtionϑ7→QZ(x;ϑ)îsôntinuousând^the întegrandⁱⁿS(t;γ;ϑ)îs^dominated^by^the

onstant1,thedominatedonvergenetheorementailsthatfor anyt^,^the^funtionϑ7→ S(t;γ;ϑ)^is^ontinuous.

Furthermore, sine for any ϑ^, |S(t;γ;ϑ)| ≤ (1/4)^γ|t| ^by ^Remark ^2, ît îs âgain â ôrollary ôf ^the ^dominated

onvergenetheoremthatthefuntionH îsôntinuousâs^well. Âpplying^Lemma³ônludes^the^proof.

Theexisteneofaglobalminimumofthefuntionϑ7→R∞

−∞S²(t;γ;ϑ)dtîs^forînstane^guaranteedîf^thereêxists ϑ0^suh^that^thedistributionofZ(ϑ0)îs^symmetriâround^0,ⁱⁿ^whihâseS(t;γ;ϑ0) = 0^forêahtând^therefore

∀ϑ∈Θ, Z ∞

−∞

S²(t;γ;ϑ)dt≥0 = Z ∞

−∞

S²(t;γ;ϑ0)dt.

Theuniquenessofonesuhϑ0^is^a^more^hallenging^problem. ^The^followingpropositionisasteptowardssolving thisquestionforalargelassoftransformations,inludingthosementionedinRemark4.

Proposition2. Assumethat(A1)^holdsând^thatX ^hasâ^positive^median. ^Letψ^beâ^familyôftransformations, satisfying(A2)ând(A3)^,^suh^that

∀x >0, ∀λ >0, ψ(x;λ) =[f(x)]^λ−1 λ

wherefîsâ^positive,ôntinuousând^stritlyînreasing^funtionôn(0,∞)^. Îf^thereêxistsâ^pair(δ, λ)∈R×(0,∞)

suhthatψ(X;λ)−δ îssymmetrially distributedaround zero,then(δ, λ) îs^theûnique ^suh^pair.

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ProofofProposition2. Sine(A1)^holdsândX ^hasâ^positive^median,^we^haveQ(x)>0^forâllxⁱⁿânôpen

neighborhoodU ôf1/2^. ^Themonotoniityoff^then^yieldsQZ(x;ϑ) =ψ(Q(x);λ)−δ^forâllx∈U^.În^partiular,

themedianofZ(ϑ)^,^whih^issymmetriallydistributedaroundzero,hastobe0andthus0 = [f◦Q(1/2)]^λ−c(ϑ)^,

where c(ϑ) = 1 +δλ^. În ^partiular, c(ϑ) îs ^positive ând f◦Q(1/2) = [c(ϑ)]^1/λ^. ^Besides, ît ^must^hold ^that QZ(1/2−s;ϑ) =−QZ(1/2 +s;ϑ)^forânys∈(0,1/2)^whihêntails^forâll ε >0^smallênough:

[f◦Q(1/2−ε)]^λ−1

λ −δ=−

[f◦Q(1/2 +ε)]^λ−1

λ −δ

orequivalently:

f◦Q(1/2−ε) =

2c(ϑ)−[f◦Q(1/2 +ε)]^λ1/λ

. ^(4.2)

Assumenowthatthereexisttwopairsϑ1= (δ1, λ1)ândϑ2= (δ2, λ2)^suh^thatZ(ϑ1)ândZ(ϑ2)âre^symmetri-

allydistributedaroundzero. Notethatitisenoughtoshowthat λ1=λ2^. Ûsing ^(4.2),^weôbtain^forâll ε >0

suientlysmall:

2c(ϑ1)−[f◦Q(1/2 +ε)]^λ¹1/λ₁

=

2c(ϑ2)−[f◦Q(1/2 +ε)]^λ²1/λ₂

.

Sinef◦Q(1/2) = [c(ϑ1)]^1/λ¹ = [c(ϑ2)]^1/λ² ând ^the ^funtionf◦Q îs ôntinuousând ^stritlyînreasing, ^this

entailsforall h >0^small^enough:

2c(ϑ1)−h

[c(ϑ1)]^1/λ¹+hiλ₁1/λ₁

=

2c(ϑ2)−h

[c(ϑ2)]^1/λ²+hiλ₂1/λ₂

.

Notingthat[c(ϑ1)]^1/λ¹ = [c(ϑ2)]^1/λ²>0^,^we^get^that^for ^allh >0^small^enough:

2−[1 +h]^λ¹1/λ₁

=

2−[1 +h]^λ²1/λ₂

.

Takinglogarithmsanddierentiatingtwie,weobtainforh >0^suiently^small:

(1 +h)^λ¹⁻²

2(λ1−1) + (1 +h)^λ¹

[2−(1 +h)^λ¹]² =(1 +h)^λ²⁻²

2(λ2−1) + (1 +h)^λ² [2−(1 +h)^λ²]² .

Lettingh↓0^entailsλ1=λ2^,^whih^ompletes^the^proof.

