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Le critère BIC : fondements théoriques et interprétationEmilie Lebarbier, Tristan Mary-Huard

To cite this version:Emilie Lebarbier, Tristan Mary-Huard. Le critère BIC : fondements théoriques et interprétation.[Rapport Technique] RR-5315, INRIA. 2004, pp.17. inria-00070685

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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Le critère BIC : fondements théoriques etinterprétation

Emilie Lebarbier, Tristan Mary-Huard

N° 5315

Septembre 2004

Unité de recherche INRIA FutursDomaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)

Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30

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¨8m©6kxdyr rjdYnYnTk>rvkxlyr t gmlyr,hjkxdohj[&fsg\rvdyaÏhp[&±>dY[&fcg!fckma :%#% t [Ir Z!k t X)fc[Ir

Mi[CrhlYkmlaclo¥¦km|pZ§ghpaq­m[90

P (M1) = P (M2) = . . . = P (Mm).«aclyrja7ygmdywQdylZ!k t X)fc[1l° [Irvh ny|jac­baqfc )xaq xª@_bkxdyr w)[Qhvhp[&W>ubnTkihpWYXIrj[1[Qh t ° ginY|pXIr&¨

1©,[Qhe¨

2©QYfsg!|p[IwpWy[)|wpWY[ t dZ\[Iaqfcfq[IdY|

Z\k t XIfq[lY[ly IwQ[CrjrjaÏhp[±>dY[fc[ewIgifswQdYf t [fsg t asrhp|jacYdohpaqkxlP (X|Mi)

ª[ewIgifswQdYfrI° kxohjac[)l>hFnygm|f8° aclxhp )m|ghpackml t [fsgt asrhp|jacYdohpaqkxle~kmaclxhj[ t d­m[Cw¡hj[IdY| t [Ir,nygm|pgmZ\XQhj|p[Ir

θi[)h t [Ir t kmlYly )[Ir

XP (X, θi|Mi)

orjdY|,hjkxdohj[Cr,fq[Cr,­gifc[)dy|pr t [θi

0

P (X|Mi) =

∫

Θi

P (X, θi|Mi)dθi =

∫

Θi

gMi(X, θi)P (θi|Mi)dθi,

kgMi

(X, θi)[Irvhfcg!­b|giasrv[IZeYfcgmlywQ[1wQkx|j|p[IrjnTkml t gil>h gmdZ\k t X)fc[

Mit [&nygi|giZ\X)hj|p[Ir

θi0

gMi(X, θi) = P (X|θi,Mi).

l|p ) Iw)|jaqh w)[Qhjhj[&aclxhj Im|gifc[1rvkxdyr fcg.¥¦kx|jZ\[

P (X|Mi) =

∫

Θi

[ g(θi)dθi,k

g(θi) = log (gMi(X, θi)P (θi|Mi)).

¾gnY|pkmygmYaqfcaqhj P (X|Mi)

[Crh1ginYnT[)fc )[. 8 !" G 99* ! Miª[.wIgifswQdYf[)zogxw¡h t [.wQ[)hvhp[

nY|pkmygmYacfqaqhj [Irvh|gi|p[)Z\[Ilxh@nTk>rjrjacYfq[xmkml\f8° gmnYnY|pkowpWy[ gifckm|r[)l§dohjacfqasrpgilxhfsg1Z\ QhpWYk t [ t ° ginYnY|pk¯zbacZ§ghpaqkxl t [F¾gmnYfsgmwQ[0

à#" C à%$

! "#"

¸¸T:¹7¸ ¸7 o¹:¸ ¼ y» #L : R

d → R!# !

L% 3

" J # R

d ! R

d u?

∫

Rd

nL(u)du = nL(u?)

(2π

n

) d2

| − L′′(u?)|− 12 + O(n−1).

[ |p IrjdYfÏhgh@|p[Irvhj[ ­gmfcgmYfc[,nTkmdy| t [Ir¥¦kmlywQhjackmlyrL±>dYa t )nT[Il t [)l>h t [

nrjkmdBr@wQ[)|jhpgmaqlY[Crw)kml t aÏhpaqkxlyr ¨:wQ¥O«lYly[Qzo[

¢ ©¡ª 6Fkmdyr f7° ginYnyfqas±>dYkmlyr aswQa¾®!fsg.¥¦kmlywQhjackml 0

Ln(θi) =g(θi)

n=

1

n

n∑

k=1

log (gMi(Xk, θi)) +

log (P (θi|Mi))

n.

