THE ASYMPTOTIC PERFORMANCE OF THE LOG LIKELIHOOD RATIO STATISTIC FOR THE MIXTURE MODEL AND RELATED RESULTS

Transcription

1 ON THE ASYMPTOTIC PERFORMANCE OF THE LOG LIKELIHOOD RATIO STATISTIC FOR THE MIXTURE MODEL AND RELATED RESULTS by Jayata Kumar Ghsh Idia Statistical I$titute, Calcutta ad Praab Kumar Se Departmet f Bistatistics Uiversity f Nrth Carlia at Chapel Hill Istitute f Statistics Mime Series N September 1984

2 ON THE ASYMPTOTIC PERFORMANCE OF THE LOG LIKELIHOOD RATIO STATISTIC FOR THE MIXTURE MODEL AND RELATED RESULTS I JAYANTA KUMAR GHOSH 2 Idia Statistical Istitute, Calcutta PRANAB KUMAR SEN 3 Uiversity f Nrth Carlia, Chapel Hill Summary. The classical distributi thery f the lg likelihd rati test statistic des t hld fr testig hmgeeity (i.e., mixture) agaist mixture alteratives. Asympttic thery fr this prblem is develped. Fr sme special cases, asympttically lcally miimax tests are als fud. It is pited ut that the mai prblem is lack f idetifiability f the usual parameterizati eve whe the mixtures are idetifiable; if e chses a idetifiable parameterisati, the there is a prblem f differetiability f the desity. AMS Subject Classificati Numbers: 62E20. 62F05. Key Wrds & Phrases: Asympttic distributi; asympttic lcal miimaxity; idetifiability; likelihd rati test statistic; mixture mdel. 1) This is e f the three examples preseted by the first authr at the Neyma-Kiefer Cferece. 2) Wrk de partly at the Uiversity f Califria, Berkeley, supprted by the ONR Grat NOOOI4-80-C ) Wrk partially supprted by the Natial Heart. Lug ad Bld Istitute, Ctract NIH-NHLBI L frm the Natial Istitutes f Health.

3 1. Itrducti. Csider a family f prbability desities ad mixtures (l-ti)g(x,e(l» + TIg(x,e(2», 0 < TI < 1. We assume the mixtures are idetifiable i the sese that if TI ~ 0, TI ~ 1 ad e(l) +8(2), the the equality (l-ti)g(x,e(l» + TIg(x,e(2» = (1-TI')g(x,e(3» + TI'g(x,e(4» (1.1) implies TI = TI', e(l) = e(3), 8(2) = 8(4) r TI = 1 - TI', 8(1) = 8(4), 8(2) = e(3). Nte that because f this, g(x,8) = g(x,8') implies e = 8'. (Bth here ad i (1.1) the relatis betwee tw desities hld almst every where with respect t the dmiatig a-fiite measure ~.) Typically i cluster aalysis e mdels data exhibitig tw clusters by pstulatig a mixture f tw desities. I this ctext it is imprtat t test whether the bserved clusters are real r merely a matter f appearace caused by radm samplig frm a hmgeeus ppulati. Frmally, detig the true desity by f, e wishes t test H ;f = g(x,e), 6 E 0 (1.2) agaist the mixture alteratives HI csidered abve with TI +0, TI +1 ad e(l) + 6(2). The idetifiability assumpti (1.1) esur~s that H O ad HI have verlap. But etheless the classical asympttic thery fr likelihd rati tests is t applicable. Of curse as pited ut i the literature, the ull hypthesis is i sme sese the budary f the parameter space f this prblem, rather tha its iterir as assumed i classical thery. Hwever, Cherff (1954) has shw hw t hadle this kid f departure frm stadard assumptis; see als Feder (1968). The real

