le critère bic: fondements théoriques et interprétation
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HAL Id: inria-00070685https://hal.inria.fr/inria-00070685
Submitted on 19 May 2006
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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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FR appor t
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THÈME 4
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)
.
[Qhjhj[nY|jkxygiYacfcaÏhp hj[)l t m[)|rnTkxdY|fc[FZ!k t X)fc[F±>dygxrvaq²³>|giaYfqkx|prp±>dY[nhj[Il t m[)|rf8° aclo¬ylYa8b[Qhm[I|prnBkxdY|hpkmdyrfc[Ir
gidYhj|p[IrIª«d!>d t [CrwQkxlyrva t I|pgihjackmlyr¾nY|p Iw) t [Ilxhj[Cr)ilYkmdyr'nTkxdYmkmlyr t Q¬ylYac|@w)[Qhvhp[,nY|pkmygmYacfqaqhj ,w)kmZ\Z\[,fcgnY|pkmygmYaqfcaqhj ±>dY[
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MAIC = argminMi
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MAIC = argminMi
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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).
«aclyrjanTkxdY|±>dY[ef7° )>gifcaÏhp S¨I]>© |j[Crhp[gifsgiyfq[xyacf¾¥:gidYh±>dY[efq[Cr t [)doz wQkb[ !w)aq[IlxhprFnY|p Iw) t gil>h1/n
|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
Z\k t XIfq[¯©¡ª@£¤f¾rvd!\hFgmfqkx|pr t [&Z\kmlxhp|j[I| ±>dY[wQ[Ir t [)dozShp[)|pZ\[Ir rjkml>h Tkm|plY Ir [)lnY|jkxygiyaqfcaÏhp mª
à#" C à%$
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£¤f,[Irvh.Yac[)l wQkmlYlbd5±>dY[rjkmdyr t [Ir!wQkml t aÏhpackmlyr t [|j ImdYfsgi|paÏhp m'f8° [CrhpaqZ§gihj[)dY| t dZ§gzoacZedYZ t [Sb|giasrv[IZeYfsgilyw)[θrpghjasrv¥:giaqh √
n(θ − θ0)L,n→∞−→ N (0, I−1
θ0)ª[±>dYa¾gmrprvdY|p[1±>dY[
√n(θ − θ0) = OP (1).
'kxdY|!fc[rj[IwQkxl t hp[)|pZ![xfq[|p IrjdYfqhpgih.[Crh\gxrjrjdY|p nBgi|.fsgwQkxl>m[)|pm[IlywQ[S[)l5nY|pkmygmYacfqaqhj t [L′
n(θ)m[)|refsgZ<>IZ\[
±>dygil>hjaqhj 1±>dY[LG′
n(θ)±>dYaµ[Irvh
E[
∂ log (gM (X1,θ))∂θ
] ¨ 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
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