¨7x©

6kxdyr lykihjkxlyr

θ?i = argmaxθi∈Θi

Ln(θi)

Aθ?i

f7° kmnynBk>rv t [&fcg!Z§gihj|pacw)[WY[Crjrjac[)lYlY[ t [Ir t )|paq­m I[Ir rv[CwQkml t [Ir nygi|jhjac[)fcfc[Ir t [1fsg.¥¦kxlyw¡hjackxlLn(θi)

[Ilθi

0

Aθ?i

= −[

∂2Ln(θi)

∂θji ∂θl

i

]

j,l

|θi=θ?i,

¨:]>©

kθj

i

[IrvhfcgjX)Z\[wQkmZ\nTk>rjgmlxhj[ t d­m[Cw¡hp[)dY| t [Cr nygm|pgmZ\XQhj|p[Ir

θiª

6kxdyr kxohj[IlYkmlyr

P (X|Mi) =[ g(θ?

i )

(2π

n

)Ki/2

|Aθ?i|−1/2 + O(n−1),

kmd[IlywQkx|j[

log (P (X|Mi)) = log (gMi(X, θ?

i )) + log (P (θ?i |Mi)) +

Ki

2log (2π) − 1

2log (|Aθ?

i|) + O(n−1).

¨8m©

¾g t a"!§w)dYfÏhp Z\gmaql>hj[Ilygil>h[Irvhf7° )­igifcdyghpaqkxl t [θ?

i

[)h t [Aθ?

i

ª«FrjubZ\nohjkmhjas±>dY[)Z\[)l>hIθ?

i

nT[)doh >Qhp|j[&|p[)Z\nYfsgmwQ 1nygm|f8° [CrhpaqZ§ghp[)dY| t dZ\gizoaqZedYZ t [&­b|giasrv[IZeYfcgmlywQ[

θi0

θi = argmaxθi∈Θi

1

ngMi

(X, θi),

[QhAθ?

i

nygi|Iθi

kIθi

[Irhfsg!Z\gihj|paswQ[ t ° aclo¥¦kx|jZ§ghpaqkxl t [ 'asrvWY[I| nTkmdY| dylY[&kmyrj[)|p­ghpackml t Q¬BlYaq[&nygm|;0

Iθi= −E

[

∂2 log (gMi(X1, θi))

∂θji ∂θl

i

]

j,l

|θi=θi

.

lS[Q´µ[QhCbfqkx|prp±>dY[n[Crhx|pgml t

log(gMi(X, θi)P (θi|Mi))

rj[wQkxZ!nTkx|vhp[FwQkxZ!Z\[log(gMi

(X, θi))b±>dYaµwQ|pk(shg¯­m[Iw

nhpgml t asr.±>dY[

log(P (θi|Mi))|p[Irvhj[wQkxlyrhgil>hIª [)Z\nYfsgmw)[)|

θ?i

nBgi|θi[Qh

Aθ?i

nygm|Iθi

t gilBr¨5©aclxhj|pk t dyaÏh!dYl

CDC Ý"EGFH.IJF

#

hj[I|jZ\[ t ° [I|j|p[)dY| [)ln−1/2 ¨:wQ¥«FlYlY[)zb[. ©Qª 6kxdyr kxohj[)lykmlyr 0

log (P (X|Mi)) =

hj[Il t ­m[)|r −∞ gC­m[Cwn

︷ ︸︸ ︷

log (gMi(X, θi)) −

Ki

2log (n)

¨:>©

+

|p[Irvhj[1Tkm|plY 90 O(1)︷ ︸︸ ︷

log (P (θi|Mi)) +Ki

2log (2π) − 1

2log (|Iθi

|) +O(n−1/2).

llY Imfcacm[Igmlxh fc[Ir hj[I|jZ\[Ir t ° [I|j|p[)dy|pr O(1)[)h O(n−1/2)

YlYkmdyr kxohj[IlYkmlBr

log (P (X|Mi)) ≈ log (gMi(X, θi)) −

Ki

2log (n).

° [Irvh t [w)[Qhjhj[ginYnY|pk¯zbacZ§ghpaqkxl1±>dY[fq[wQ|paÏhpX)|p[¢,£j [Crhacrprvd¾ª¾fcdyr¾nY|j CwQasrv IZ\[)lxhCCnTkmdY|¾fc[Z\k t XIfq[Mi

acfxw)km|p|j[CrvnTkml t®!f8° gmnYnY|pkCzoacZ\gihjackml t [ −2 log P (X|Mi)

[Qh[Irvh t kmlyw t Q¬ylYa¾nygi|;0

BICi = −2 log (gMi(X, θi)) + Ki log (n).

¨&m©[&Z!k t X)fc[&rj )fc[Iw¡hpaqkxlYlY 1nygm| w)[&w)|jaqhjXI|j[1[Irvh

MBIC = argminMi

BICi.

b O 6Fkmdyr\gC­xkmlyre¥:giaqh\f8° Wbu>nTkmhjWYXCrv[±>dY[fsgfckxa t [Cr!Z!k t X)fc[IrP (Mi)

[Crh\dYlyaÏ¥¦kx|jZ\[mª5¾gnY|jasrj[[)lwQkxZ\nohj[ t ° dylY[Faclo¥¦km|pZ\gihjackml %#"%@rjdY|fc[IrZ!k t X)fc[Ir[Crhhjkxdohj[)¥¦kmasrnTkxrprjaqYfc[mxkxl§dohjacfcacrj[gifckm|r@fq[wQ|paÏhpX)|p[ Z\k t aq¬y 0

−2 log (gMi(X, θi)) + Ki log (n) − 2 log (P (Mi))

b O ° [I|j|p[)dy|[)l O(n−1/2)t gmlyrf8° )xgmfcaÏhp ¨

6©[Irvh1lY )xfqacm[CgiYfc[efckm|rp±xdy[

nhj[)l t ­m[)|rf8° aclo¬ylya7ªe@gi|

wQkxlxhp|j[f7° [)|p|j[IdY| t ° gmnYnY|pkCzoacZ§ghpaqkxl&[)l O(1)nT[IdohnT[)|jhjdY|pT[)|fc[,wWYkmaqz t deZ\k t X)fc[¬ylygifbZ<>IZ\[,rja>fq[Cr t [)dYz&ny|j[)Z\ac[)|r

hj[I|jZ\[CrrvkxlxhenY|j InTkml t I|pgmlxhr&±>dygil tn[Irvhx|pgml t nYdYasrj±>d° [)fcfc[§[Crh.rvuorvhj IZ\gihjas±>dY[mª 6 IgmlYZ\kmaclyr&nTkmdy|ewQ[)|jhpgmaqly[Ir

t asrhp|jacYdohpaqkxlyr @ %#"%rjdY|.fc[Irenygm|pgmZ\XQhj|p[Irθi@fq[Shj[I|jZ\[ t ° [)|p|p[)dY|enT[Idoh<>Qhp|j[SnYfcdyr.nT[QhpaÏh§±xdy[ O(1)

ª gmlyrefc[w)gxr t ° dylY[ t asrhp|jacYdohpaqkxl ZedYfqhjaclYkx|jZ§gifc[enygm|[Qzo[)Z\nYfc[mµfq[ehj[I|jZ\[ t ° [I|j|p[)dy|F[Irvh t [ef8° km| t |j[ t [ O(n−1/2)

¨7g¥ hp[)|pu¨.)3)m]>©v©Qª

T.' /'%,'01Q * '' '

° dYlY[ t [Cr t a"!wQdYfqhj Cr t dwQ|paqhjX)|p[ ¢,£j[Irvhrvkxl!aclxhp[)|pnY|j )hpgihjackmlª'g&±>dY[Irvhjackml\[Irvhfsg&rjdYac­gil>hj[;0±xdy[)fy[Irvhfc[ Z\k t XIfq[±>dY[&f8° kml<wWY[)|wWY[®§rv Ifq[Cw¡hpaqkxlYlY[)|nygm| fc[ewQ|paÏhpX)|p[&¢,£jw)[&lYac­m[Igmdyfq[Cr lYkmhjackmlyr t [&nY|pkmBgiYacfqaqhp @ ; %#"%,kmd G %#% t ° dYl<Z\k t XIfq[ervkxlxhFnT[)d<[QzonYfcaswQaqhj[Ir[)hFlY[ t kmlylY[)l>hFnygmrdYlY[a t )[aclxhpdYaqhjac­m[ t [ew)[e±xdy[¢,£j w)kmlyrja t X)|p[>Qhp|j[FdYl ¡Tkml ,Z\k t X)fc[mª@[IrwQkxlyrva t I|pgihjackmlyrgmrjubZ\nohjkmhjas±xdy[IrnY|j Crv[Ilxhp )[IraswQaB­mkxlxhlYkmdyrnT[)|pZ\[Qhvhp|j[ t ° aclxhp[)|pnY|j )hj[I|wQ[)hvhp[lYkmhjackml t [Z\[)acfcfq[IdY|Z\k t X)fc[m t [ t Qhp[)|pZ!aclY[I|w)[±>dY[f7° kml[)l>hj[Il t nBgi|¾ny|jkxygiYacfcaÏhp G=%%#"% t ° dYleZ!k t X)fc[mZ§giasr'gmdyrprva t [,nY|j CwQasrv[I|[Il!±xdy[)forj[)lyr¢,£j [IrvhdYl!w)|jaqhjXI|j[ )wQkmlyrjasrhgil>hnTkmdy|'fsg t acZ\[)lyrjaqkxl ª@[Qhjhj[ aql>hj[)|pnY|p QhpgihjackmllYkxdyrnT[I|jZ\[Qhjhj|g.gidBrjrja t [ t asrpwQdohp[)| fcg.lY Iw)[Irprvaqhj t [f8° WbubnTkihjWyXIrj[ t ° ginYnBgi|jhj[)lBgilyw)[ t d­b|giaTZ\k t X)fc[1®efsgefcacrvhp[ t [IrZ\k t XIfq[Cr wQkmlBrva t )|p IrIª

à#" C à%$

! "#" &

!#"$

6kxdyre|p[)ny|j[IlYkmlyraswQa,fsg |p[)Z§gi|±>dygiYfc[nY|j Crv[Ilxhpgihjackml t [wQ[Qhjhj[lYkmhjackmlnY|pkmnTk>rv I[§nBgi|!¢,dY|jlyWygiZ gil t «Fl t [)|rjkml¨7mm>m©¡ª'6kmdBr rjdYnYnTkxrjkmlyr fq[Cr wQkxlYlygmacrprjgmlywQ[Cr,rjdY| fc[&wQ|paqhjX)|p[1«F£j½[)hrpg t )Z\kxlyrhp|pgihjackmlgmwI±>dYacrj[IrIª

FginYnT[)fckmlBr±>dY[efcg t [Ilyrvaqhp e®§[IrvhjacZ\[)|F[Crhfª l rvdYnYnTk>rv[e±>dY[fc[Ir

mZ\k t XIfq[Cr

M1, ...,MmrjkmlxhF[IZeTk(shj Cr)ª

¾gnyrj[)d t ki² t asrhgilyw)[ t [&%1dYfqfcygxwp¶>²¤[)acYfc[)|\¨:gmnYnT[)fc )[ t gilyrfsgrjdYaqhj[ t acrvhpgmlywQ['%©F[Ilxhj|p[ t [Idoz t [)lBrvaqhj Crf[Qh

g[Irvh t Q¬ylYac[&nygi|;0

dKL(f, g) =

∫

Ω

log

(f(x)

g(x)

)

f(x)dx.

gm|giydyr t [&lYkihghpaqkxlokml t )¬ylYaqhfsg t asrhgilyw)[(%& t [fgidZ\k t XIfq[

Minygi|;0

dKL(f,Mi) = infθi

dKL(f, gMi(., θi)).

¨-x©

dYasrp±xdy[fc[Ir Z\k t X)fc[Ir rjkml>h [)ZeTk"(chp IrIofcg t asrvhpgilBwQ[)%&[Irvh dYlY[¥¦kmlywQhjackml t IwQ|pkmasrjrpgil>hj[ t [1fcg t aqZ\[)lyrjackmlKiª l

lYkmhj[Mt

fq[Z\k t X)fc[®nygi|jhpaq| t dB±xdy[)f wQ[Qhjhj[ t asrhgilyw)[Sly[ t acZ!acl>dy[nyfqdyr+*ªd nTkmaclxh t [­bdY[ t [fsg t asrhgilyw)[KL

Mt

t kmaqh >Qhj|p[nY|p Q¥¦ I|j e®\hjkxdyrfc[Irrvkxdyr²³Z\k t XIfq[CrMii = 1, ..., t − 1

nYdYasrj±>d° aqf[CrhnYfcdyrFnY|jkYwpWY[ t [fª gm|

giacfcfq[IdY|prIMt

t kmaqh&gidyrprva >)hj|p[.ny|j )¥¦ )|p !®hjkxdyrfq[CrZ!k t X)fc[Ir t ° kx| t |j[!rvdYnT I|jac[)dY|rMii = t + 1, ...,m

nYdYasrp±>d° acfsrrjkmlxh,nYfcdyr,wQkxZ\nYfqas±>dY Ir,±>dY[

MtrjgmlyrnTkmdy|,gidohgil>h>)hj|p[FnYfcdyrny|jkowWY[Ir t [

f0@wQ[CrZ\k t XIfq[Crrjkml>h t kxlywFrjdY|g~dyrhp IrIª

6kxdyrgifcfckmlyrFZ!kxlxhp|j[I|±>dY[efc[.wQ|paqhjX)|p[e¢,£j [IrvhwQkxlyrvasrvhpgil>hFnTkxdY|w)[eZ\k t X)fc[enygm|vhpacw)dYfqac[)|C t CrvacmlY enygm|¢,dY|plYWygiZ[QhF«Fl t [)|rjkmlw)kmZ\Z\[1fc[1Z\k t XIfq[ )±>dygmrjaq²8­b|pgia ª''kxdY|

nrjdYnYnTkxrj 1m|gil t bkmlrI° aql>hj )|p[Irprj[1®!fcg t aÏ´µ )|p[)lyw)[90

BICi − BICt, i 6= t.

b §» yi < t.° gmnY|jXCr&¨&m©¡Ykmlg 0

BICi − BICt = −2 log(gMi(X, θi)) + 2 log(gMt

(X, θt) + (Ki − Kt) log(n)

= 2n

[

− 1

n

n∑

k=1

log(gMi(xk, θi)) +

1

n

n∑

k=1

log(gMt(xk, θt))

]

+ (Ki − Kt) log(n)

= 2n

[

1

n

n∑

k=1

log

(

f(xk)

gMi(xk, θi)

)

− 1

n

n∑

k=1

log

(

f(xk)

gMt(xk, θt)

)]

+ (Ki − Kt) log(n).

[Cr t [Idoz t [I|jlYacX)|p[Ir\rjkmZ\Z\[Ir\rjkml>h t [Cr![CrhpaqZ§gihj[)dy|pr\w)kmlyrjacrvhpgmlxhr t [Ir§±>dygil>hjaqhj IrdKL(f,Mi)

[)hdKL(f,Mt)

|p[IrjnT[Iw¡hpaq­m[IZ\[)lxh¨7w¡¥@ acnYfc[)u¨v*)3)>m©j©¡ª'kmdY|

nm|gil t okmlg t kxlyw<0

BICi − BICt ≈ 2n[dKL(f,Mi) − dKL(f,Mt)] + (Ki − Kt) log(n).

[Qhjhj[ ginynY|jk¯zoaqZ§gihjackmlYac[)l ±>dY[ t )hj[I|jZ\aclYacrvhj[x rvd!\h½®[QzonYfcaswQaqhj[)| fq[ wQkxZ!nTkx|vhp[)Z\[)l>h gmrju>Z\nohpkihpac±>dY[ t [BICi−BICt

0¾fq[nY|p[)Z\aq[I|¾hj[I|jZ\[ t kmZ\aclY[[)hhj[Il t ­m[)|r+∞ g¯­m[Iw

nª l[)l t t dyaÏh t kmlBw±>d° gmrju>Z\nohpkihpac±>dY[IZ![Ilxh

fc[Ir Z\k t X)fc[IrMii = 1, ..., t − 1

rjkml>h t asrp±xdBgifcaϬy Cr nygi| fq[wQ|paqhjX)|p[1¢,£jª

, >.-b » Y<i > tgilyrFwQ[ew)gmr fs®Yykml|p[Iw)kmlYlBg=(sh t gmlyr fc[1hp[)|pZ\[

2 log(gMi(X, θi)) − 2 log(gMt

(X, θt))fsg§rvhpghpacrvhjas±>dY[ t dhp[Irvh t d

/ C ÛÜØÙ.0Iò¯Û1 MtÛ2IâÄã:Ú7Û@Ú43¡ò6543¡òCÙ7ãvÔi×Cò¯âäã70)ò¯Û@é98 3¡ÝØ,ØòÜ3¡âäݯã dKL(f, Mi) ≤ dKL(f, Mm) ×:3¡òCÙÚ73¡òCÚ i

CDC Ý"EGFH.IJF

- #

|ginYnTkx|vh t [e­>|giasrj[)ZeYfsgilyw)[1nTkmdY| t [)dozZ\k t X)fc[Ir [IZeTk(shj IrIB±>dYarjkmdBr f7° W>ubnTkihpWYXIrj[Ho

rjdYaqhgmrjubZ!nYhjkihpac±>dY[IZ\[)lxhdYlY[&fckma t d Wyaϲ!®

(Ki − Kt)t [)m|p Ir t [1fcacT[)|jhj mª lg t kxlyw<0

BICi − BICt ≈ −χ2(Ki−Kt)

+ (Ki − Kt) log(n).

° [IrvhaswQa'fc[!rv[CwQkxl t hj[)|pZ\[.±>dYa t kxZ\aqlY[.[)hhj[)l t ­m[)|r+∞ g¯­m[Iw

nµfq[CrFZ\k t XIfq[Cr

Mii = t + 1, ...,m

rjkmlxh[)dozgidBrjrja t acrp±>dygifcaq¬y IrIª[Fhj[)|pZ\[F[Il

log(n)~kmdy[ t kxlywdYlS|87mfc[ ¥¦kxl t giZ\[Ilxhpgmf 0aqfµgxrjrjdY|p[F±>dY[Ffc[w)|jaqhjXI|j[¢,£j nT[)|pZ![)h

t [!wQkxl>­m[)|pm[I|­m[)|rFfq[!±>dygmrjaϲ³­b|pgia¾Z\k t XIfq[xª1° [Irvhw)[Qhjhj[!wQkxl>­m[I|jx[)lywQ[­m[)|rFfq[.Z\k t X)fc[.±>dygxrvaq²8­b|gia7BZ9>)Z\[!r)° aqf'[Crh[)ZeTk"(chp t gmlyr dYlZ\k t XIfq[1nyfqdyr x )lY )|gif8o±>dY[&f7° kmlgmnYnT[)fcfq[wQkmlBrvasrhgilyw)[nTkmdY| fsg t acZ\[)lyrjaqkxlª

£¤fylykmdyr[IrhZ§giaclxhp[)lygmlxhnTkxrprvacYfc[ t ° aql>hj[)|pnY|p Qhp[)|wQfsgiac|p[)Z\[)lxhwQ[±>dY[f8° kxl\[)lxhp[)l t nygi|nY|pkmygmYacfqaqhj G%!%#%t dZ\k t XIfq[

Miªfqfc[!rI° [IrvhjacZ![!®nBgi|jhjac| t [Ir t aq´T I|j[IlywQ[Ir

∆BICi = BICi − BICminµk

BICmint IrjaqxlY[efsg

nYfcdyr nT[QhpaÏhp[&­gmfq[IdY| kmBrv[I|j­m I[ t [¢,£jrvdy| fq[CrmZ\k t X)fc[IrIª lg 0

P (Mi|X) =exp(− 1

2∆BICi)∑m

l=1 exp(− 12∆BICl)

.

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2 L′′(u?) [ no((u−u?)3)du¨vCm©

nYdYasrp±xdy[1nygm| W>ubnTkmhjWYXCrv[xL′(u?) = 0

ª 6kxdyrwpWY[I|pwWYkxlyr ®.¥:giac|p[&ginYnygm|pg(chp|j[fc[Ir Z\kmZ\[)l>hpr t ° dYlY[fqkxaO>gidyrprvac[)lylY[mª'kxdY|w)[)fsgYolYkxdyr t I­m[)fckmnYnTkxlyr,fc[rv[Iw)kml t hj[I|jZ\[&[QzonTkmlY[Ilxhjac[)f¾rjkmdyr f8° aclxhp )m|gifc[[Ildohpaqfcacrpgil>h

[ x = 1 + x +x2

2+ o(x3).

° aclxhj Im|gifc[ t gilBr f8° [)zbny|j[Irprjaqkxl¨12©,­igidoh

∫

R

[ n (u−u?)2

2 L′′(u?)du

+

∫

R

[o(n(u − u?)3) + o(n2(u − u?)6) + o(n3(u − u?)9)

] [ n (u−u?)2

2 L′′(u?)du.

lnTk>rjgmlxh

σ =1

√

−nL′′(u?)

[Qhv =

(u − u?)

σ,

¨vI>©

lYkxdyr g¯­mkmlyr,nTkxdY|i ≥ 0

∫

R

(u − u?)i [ n (u−u?)2

2 L′′(u?)du =√

2πσσi 1√2π

∫

R

vi [ − v2

2 dv.

CDC Ý"EGFH.IJF

)] #

l |j[CwQkmlylyg=(sh\fq[<Z!kxZ\[)lxh t ° km| t |p[it ° dYly[­igi|pacgmYfc[gifc Igihjkxaq|p[

Vt [<fqkxa xgidBrjrjaq[IlYlY[wQ[Ilxhj|p )[|p t dYaqhp[<®dylY[

wQkxlyrvhpgil>hj[1nY|pXIrIª@[Cr Z\kmZ\[)l>hpr t ° kx| t |j[acZ\nygiac| Qhgil>h lbdYfsr)YlYkxdyr kmYhj[)lykmlyr∫

R

[ nL(u)du =[ nL(u?)

√2πσ

[

E[V 0] +

∫

R

o

(v6n2

σ6

) [ − v2

2 dv

]

.

.° gmnY|jXCr,fsg t Q¬BlYaÏhpaqkxl t [σ¨13©QYfc[hj[I|jZ\[ t ° [)|p|p[)dY| [Irvh t ° kx| t |p[[)l

1/n[Qhaqf[Irvh ¥:gxwQacfq[ t [&­mkmac| ±>dY[1fc[Ir,hp[)|pZ\[Ir

t ° [)|p|j[IdY|r,rjdYnT I|jac[)dY|r rjkml>h t ° km| t |j[aclo¥¦ I|jac[)dY|kmd )>gif®1/n

ª° aql>hj Im|gifc[\¨11© t [)­bac[)l>h

∫

R

[ nL(u)du =[ nL(u?)

√

− 2π

nL′′(u?)

[1 + o(n−1)

].

¨v)]b©

[&±>dYa¾w)kmlywQfcdohfcg!ny|j[IdY­m[ t [1fsg!nY|pkmnTk>rvaqhjackml2.1ª

[&|j CrvdYfqhpgih [Crhkmohp[)l>dnTkmdY| dYlY[1¥¦kmlywQhjackmlL±>dYalY[ t InT[)l t nygmr t [

nª_oaw)[1l° [Irvh nygxr fq[w)gmrIofq[&|j IrjdYfqhpgih

l° [Irvh!nYfcdyr!rja, )­ba t [)lxhCª l [Q´µ[QhI[)l5[Q´µ[IwQhjdygmlxh!dyl t )­m[IfqkxnYnT[)Z\[)l>h t [V'gCubfckm| t [L(u)

gidYhjkmdy| t [L(u?)