Wenotethatthisresultrequiresthe medianofX ^to^be^positive. ^For^some^families^suh^as ^theBikel-Doksum family(1981),

∀x∈R, ∀λ >0, ψ(x;λ) =sgn(x)|x|^λ−1

λ , ^with sgn(x) =









1 ^ifx >0,

−1 ^ifx <0, 0 ^ifx= 0,

(4.3)

thisassumptionmayatuallybedropped,asshownbyCorollary1below. Thispartiularfamilyoftransforma-

tions,whihoinideswiththeBox-Coxfamilyoftransformationsforpositivevaluesofxândλ^,îs^theône^we

shallonsiderinoursimulationstudy.

Corollary 1. Let ψ ^be ^the Bikel-Doksum family of transformations. Assume that (A1) ^holds ^and ^that ^the

distributionof X îs ^not^symmetri âround^zero. Îf^there êxistsâ^pair(δ, λ)∈R×(0,∞) ^suh^that ψ(X;λ)−δ

issymmetrially distributedaround zero,then(δ, λ) ^is^the^unique ^suh^pair.

Proof of Corollary 1. Werstnotethatfor anysuhpairϑ= (δ, λ)^,^thenδ6=−1/λ^. Îf îndeed^we^had ^that δ =−1/λ^, ^thenûsing^(4.3), ^the^random ^variablesgn(X)|X|^λ ^would ^be ^symmetri. ^This^wouldîmply,^for âny x≤0^,^that

P(X ≤x) =P(sgn(X)|X|^λ≤ −(−x)^λ) =P(sgn(X)|X|^λ≥(−x)^λ) =P(X≥ −x).

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ThenX ^would^besymmetriallydistributedaround zero, whihis aontradition. Moreover, wemay assume withoutloss ofgeneralitythat themedianQ(1/2) ^ofX ^isnonnegative: ifindeedthis isnotthe ase then−X

hasanonnegativemedianand,lettingδ^′=−(δ+ 2/λ)6=−1/λ^,^the^random^variable ψ(−X;λ)−δ^′=−[ψ(X;λ)−δ]

issymmetriallydistributedaroundzero. Finally,sine(A1)^holdsând(A2)ând(A3)âre^satised^for^the^Bikel-

Doksumfamily,wehaveQZ(x;ϑ) =ψ(Q(x);λ)−δ ^by^(4.1). ^SineZ(ϑ) îs symmetriallydistributed around zero,wemusthave0 =Q(1/2)^λ−(1 +δλ)^. Êspeially^,^the^medianQ(1/2) = [c(ϑ)]^1/λôfX îs^positive. Âpplying

Proposition2onludestheproof.

5 A Monte-Carlo simulation study

5.1 Finite sample performane of the presented tehnique

Inthissetion,wepresenttheresultsofaMonte-Carlostudyondutedtoassesstheperformaneofourmethod.

Inwhatfollows,thetransformationfamilyonsideredistheBikel-Doksumfamily(4.3). Thefollowingestimators

areompared:

• ^our^estimator^(3.1),^denoted^byMγ^,^withγ∈ {1,2}^;

• ^the^estimator

arg min

ϑ∈Θ

Z ∞

−∞

"

1 n

Xn

k=1

sin(tZk(ϑ))

#2

e^−|t|dt

whihorrespondstousingtheECFwithanexponentialweightingfuntion(seeYeoandJohnson,2001),

andwillbedenotedbyEECF;

• ^the^Gaussian^maximum^likelihood^estimator^(GMLE),^assuming^that^the^target^symmetridistributionis Gaussian. Whilethisestimatoratuallyattemptstotransformtonormality,weinludeitforomparative

reasons. Theshapeestimatorisbλând^the^loationêstimatorîsbδ(bλ)^where λb = arg max

λ∈Λ

(

−n

2log(cσ²(λ))−1 2

Xn

k=1

(ψ(Xk;λ)−bδ(λ))²

σc²(λ) + (λ−1) Xn

k=1

logXk

)

= arg max

λ∈Λ

(

−n

2log(cσ²(λ)) + (λ−1) Xn

k=1

logXk

)

with bδ(λ) = 1 n

Xn

k=1

ψ(Xk;λ)

and cσ²(λ) = 1 n

Xn

k=1

(ψ(Xk;λ)−δ(λ))b ².

Togetagraspofhowtheseestimatorsbehaveinpratie,weusethefollowinggeneratingalgorithm: foragiven

n−independentsampleY1, . . . , Ynôf^randomôpiesôfâ^symmetri^random^variableY^,^we^pik^(known)^values

ofλândδând^weônsider^then−independentsampleX1, . . . , Xn^suh^thatXk=τ(Yk+δ;λ)^where τ(y;λ) = sgn(λy+ 1)|λy+ 1|^1/λ

istheinverseoftheBikel-Doksumtransformation. Withthisnotation,wethushaveψ(Xk;λ)−δ=Yk^whih^are

symmetrirandomvariablesandwemayapplyourvariousproedurestoassessthequalityoftheestimationofλ

andδⁱⁿêahâse. În^what^follows,λîs^pikedⁱⁿ^the^set{1/4,1/2,3/4}^,δ= 1ând^the^symmetridistributions onsideredarethefollowing: