Large deviations for Bayes discrimination of a finite number of simple hypotheses
We consider the problem of discrimination of a finite number of simple hypotheses in the general scheme of statistical experiments. Under conditions of the validity of theorems on large deviations for the logarithm of likelihood ratio, we investigate the asymptotic behavior of probabilities of error...
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| Date: | 1999 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
1999
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4735 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510899585941504 |
|---|---|
| author | Gabriel', L. A. Lin'kov, Yu. N. Габриель, Л. А. Линьков, Ю. Н. Габриель, Л. А. Линьков, Ю. Н. |
| author_facet | Gabriel', L. A. Lin'kov, Yu. N. Габриель, Л. А. Линьков, Ю. Н. Габриель, Л. А. Линьков, Ю. Н. |
| author_sort | Gabriel', L. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | We consider the problem of discrimination of a finite number of simple hypotheses in the general scheme of statistical experiments. Under conditions of the validity of theorems on large deviations for the logarithm of likelihood ratio, we investigate the asymptotic behavior of probabilities of errors of the Bayes criterion. We obtain the asymptotics of the amount of Shannon information contained in an observation and in the Bayes criterion. |
| first_indexed | 2026-03-24T03:04:20Z |
| format | Article |
| fulltext |
YLIK 519.21
I O , H. d'IHHbKOB, J-L A, Fa6pne, nb
(HH-T nparot. Ma'reMa'mgrl x Mexalll, IXl4 HAH YKpamlta, J~Ollell, K)
B O d I b l L I H E Y K d l O H E H I ' L , q I I P H B A ~ E C O B C K O M
P A 3 d l H t I E H H H K O H E t l H O F O q H C J ' I A I ' I P O C T b I X F H I I O T E 3
We considei the problem of testing of a finite number of simple hypotheses in general scheme of
statistical experiments. Under condition of the validity of theorems on large deviations for logarithm of
likelihood ratio, we investigate the asymptotic behavior of probabilities of errors of the Bayes criterion.
We obtain asymptotics of theShannon information containingin an observation and in the Bayes
criterion.
Po3v~lanyro zaJtaqy po3pimlemla cxinqemtoi Kian,Koc'ri npoc'rvtx vinoTe3 B 3araJlhlliPl exeMi cTa-me-
TttmmX exeneprt~e~rriB. B yMonax cnpm~e]tJmBocTi TeOpeM npo BeJmxi ~d;txHJtetma ]t.n~ ~rlorapnqbMy
BiltHOtUemm npanllonol.ti611ocTi iLocJtil~meno acrtMn'roTaqHy nol~eltiltgy ltMO~fipnoc-reI1 n0MH./IOK
6atteeonchgoro Kprn~epito. O]tep~ano aet4MnTorrmy Kian,gocri metmolfiBCi,KOi iltdpopMalxii, .aKa
Micrrrrbcg y cnoc'repe~.emfi "ra y 6attecona,KoMy gprrrepi L
1. Bue~aeane. l-lycTb (~'1, .T,, P) ~OCHOBH.Oe sepOaTHOCTn0e npocTpaBCTBO, (X t, ~t ,
{ P/, P~ . . . . . P~ } ), t ~ R+, - - ceMe~iCT~O CTaTHCTHqeCKnX ~KcnepHMeHTOB, nopox-
aaern0e Ha6~o~eHrtaMa npOa3BOZbHOR npHpoz~a ~t, 3amtcamaMa OT c~yaattHoro
napaMe'rpa 0 co 3HaqerlHaMrI rt3 KOHeqHoro VtHO~KeCTBa O = {Yl, Y2 . . . . . YN},
2<'N<*,,, r ae P/(A) = p { ~ t e A / 0 =Yi} z z z Bcex A a ~t, i = 1, 2 . . . . . A/J1].
PaCCMOTprlM 3a~a'~y npoaepKH rrmoTe3 Hi = { 0 = Yi }, i = 1, 2 . . . . . N, no pe3y~u, Ta-
TaM Ha6~IIO/IeHH~I ~t.
l Iycrb ~ = (~:t, re2 . . . . . ~N), r ae x i = P ( H i ) , i = 1, 2 . . . . . N, ~ anpaopr t~e Bepo-
' N XL, , a ato-
m a z 6eayc~OBH0e pacnpe~e~eHHe aa6.rllO/~eHHti ~t. Oqesrt~ao, p/t << p t IMI$1 Bcex
i= 1, 2 . . . . . N, rt n y c ~ 8~(x)= d P / / d p t ( x ) , x e X ~, ~ n p o r t a B o a n a z Paao-
Ha--HrtzO~aMa ~epta P/ OTHOCHTeflhHO ~epta pt . J~t~t Bcex ~ ~ R 1 aBe~eM
BesH~HHy H/j (~) = H (e, p/t, /ff ) = Et(8~)r (8~)1-~, Ha3mBaeMy~o HHTerpa~oM Xe~-
JlHHFepaIIOp$1~a E ,/~Jl~IMep p/t H pjt [1], v~c Et~MaTeMaTI, tqeCKOC O.h'KH/~aHHe
no ~epe p t , 8~ = ~ ( ~ t ) . K P OMr TOrO, ~BeaeM OTHOtlleHrte rlpaBaorlo/~o6H.a
z b = 5 ~ ( x ) / s r j ( x ) a z ~ acex i , j , c,orra~ 0 / 0 = 0 , H noao~rtM A~j (x)= ln z#t (x).
J~Iz pa3.rta'aeHHz ranoTe3 Hi, i = 1, 2 . . . . . N, rio Ha6.rno~emtio ~t BBe~eM 6a-
~eco~cKrtil KpaTepH~t 5~, n o z a r a a 5~ t (x) =Yi, ecmt Atij (x ) >_ I n ( ~ j / x i ) a z a Bcex
i* : j [2]. J~ag KpaaxocTrt 6y~;e~ nrtca'rb 5~ t (~t) = 5~. qepe3 et n = P { 5~ t ~: 0 }
6yaeM O603HaaaTb BepOaTHOC'n, omH6KH nprmzTHa rHnore3 Hi, i = 1, 2 . . . . . N, c
noMombto 6a~ecoBcKoro Kprrrepr~a 5~.
B HaCTO~IIJ.ICI~I pa6orc nccnc~tyCM acrtMrrroTHqCCKOe nOBO/XeHHC npH t --~ o, Bepo-
ffrHOCTtt oma6Ktt e~ t 6atlecoscKoro KpttTep~z 8~ B ycnoaazx , Korea cnpaBe~n~-
Bbl TeOpeMM O 6OJIbUItlX yK.rlOHCHtDIX/]J'l~l JIoFapHCl)MoB OTHOIII~HHJI IIpaB/~OIlO/]O6H~
A~ ,~.rtJi Bcex i *:j, i , j = 1, 2 . . . . . N, A~ = A~ (~'). J~.rt~l KOHKpeTH~X aa6~mo~eHHfl
~t ~Ta za~a,~a rl3yqa.rlaCb MH0rHMrl aBTopaMa (CM., HanprtMep, [3-5]). OTMeTHM
Im6OTy [6], s KOTOpOR pacc~aTptmaeTc~ o6ma~ rno,ae~b Ha6.rtlO~eHait, HO no~araeT-
ca N = 2 H S OTJ'IHqHe OT HaCTO~IIL~eI:t pa6oT~ rtcno~bayeTca He MeTO~ 60~[btttHX
yxaonermlt, a c~oiacT~a paccToamm PermH Ha pa6or [7, 8].
IIOBe/~eHHe nepOKrHOCTt~ OtttH6Ka ef TCCHO CBZaaHO C notaeReHHeM KO~tHqecTBa
IO. H. 31HHbKOB, J'l. A. FABPHEJIb, 1999
1360 ISSN 0041-6053. YKp. ~vam. ~'vpa.. 1999. ra. 51. N ~ 10
BO.,rlblIIHE YKJIOHEHH,q IIPH BA~ECOBCKOM PA3fIHqEHHH... 1361
Inenrlonol~cKofl rlHdpopMattriH I (~ t, 0), Co/lepx<atuelaCa B rla6yno/IeHrm ~t o napa-
MeTpe O, H KOal4qec'raa nHClOopMallrlrl I ( ~5~ t, O), coaep~Kalllei~ca B KpHTepHl, l ~ O
napaMeTpe 0 [4, 5]. Ha CBOilCTa KOZHqeCTBa rmqbopMamm HMeeM [9, 10]
r~e
I(~ t, 0) = H ( 0 ) - EH(O/~t) , (1)
N
H ( 0 ) = - ~ x i lnxi
i=1
m~-rponrla pacnpe~e:aemta riapaMeTpa 0, a
N
E H ( O / ~ t) = E ( - Xr~i (~t)lnr~i(~t))
i=1
cpe~tnaa ycJloBnaa ~HTponrla pacnpe/~eaermz rlapaMeTpa 0 riprt ycsiomm, q T O ~t
rlaBeCTHO. 3/1ec~, ~.(x) = P { 0 = Yi/~t = x } - - anocTepnopHoe pacripe~e~eHrm riapa-
MeTpa 0, a E - - MaTeMaTaqecKoe O~Kri~aHrle rio Mepe P. AHanor~tqHo Ha CBOP~CTB
KO~nqecTBa naqbopMamm 14MeeM [9, 10]
l ( 5 t ~, 0) = H ( 0 ) - E n ( 0 / 5 ~ t ) , (2)
r~e
N
i=l
cpe/mJ~a yc,aoanaa arrrponrla pacnpe/Ie.nemia napaMe'rpa 0 npri yCJmBHn, q r0
aHa,~eHrle St ~ naaec'rno; ~i(Y) = P { 0 = y i / 8 ~ = y} , y ~ O . 3aMeTriM, qTO
at~noJmaeTca HepaBeHcrao I( $~, 0) _< I (~ t, 0), H, aaaqrrr, rla paaeacTa (1) n (2) no-
~qyqaeM
EH(O/~ t ) < E H ( 0 / $ ~ t ) . (3)
Ilpri ttocTaTOqrtO caa6htx ycJtomIax I ( ~ t, 0 ) --9 H ( O ) tl I( ~ ~ , 0 ) .-.4 n ( o ) npri
t - -+ o. [I , 4, 5], a , anaqnT, B c n ~ y paBeacTB (1) r i (2) E H ( O / ~ t ) .---> 0 ri
E H ( 0 / 5 ~ t) ~ 0 npH t ~ o0. H c c m g o a a n a m cKopocTH Cxo~mMocTH Z Hya~o B aTHX
COOTHOIIIeHH.qX/I.rLq KOHKpeTHblX CXeM Ha6.rllOgeHHl~l ~t nocBameHu pa6oTU Mrlorrtx
aBTopoB (CM., HanpuMep, [4, 5]). B uacToamela pa6oTe riccJie~yeM CKOpOCTrt crpeM-
hernia K Hydro EH(O[~ t) ri EH(0/~5 t ) nprl t ~ oo B o6mefl cxeMe Ha6Juo/xenrllt
~t B ycJloBaax, Korea cripaaegaHBa "reopeMa o 6o~n, tmtx yi~.noitenrlax ~ n a A ~ ripri
Bcex i c j .
2. ACrlMnTOTrlKa llep0aTU0CTei~ OmH60K 6alteconcKltX KpnTepHeil. ~.rlz Hc-
c.rle/IOBaHrDl BepoJrmocTefl omrl6OK 6ataecoscKoro Kpl4Teprl.q ~ Bse~eM c~e~ysa-
mile yC~OBHa npH i ~ j :
At/. Jlcta mo6oro e e R t cymeerBycT npeaea
lim V t I l n H b ( r = ~:ij(e),
t--'r
r~e V t - - HeKOTOpaa no~omrrrem,Haa qbyHZUaa TaKaz, wro Vt "~** npH t --r **, a
Kij ( e ) - - co6eraennaa manyK~az qbyHrtala, /mqbqbepemmpye~taa Ha (e~, S~), r~e
ISSN 0041-6053. Yrp. ~uzm. ~y. pu., 1999, m. 51, N~IO
1362 IO. H. J'IHHbKOB, J'l. A. FABPHEJ'Ib
H TaKa.~[, HT0
+ = sup{e :r./j(g)<**}, s = inf{e :g/j(e)<**}, e~/
- + > I. EcJm s yCJ'IOBHH Aij BIaIIIOJIHJIeTC.q CTpOFOC Hcpa- OqcsH~HO, Eij <--0 H E i j _
-- �9 /~ "I=
BCHGI'BO E/j < 0, TO onpc~aemeHa IlpO143~OAHaJl KU(0) = 7 , a B cnyqae e U > 1 --
, l BBC,~OM BO,n14qHHbl npOH3BOAHa~ K~/(1) = Yij"
r ~ =
+ 1) + e/j 1).
H H ~ e HaM noTpc6yeTc~ cne~ymmaa TeopeMa o 6onbmax yK~OHeHrI~X ~n)I no-
rapadpMa OTHOmeHaS npas/I0nono6az A ~ npn t ~ oo.
TeopeMa 1. Ec, au etano.anaemca yc~ooue Aij c F 0 < 7~j, mo ~.,z modozo y e
(r ~
lim V~ -t ln P] ( V; ' A~ > 7) = lira V ; ' lnP] ( V; ' A~ > 7) = - l~/(V),
t . . . ) c ~ t---y *~
zae l/j(7) = supceRI (~7-- ~q(~)) - - npeoSpa#osanue fleacanapa-~enxe.aa qbynr-
~4uu zq(e).
~0Ka3aTCJII~CTB0 TeOpeMla I btO)KH0 Haffr14 B pa6oTe [11].
3a~enasue 1. HeTpy/mo 3aMeTaTS, aT0 H/~ (~) = H~j ( 1 - e ) np~ i * j .
HOaTOMy npH BblIIOJIHCHHH yCJ~OBH~ A i j TaK;~e BMHOJIH$ICTC$1 H yCJ'IOBHe Aj i ,
np14tleM Kji(•)= Kij(1-e). OTclojIaaerKo InaBeCT14. tlTO / j i (7) = l j i ( - -7) + 7" J~JIs
aToro ]IOCTaTOqHO npmdeam'b paccyer,,tteHrta H3 ]10Ka3aTeJrbcTBa TeOpeMra 1 B pa60Te
[12] a n . paccyz:nXeHaa npa suso~e CnC~XCTSHJI 2 I43 pa~OTU [13], r~ae HCO6XO~HMO
IIOJIO~KHTb Vt= ~', ~t _ p/t, At = Ate, K(e) = Kij(E), K(e) = Kji(E), l('y) - [ij(~[),
=
CJlc]~y~ottxa.q TCOpeMa OrtHCblBaCT rlOBC/ICHHC BCpO~THOCTH OLttH6KH et ~ 6ailcCOB-
czoro KpHTcp~s 5~ npH t --~ o, S cnyqac sunon~eH14a ycnos14~ Aij.
TeopeMa 2. llycrab e~mo.ana~omca yc:zo~ua Aij c r 0 < 0 < r I npu scex i ~ j,
i, j = I, 2 ..... N. Toz3a u~,eem ~#ecmo coomnomenue
lira ~'[ Ine~ = - rain I/j(0). (4)
t'-+ ** i ;~ j
)7{o~.?ame~bcmeo. OqeaH]IUO. cIIpaBC]~J'IHBO paBeHCTBO
e~. = P ( 5 ~ , O ) = ~ t , P / ( 5 ~ , y , ) = ~ , P/ A~,>In ~-L . (5)
i=1 i=l i ~:j
Orclo~a no~yqaeM e~e/Iy~otuylo OtteHKy caepxy:
e~ S Z s i Z P / A~ii>-ln ~ �9 (6)
i=1 j~i K]
TaKKaK u i) -->0 npn t--~** 14 F ~ < 0 < F~, TO BC14JIyTe, OpeMM 1
ISSN 0041-6053. Yrp. ~tam. ~.'vp.., 1999, m. $.1, N ~ 10
SOJIbLLIHE YKJIOHEHH~I HPH SAII~ECOBCKOM PA31[HqEHHH... 1363
lira ~ q In P/t /A~i-> In ~/ I =-[ j i (O) . (7)
t-->** K j
Yqnq1~Ban, ,fro ycnosne Ai) c r ~ < 0 < F~. sunosmnc'rca npH BCeX i #j,. Ha (7)
cslejxyeT, '~TO ~SXa nK)6oro e > 0 cymecvsyev to = t0(Z), O~HO aria scex i #j,
TaKoe, qTO nprl BCeX i # j ~na t > to cnpaaeamma OReHKa
Pit ( A~i>-lnl~i )<- -exp ( ( - l j i (O) + e)~t). (8)
rcj
TarriM o6paaoM, Ha (6) H (8) rlpri t > to noJtyqaeM ouermy
N
i=1 j~i t~3
OTClO/~a, yunT~,man npOH3B0~U, HOCTb e, nonyqaeM
lim sup V~ -l In e~ < - min l/j(0). (9)
t--->o* i~j
OqeBHaHO, cymeCTByeT napa qnce,n i' H j ' , j ' ~ i ' , raKa.,q, qVO l],i,(O ) =
= min ~i(0). Tor~ta Ha paBertc'raa (5) nony,4aeM ouermy CHHay
ir
e~ >- rq, Pit,(A~,r >-- ln(nr/r~j,)).
yqwruaan COOTnOtUeHHe (7), o'rc~o~a nonyqaeM
lim inf V~'~ln e~ - - rain I f i (0). (10)
t-)o* i~j
O6a,e~tnHha HcpaBeHCTBa (9) H (10), BI~/BOaHIM I'ICKOMO~ COOTUOttteaHe (4): TeopeMa 2
aoKa3aHa.
3a~e~tanue 2. TaK KaKBCHJly 3aMeqaHHJt 1 Brano.rlHeHHe ycJIOBn~l A i j .BJICqeT
BI,~rlO,qHeHHe ycJ~OBl,ln Aj i, TO a TeopeMe 2 aOCTaTOqHO Tpe6oaaTb, qTO6rZ abtrlO.rlnJl-
J'IHCb yCJ-IOBH~I A i j npH BCeX i > j (HnH, wro aKBHBaJIeHTHO, IIpa Bcex i < j ). KpoMe
TOFO, TaKKaK //j(0) = #i(0), TO min I~(0)= ~i>'nl~y(0) = m<inI/~(0), H, :~a.wr, s
i~j
(4) BMeCTO min MO~HO B3n ' rb min H~a min .
i#j i<j i>j
3a:,te~tanue 3. OqeaHaHO, yTBep~K~eH/4C TeopeMM 2 0 c T a e T c s B crLne 14 a
cnyaae anpHopnoro pacnpe~eneHria ~t = (~t,r~2, . , t . . , ~ ) , 3amlc~mero OT t TaK,
v/TO ~!/71 In (HI/ n~) ---> 0 npH t ---> o. Ann Bcex i # j .
TeopeMy 2 MO:~HO cqbop~y:BtpoBarr~ B c.neay~omeM ~KsnlBa.nen'rnoM r~rtae.
TeopeMa 3. llycmb~monn.~tomc~tycnoou.~ Aij c F 0 < 0 < F 1 npu scex i , j =
= 1, 2, , N, i ~j. ToeOa u~teem ~tecmo coomnouteRue
l imvt~ln e~ = max rn~n (11) t--'>o* i~j Kij (e).
~o~a.mamea~meo. H3Bec'rHo,'aTO lij(7) = Ye0(Y)- z(/(e0(Y)) as, J, nxo6oro
7e (7/y,7~),- + rae e0(7) -- mo6oe pemeaHe ypaBaem~a ~:#(~)' = 7 [II]. Tax KaZ
0 e (y~j,y/~), TO OTCxoaa cneayeT, ,fro Iij(O) = - Ir 0 (e0(0)). C ~mpyro{t cwOpOHm,
noczosmzy ~:~j (e/y(0)) = 0, TO ~(/(EU(0) ) = rain Ir 0 (e). TaznM 06pa~oM, COOTnO-
mealie (11) ebrreKaeT HiS COOTHOmCHHa (4). TcopeMa 3 aoza3a~a.
ISSN 0041-6053. YKp. ~tm. ~'vlm., 1999 m. 51, N~IO
1364 IO. H. J"IHHbKOB, .rl. A. FABPHF_JIb
3a~e ,mnue 4. Ec.n, ~t = (~l, ~2 . . . . . ~t), rite ~ , ~z, . . . . ~t - - ltc3aBHcHMblC
o~naxoBo pacnpeite~eHHue cay,aalaHue aemtqnma, TO COOTnOmetme (11) xopotuo
rlaaeCT~O [3], nprlqeM m~n ~r (e) ecru, D-pacxo~citemte Heac.ay ItayM,'a pacnpezte-
ncrlrIJlt~ BeJIHqHHH ~1, COOTBCTCTByIOII~HMH rrlrloTeaaH Hi rl /-/j [5].
3a~e,~anu~ 5. YqHTUaa~ 3aMeqaaHa 1 a 2, a pasencrae (11) a~ecTo max
M0~KHO Baffrb max n~a m.~.
i<j t>J
3. AenHnroTmcu KO~Hqeeraa HHCbopMauHH. Hrmce Ha~ noTpe6yeTca c~e-
ItylOird;aJ~ ,rleMMa.
J-IeHMa 1. CnpaeeO~uobt ot4enKu
z0e
EH(O/~ t) >_. - I n ( 1 - e~),
EH(0/~5~) < h ( e ~ ) + e ~ l n ( N - 1 ) ,
(12)
(13)
h(x) = - x l n x - ( 1 - x ) l n ( 1 - x ) , 0 _< x ~ 1. (14)
HepaaeHCTaO (12) ItOKa3aHO a pa6oTC [ 14], a uepaBcHcrao (13) - - a pafoTC [ 15].
I/Ia YlCMMFI 1 BbtTeKaeT cne~ymmce yTBep~K/IeHae.
C~eOcm~ue 1. tt~tetom ;~tecmo u;un.auratcuu
lira EH(0 /~ t ) = 0 r lim E H ( 0 / S t ~) = 0 r lira e~ = 0. (15)
l.--),o t--.>r l--->oa
PIHn~HKasma HeMeit~eHno c~e~ymT ua oueUOK (12) a (13) n uepaseucraa (3).
3age,~anue 6. Ecmi N = 2, TO HMIHIHKalIHH ( 1 5 ) ~ a l O T Heofxo~aHrae a ]~OCTa-
ro,mue'yc~osrm nosmotl acHMnTOTriqecKoia paaitemlMOCTrI ceMeiaCTB Hep (P/) n
(P~) nprt t ---> ** a TepMmmX EH(O/g t) a E H ( 0 / 5 ~ ) (cH. [1], TeopeMa 2.2.1).
C~eity~omaa TeopeHa ItaeT acHMnTOTrIKH ~t~a cpeitmtx yc~oBmaX auTponn~
EH(OI~ t) rt E H ( 0 / ~ ) npn t--->**.
Teopesla 4. llycmb ~ano,an~tomc~ yc,~osu.~ A 0 c F ~ < 0 < F 1 npu ocex i, j =
= 1, 2, . . . . N, i v~j. TozOa u~teem ~tecmo coomnou~enue
l i r n v t I In EH(O/~ t) = t l imvt t I n E H ( 0 / 8 ~ ) = - imi~n I/j(0). (16)
,llora.~me,a~mso. B criay coo'momearla (4) aMeeH
) i m v ~ l l n l n ( 1 - e~) = ) inavt l In e~ =-.~t~nlij(O). (17)
Ha OReHKH (12) ~t coo'momemia (17) c.rleityeT
lira inf V~-t In EH(O/~ t) >_ -rain I#(0). (18)
t--~ ** i ~ j
AHa.noraqno B cH.n T COOTHOmeHHa (4) HMCCM
lira V~-I In (h (et n) + e~ In ( N - 1 )) = -rain I~(0). (19)
t"'>-- i ~ j
I/I30ReHKH (13) H cooTaomeurm (19) no.rwqaetd
limsup V~ "l In EH(O/6~) <- -min 1~/(0). (20)
I-.-)~
O6~r (18) H (20) rt yqHTrasaa HepaseneTso E H ( O / ~ t) < E H ( O / ~ ) ,
no.ay~aeH COOTnOmemie (16). TeopeHa 4 Itoxaaana.
Ana.aorH,mo Te0peHr 3 yTsep~iteHHe (16) ~ o ~ a o aarmcaT~ n c~eitymme~
~KBI, IBaJICI-ITltOM BI ,~ r
ISSN 0041-6053. YKp. ~tam. ~. pu., 1999, m. 51. N ~ 10
BO.FIBIIIHE YKJIOHEHH~q rlPH BAI~IECOBCKOM PA3JIHqEHHH... 1365
Teope~a 5. B yc,wousLx meope~t 4 u~wem ~tecmo caei~vou4ee coomnotuenue:
~ i m v ~ l l n E H ( O / ~ O = ~ i m v ~ ' I I n E H ( 0 / ~ ) = m~x rr~n~0(e). (21)
~OKa3aTeJIbCTBO TeOpeMbl 5 arla~orHtlHO ZloKaaaTe~bCTBy TeopeMra 3 H nOaTOMy
onycKaeTc~.
3a:~euanue 7. B CH~y 3aMe~laHna 2 B TeopeMax 4 ~ 5 ]IOCTaTOqrIO Tpe6OBaTb,
qTO6bI yCJIOBHR Aij BbIIIOJIHRJIHCb 3IHIIIb ~3~Jl BCCX i > j (H2II4, qTO ~KB14BaJIeHTHO,
.rlHlllb /~JIJ~ Bcex i < j ) . KpoMe TOFO, B C14JIy 3aMeqart14~ 2 I4 5 B COOTHOII/eH1414 (16)
BMeCTO min Mo~,x14o B3$ITb min 14zm min, a B COOTHOIlleI-I1414 (21) BMeCTO max
i~j i<j i>j i~j
MO~KI-IO B3~Tb max 14zt14 max.
i<j i>j
3a~e~tauue 8. Eczm ~' = (~1, ~2 . . . . . ~t), r ze ~1, ~2 . . . . . ~t u He3aB14C14MHr
O~14HaKOBO pacnpe/le.neHa~,le c~yqa~Hue Be.r114qHHbl, TO COOTHOmeH14e (21) xopotUo
14aBec'mo [51.
Y,~rn]aBa.a par~eaerBa (1) ri (2), 14a Teope r~ 4 no~y'~aeM c~e~ymttlee y'mep~t~e-
H14e O rIOBeaeH1414 KOJIHHeCTB rlHqbopMaI11414 I(~ t, O) H I(5~,0) rrpH t ~ **.
C.aeDcmaue 2. B yc.~ooustx meope~tbt 4 u~teem ~tecmo coomnoutenue
lim V t I In [ H ( 0 ) - I ( { t, 0 ) ] = l i m v , t i n [ H ( 0 ) - l(St~,0)] = ~ n Iq(0),
m.e. npu t---> ~,, (zOect, p = min lq(0))
i~j
I(~t,.O) = H ( O ) - e -p%(l+~
I(St~,0) = H ( 0 ) - e -0%(1+~
rlonyqeHHb~e B TeopeMax 2 - 5 yTBep~eHH~ n o6tuela cxe~e CTaT14CT14qeCK14X
aKCrlep14MeHTOB HO~KHO np14Memrrs K qaCTHBIM Mo~eJIJ~M CTaT14CT14tleCKHX ~Kcnepn-
MeI-rroB.
4. qaCTH~ae eJ~yqan. PaCCMOTp14M np14~e14en14e TeopeM 2--5 K qaCTHHM
n o ~ e n z n cTa'racTra,aecK14X aKcnepm~eaTOB.
C, ayea~ N = 2. HycT~, N = 2 14 BSmO~mJ~eTcz ycnoa14e A 2 t c F~ < 0 < F~.
Tor~la 143 TeopeM 2 14 3 B C14J~y aaMe,~aHHJ~ 2 nony,~aeM
lim v~qln e~ = -12~(0) = rain ~:9l(e)
(cp. c COOTHOtUeHHeM (6.1) 1,13 pa6oT~ [6]). ~a.aee, 143 Teope~ 4 H 5 B cnny 3a~e,aa-
aria 7 cne~yeT
lirn V t t In EH(O/~ t) = }imv~ t I n E H ( 0 / 5 ~ ) = -121(0) min ~:21(e)
(cp. c qbop~yno~t (2.2) 143 pa6o'na [4]).
llpm4eec nop~am,no~ aemopezpeecuu. I'IycTs {r = ({~, {2 . . . . . { t ) , t = t, 2, . . ,
na6mo/lemm npouecca aBTOpeFpeCCHH, 3a/Ial'-IHOF0 C rlOMOlI~hlO peKyppeH-moro
coo'rHomenaa ~t=O~t_l +COt, t= 1,2 . . . . . r~ae ~0=0 , O e R 1 -HerlsSeCTWai~ cny-
qataHma napa~eTp, 0}, ~ Heaanacr~Mr~e eraH~apwmar rayccoscgHe Be~atmH~a,
ne3aB14C~Im14e OT rmpa~evpa 0. IIycTh napa~erp 0 IIpHH14MaeT 3Haqem~a Yb Y2 . . . .
. . . . YN, npr~e~ lYi l< 1 ~lJ~a Bcex i = 1,2 . . . . . N. Bae~eM qby,Ktm14 bij(e) = - Y 1 -
1+ 2 + , 2 2x RI" 1 2 -F . ( y i - y j ) H c i j (E ) - Y1 e t Y i - Y j ) , e r rIycTb e 0 H e~/--KOpHH
c(/(e)-4b/~(e) = 0, pa~Hme ypaBHeH14~[ 2 2
ISSN 0041-6053. YKp. ~am. ~.'vpn., 1999, m. 51, NelO
1366 10. H. ~IHHbKOB, Jl. A. FABPHF_JIb
= ( l + y : ) 2 = 0 - y : ) 2
(Yi - Yj)(Yi + Yj + 2 ) ' (Yi - y j ) ( 2 - Yi - Yj)"
Ha TeopeMra 1 p a 6 o ~ [ 16] BhrreKaeT czmzxy~omHlt peayzmTaT.
d - l e ~ a 2. ]lna ocex i ~ j o~monnatomca ycnooua AO , npu~em ~llt=t u
1In cij(e)+~jc~j(~')-4bi~(e) ~ [ e / ~ , ~ ] ;
~:/j (e) = 2 2 ' r
e 0 e
4- HeTpygHo aaMeTnTt,, wro e~j < 0 n e/j > 1 ~.a.a acex i ~ej. C.neaoaaTe.nbHO,
o 1 7U < 0 rt ~(U > 0 ~JZBCOX i ~ j n , F O = ' Y H F 1 = T# a -rlJIBcex i s j . Ho 0 1
3Haqwr, yc~oBrm F ? < 0 < F l B u n o ~ n a e x c z a : l a Bcex i ~ j .
Ha m m m a 2 n o a y q a e m t / j ( 0 ) = - ~:q(eg(0)) , nae
Yi - Yj - YiYj (Yi + ) ) )
eij(O ) = 2
(Yi - Yj)(4 - (Yi + yj)2)"
Ecar t yi+Yj=O, TO ei j (0) = 2 - ' t , , , anaqaT, [ i j ( O ) = - K i j ( 2 - 1 ) = In ( l + y / 2 ) 1/2. B
qaCTHOCTH, ecm[ N = 2 n Yl + Y2 = 0 , n o ~ y a a e ~
l im t - I In e~ = l im t - I In E H ( O / { t) = l im t - l In E H ( 0 / 5 ~ ) = - I n (1 + y~)l /2.
t - - )*o t--->oo l---)oo
H c n o ~ y a r e o p e My na pa6oaaa [17], n e r p y ~ n o noKaaaa~, qrO y c a o B n e l y i l < 1
~ o m n o saMemrr~ y c a o a a e u lYil <1 ~t:ta ncex i ~ j , npnaeM y c z o B n e F O < 0 <F~.
n aTOM c ~ y a a e 6y~teT manO:maaa, c a n p a ncex i ~ j . H a CaMOM/~e~e; IlyCTb Yl <
< yZ< .. . <YN. Tor~aa y~ > -1 a yN< 1. I l y c ~ Yt = - 1 . Ha [17] caet~yeT, ~TO e~ =
r o 4- = 0 , = - 7 ~ = - ~ * a a a n c e x i = 2 , 3 . . . . . N . B c a z y a a ~ e q a ~ a a I eli =
= 1 - e ~ = 1 n F~ti = q(l + = - ? ~ = ** n a a Bcex i = 2, 3 . . . . . N, A n a . a o r a a n o , e c z .
YN = 1, To e .V = 0 a F O = ~'~v = - ** n : t a s c e x i = 1, 2 . . . . . N - 1. 3HaUnT, n cn-
Jay aaMeqarm.a 1 TaKme ~Vi = 1 -- ~Ni = 1 rt I't~ i = T~Vi = -- ~,V = ** ~stJ~ ~cex i =
= I , 2 . . . . . N - I . OTclo/la c.rle]~yeT, qTO yc.rIOBtle F ? < 0 < I'/} SblrIO.rII-I~leTC~q ]~JI~l
BCeX i ~ j .
E c : m Yt = -1 a ylv= 1, TO, ~ a x H Bume, noz[y 'aae~ fiN(O) = l m ( 0 ) = = In ~r~.
Ec.rm, Xpoue TOrO, N = 2, T0 aMeeT ~ e c r o cooTnomerme
l i m t - l In e~ = l im t -~ In E H ( 0 I~ t) = l i ra t - l In E H ( 0 [ 5~ ) = - In -~/2.
HcTpy /mo a a ~ r r m , , wro c c a u lYil > 1 r ipe HCKOTOpOM i , TO yczm~mc F ~ < 0 <
< F~, s o o 6 m e r o a o p a , npH n e x o r o p u x i ~ej He ~rano.,maeTC~.