®f8° kx| t |p[

4YkmlkmYhjac[)lxhf8° [QzonY|j[Crjrjackml[QzonYfcacw)aÏhp[ t dhp[)|pZ![ t ° [I|j|p[)dY| [)l O(n−1)

¨ t gilyr¨14©v© ±>dYaO[Crh0

1

n

[5

24

L′′′(u?)2

L′′(u?)3− 1

8

L′′′′(u?)

L′′(u?)2

]

+ o(n−2).

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|p[Irvhj[)l>hFTkx|jly Ir[)l

n¨v¨:V ac[)|plY[Iu gil t %&g t gilY[*)3-xx©j©¡ª&£¤f@rv[I|pg t kmlBwelY CwQ[Irprpgiac|j[ t [\nTkxrj[)| t [Ir1w)kml t aqhjackmlBr t [!|p )xdYfsgi|paÏhp .rvdY|1fsg

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£w)alYkmdyrwpWy[)|pwWYkxlyr&®ginYnYfcas±xdy[)|&fsgnY|pkmnTkxrjaqhjackml ®fsg¥¦kmlywQhjackmlLn

¨3©1±>dYa t InT[)l tt [

nª@6Fkmdyr&nTkmdY­mkxlyr

rjdYnYnTkxrj[)|±>dY[fsgwQkxl>­m[I|jx[)lyw)[ t [Cr ¥¦kxlyw¡hpaqkxlyr t ° aclxhp )|8>QhprF­m[I|pr t [Ir±>dygmlxhjaqhj CrnTk>rjrj t gilxh t [CrTkxlYlY[IrFnY|jkxnY|paq )hj Irrjd!\hIª@gilyrlYkihp|j[FwIgmrIxkml t asrvnTk>rv[ t [FfsgwQkxl>­m[)|pm[IlywQ[ [)lnY|jkxygiyaqfcaÏhp t [

Ln­m[)|rdYlY[±>dygil>hjaqhj ±>dYaBlY[ t )nT[Il t

nygxr t [n¨:w¡¥'«lYlY[)zo[«©Qª@h nBgi|fc[Z9>)Z\[|giasrvkxlYlY[)Z\[)l>hI>kmlSkmohpaq[Ilxh¥:gxwQacfq[IZ![Ilxh fcg.wQkmlb­m[)|pm[IlywQ[ t [Ir t )|paq­m I[Ir

t [&fcg.¥¦kxlyw¡hpaqkxlLnª

° kmo~[IwQhjaq¥[Irvh aswQa t [ t kmlYly[)| f8° kx| t |p[ t [Ir [)|p|p[)dY|r t ° gmnYnY|pkCzoacZ\gihjackml t [θ? nygm| θ

[Qh t [Aθ?

nygi|Iθ

ª

º ¸7 o¹:¸ θ?

θ

lwWY[)|wpWY[&®!Z\kxlxhj|p[)|±>dY[√

n(θ? − θ) = OP (1). l t Iw)kmZ\nTkxrj[1w)[hj[I|jZ\[&[)lfsg\rjkmZ\Z\[ t [ t [)dozShp[)|pZ![Cr

√n(θ − θ0) +

√n(θ0 − θ?),

kθ0[Irvh1f8° dylYac±>dY[\Z§gzoacZedYZ t [

E[log (gM (X1, θ))]¨:f7° dYlYaswQaqhj §[Qzoacrvhj[\rjkmdBrfsgwQkml t aÏhpackml t ° a t [Ilxhjaq¬BgmYaqfcaqhj t d

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! "#" C

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n(θ − θ0)L,n→∞−→ N (0, I−1

θ0)ª[±>dYa¾gmrprvdY|p[1±>dY[

√n(θ − θ0) = OP (1).

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n(θ)­m[)|refsgZ<>IZ\[

±>dygil>hjaqhj 1±>dY[LG′

n(θ)±>dYaµ[Irvh

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] ¨ t )Z\kmlyrvhj|ghpackmlSrjaqZ\acfcgmaq|p[®.wQ[Ifqfc[nY|p Irj[)l>hj )[[Il«lYly[Qzo[«©¡ªdYasrp±xdy[!fq[CrWbu>nTkmhjWYXCrv[Crrvdy|1wQ[)hvhj[!fcacZ\aÏhp[!kml>h t ~v® Qhj !nTk>rv I[IrnTkmdY|kxohj[IlYaq|1fq[!|p IrjdYfÏhgh1rjdY|

θOkmlkxohjac[)l>h1fsg

wQkxl>­m[I|jx[)lywQ[[)lny|jkxygiYacfcaÏhp t [θ? ­m[I|pr θ0

¨7w¡¥[IZ\Z![5.10

t gilBr¨gml5[)|gmgm|vh*)3)-x©v©1[)h.fsg lYkm|pZ§gifcaqhj gmrjubZ\nohjkmhjas±>dY[\¨7w¡¥@V WY Ikm|pX)Z\[

5.21t gmlyr&¨gil<F[I|gxgi|jh1*))3->©v©¡ª

º ¸7 o¹:¸ Aθ?

Iθ

«¾­igilxh t [wQkxZ!Z\[IlywQ[I|fsg t IZ!kxlyrvhj|ghjackml¾ lYkmdyr t )Z\kml>hj|pkmlyrS±>dY[ √n(Aθ − Iθ)

[IrvhBkx|jly <[Il½nY|pkmygmYacfqaqhj xª.° gmnY|jXCr fq[Cr t )¬ylYaqhjackmlyr t [

Aθ¨4©¡

LGn(θ)[)h

Bn(θ)¨10©¡ykml Iw)|jaqh±>dY[

Aθ − Iθ = LG′′n(θ) − Iθ +

1

n[log (P (θ))]′′,

k fsg t )|pac­m )[5rvacmlYaq¬y[ t I|jac­m )[nygi||ginYnTkx|vh®θª llykihpgmlxh ±xdy[

E[LG′′n(θ)] = Iθ

.rjkmdyr fcgw)kml t aÏhpaqkxl±>dY[

E

(

‖[

∂2 log (gM (X1,θ))∂θj∂θl

]

j,l‖2

)

< ∞ nygm|§fc[hjWy )km|pX)Z\[ wQ[)l>hj|giffcaqZ\aqhj[x kml½gfcg w)kml>­x[)|pm[)lyw)[[)l½fqkxa t [√

n (LG′′n(θ) − Iθ)

ywQ[&±>dYaacZ\nYfcac±>dY[±>dY[

LG′′n(θ) − Iθ = OP (n−1/2).

kmZ\Z\[ 1√n[log (P (θ))]′′

wQkxl>­m[)|pm[nY|p[Irp±xdy[1rY|j[IZ\[)lxh­m[)|r0Ykmlkxohjac[)l>hF±xdy[

√n (Aθ − Iθ) = OP (1) ∀θ.

¨¯m©

[rv[CwQkml t |p IrjdYfqhpgh±>dYa­g!lYkxdyr rj[)|p­>ac| [IrvhFwQ[)fcdYa t )Z\kxlxhj|p t gmlyr fsg!nygi|jhjac[1nY|p IwQ t [)l>hj[&±>dYa[IrvhF±xdy[

θ? = θ + OP (n−1/2).¨vIx©

l[Q´µ[IwQhjdygmlxh dYl t )­m[)fckmnynB[IZ\[)lxh t [1V'gCubfckm| t [Aθ?

gmdohjkxdY| t [Aθ

®.f7° km| t |p[2onYdYasrj±>dY[

θ? [Qh θrvkxlxh nY|pkowWY[Ir

±>dygil tn[Crhm|gil t bkxlnT[Idoh CwQ|jac|p[1±>dY[

√n(Aθ? − Iθ) =

√n(Aθ − Iθ) +

√n(θ? − θ)A′

θ+ o(

√n(θ? − θ)2).

l|j[IZ§gi|±xdy[\±>dY[A′

θ= L′′′

n (θ)[Qhekml|pgmnYnT[)fcfc[§±>dY[§fsg<wQkxl t aÏhpackml ±>dY[wQ[)hvhp[±>dygmlxhpaÏhp rjkmaqhTkm|plY )[§[)l

n[Irvh

t [IZ\gml t )[&nTkmdY| kxohj[IlYaq|f7° km| t |p[[)l1/n

t gmlyr f8° gmnYnY|jk¯zoacZ\gihjackml t [&ginYfsgmw)[mª@[&|j CrvdYfqhpgih¨15© |j[Crhp[1­>|giaOnTkmdY|

θ = 誣¤f[Il ­>ac[)l>h±>dY[efq[enY|p[)Z\ac[)|hp[)|pZ\[[Crh t [.f7° km| t |p[ t [ OP (1)

ª@gi|!¨16©¡µkml gS±>dY[fc[.rj[IwQkxl t hj[I|jZ\[e[Irvh

t [&f7° km| t |p[ t [ OP (1)[QhF±>dY[1fc[ t [)|plYaq[I|&¨:[)l

oP (n1/2)©,[CrhlY )xfqacm[CgiYfc[mª lkmohpaq[IlxhFgifckm|r ±xdy[

√n(Aθ? − Iθ) = OP (1).

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Unité de recherche INRIA FutursDomaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France)

Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

ISSN 0249-6399