1 . . / /un~o~ /O. H. Acu~mxrmttecxHr ~r e'ra'raeraxt~ c~y,mflt,ax npmteceon. - Kava: HayK.
/l)'~xa. 1993. -256 c.
2. ~opo~xoa A.A. MaTcraa'mqecxaa etaT~emxa. - M.: HayKa. 1984. - 472 c.
ISSN 0041-6053. Yrp. ~eam. :,rypn.. 1999. m. 5 J , N e 10
BOY[IMLIHE YK.JIOHEHH,q HPH EAf lECOBCKOM PA3.) 'IHqEHHH... 1367
3. ChernoffH. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of
observat ions / /Ann. Math. Statist. - 1952. - 23, No4. - P . 4 9 3 - 5 0 Z .
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Colloq. Inform. Theory. -Budapest , 1968. - 2. - P. 3 4 3 - 3 5 7 .
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P. 4 8 9 - 5 0 1 .
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1990. - 26, Ng4. - P .273-288 .
7. Liese F., Vajda I. Convex statistical d i s t ances . - Leipzig: Teubner, 1987. -224 p.
8. Kraft 0., Plachky D. Bounds for the power o f likelihood ratio test and their asymptotic properties
/ /Ann . Math. S ta t i s t . - 1970 . - 41, N~5. - P . 1646-1654.
9. ~o6py:uun PJI. O6ma~ qbopMy~uponKa ocnonnott "reopeuu IIlennoHa n TeOpt4H anqbopMatta•//
YcnexH Man'. IiayK. -- 1959. -- 14, N'g6. - C. 3 -104 .
10. fluncrep M.C. HHqbop~aaa~ n m~dpopMmmom~a~ yc ' rof lqnnocn, c .nyqafln~x ae .qntmn H
npotteccon. - M : Ha~-ao AH CCCP, 1960. - 203 e.
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- P. 263-273 .
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acypIL -- 1993. -- 45, N ~ 11. - C. 1514-1521.
13. flultbl~Oe IO.H,, MeaaeOeoa M.H. TeopeM~ o 6oar~mnx yK~otmtmax a aadlaqe npoaepKH/Iayx
npoc'r~x t r ine ' tea/ /TaM m e . - 1995. - 47, Ng2. - C. 227-235 .
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C. 381-382 .
l'loJIyqellO 25.12.97
ISSN ~1-6053. Yrp. ~lam. ~.-~lm., 1999,m. 51, Ne lO
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| id | umjimathkievua-article-4735 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:04:20Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/31/7a836c2dd70b58316ec0830dd8d95c31.pdf |
| spelling | umjimathkievua-article-47352020-03-18T21:12:54Z Large deviations for Bayes discrimination of a finite number of simple hypotheses Большие уклонения при байесовском различении конечного числа простых гипотез Gabriel', L. A. Lin'kov, Yu. N. Габриель, Л. А. Линьков, Ю. Н. Габриель, Л. А. Линьков, Ю. Н. We consider the problem of discrimination of a finite number of simple hypotheses in the general scheme of statistical experiments. Under conditions of the validity of theorems on large deviations for the logarithm of likelihood ratio, we investigate the asymptotic behavior of probabilities of errors of the Bayes criterion. We obtain the asymptotics of the amount of Shannon information contained in an observation and in the Bayes criterion. Розглянуто задачу розрізнення скінченної кількості простих гіпотез у загальній схемі статистичних експериментів, в умовах справедливості теорем про великі відхилення для логарифму відношення правдоподібності досліджено асимптотичну поведінку ймовірностей помилок байесовського критерію. Одержано асимптотику кількості шеннонівської інформації, яка міститься у спостереженні та у байесовському критерії. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4735 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1360–1367 Український математичний журнал; Том 51 № 10 (1999); 1360–1367 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4735/6171 https://umj.imath.kiev.ua/index.php/umj/article/view/4735/6172 Copyright (c) 1999 Gabriel' L. A.; Lin'kov Yu. N. |
| spellingShingle | Gabriel', L. A. Lin'kov, Yu. N. Габриель, Л. А. Линьков, Ю. Н. Габриель, Л. А. Линьков, Ю. Н. Large deviations for Bayes discrimination of a finite number of simple hypotheses |
| title | Large deviations for Bayes discrimination of a finite number of simple hypotheses |
| title_alt | Большие уклонения при байесовском различении конечного числа простых гипотез |
| title_full | Large deviations for Bayes discrimination of a finite number of simple hypotheses |
| title_fullStr | Large deviations for Bayes discrimination of a finite number of simple hypotheses |
| title_full_unstemmed | Large deviations for Bayes discrimination of a finite number of simple hypotheses |
| title_short | Large deviations for Bayes discrimination of a finite number of simple hypotheses |
| title_sort | large deviations for bayes discrimination of a finite number of simple hypotheses |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4735 |
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