4 prblem is that thugh the mixtures are idetifiable, the parameters 8(1) 8(2), are t s. If the alterative hypthesis H 1 is true ad the true desity is writte as f(x,ti,8(1).8(2» the there is exactly ather set f parameter values, amely (1-TI,8(2),8(1», which will give exactly the same desity; it will be see i Secti 5 that this kid f uiqueess is t hard t take care f. Hwever, if H O is true ad the true desity f is g(x,8(0»,8(0) fixed, the the same desity is TI, represeted by three curves: TI = 0 ad 8(1) = 8(0) r TI = 1 ad 8(2) = 8(0) r 8(2) = 8(0) ad 8 (1) = 8(0). Ather way f expressig this fact is t bserve that we ca pass t the e dimesial space f H O by specifyig ly e c-rdiate at a time -- ad t tw -- i the three dimesial space f H 1 Of curse e ca try a parametrisati which is idetifiable, i.e., e which sets up Euclidea parameters i e t e crrespdece with the mixig distributi. The the prblem becmes e f lack f differetiability f the desity with respect t these parameters, at pits i the space f H ' Fr example, we may try O A = (1-TI)8(0) + TI8(1), A = the 8 crrespdig t Mi(TI,l-TI) (with 1 2 a suitable cveti fr TI = ~), ad A 3 = {Mi(TI,1-TI)}{2A 2-8(0)-8(1)}. We shall retur t this prblem after csiderig i detail a similar but simpler e which may be called the case f strgly idetifiable mixtures. Here e csiders tw families f prbability desities gl(x,8 1 ) P1 P2 ad g2(x,8 ), 8 E 01 c Rad 6 E 02 cr. It is tatia11y cveiet t replace TI by 8 0 i the mixture f(x,8 0,8 1,8 2 ) = (1) (1-8 0 )gl(x.8 1 ) g 2 (x,8 2 ). If 8 0 ~ 0 r 1, we assume that f(x 8(1) 8(1) 8(1» = f(x 8(2) 8(2) 8(2) implies 8(1) = 8(2) where '0'1'2 '0'1'2 8 = (8 0,8 1,8 2 ), Such mixtures may arise as mdels f partial slippage, -2-

5 ctamiati r cluster aalysis with sme ifrmati abut the directi f additial clusterig. We wish t csider the ull hypthesis (1.3) agaist the strgly idetifiable mixture alteratives. Nte that if H O is true the parameters are still t idetifiable. Here is a example f this srt which will be wrked ut i detail i Secti 4. Example 1. Here N(e,l) stads fr a rmal desity with mea e ad variace e. T mtivate ur mai result i the strgly idetifiable case, we must make a few geeral remarks abut the asympttic behaviur f the mle (maximum likelihd estimate) whe the parameter is t idetifiable. Suppse, i fact, all f Wald's (1949) cditis fr the csistecy f mle hld except fr idetifiability f e. Suppse als, t fix ideas, that the parameter space is the three dimesial Euclidea space ad all pits -idetifiable frm the true value eo lie a curve f. The best that e ca hpe fr is that the maximum f the likelihd will evetually be attaied i a eighburhd f this curve. Actually, Reder (1981) has bserved that essetially Wald's prf uder Wald's cditis (sas idetifiability) guaratees this; Reder calls it cvergece f the mle i the tplgy f the qutiet space btaied by cllapsig r it a sigle pit. This geeral fact has the fllwig implicati i the strgly idetifiable case. Whe the true desity is gl(x,e~), the first A A A tw cmpets f the mle e, amely e ad e l, will cverge almst -3-

6 surely t their true values e = 0 ad Of curse there is true value f e z t which A e Z ca cverge. (I fact uder the assump- A tis made fr Therem Z.l i Secti 3, it ca be shw that e Z cat cverge almst surely t a cstat.) The precedig facts will t be used explicitly i the sequel but they mtivate what is de here. Amg ther thigs e sees that whe H O is true, e cat cfie atteti t a eighburhd f a sigle pit i rder t maximise the likelihd. This meas that the usual quadratic apprximati t the likelihd is available ly with respect t the first tw cmpets eo ad 8 1 fr which the mle is csistet. Hwever, uder certai assumptis, we ca still utilise these partial quadratic statistic is distributed as a certai fuctial where W= sup{t( z )} ad T(e) is a Gaussia prcess with zer mea ad cvariace kerel depedig the true value e~ uder H O I Remark 2.1 we prpse a family f ther tests with simpler limitig distributi. Nte that ur treatmet des t fllw frm Cherff (1954) r Feder (1968), because they were able t explit the existece f a csistet sluti f the likelihd equati i the idetifiable case. T fllw their apprach, e wuld have t develp first results i slutis f the likelihd equati i the -idetifiable case. This ca be de but use f techiques similar t thse f Reder (1981) seemed aesthetically mre satisfactry. The fact that the likelihd is t lcally apprximable by a quadratic has ather repercussi. The prf f asympttic lcal miimaxity f the likelihd rati test via apprximati by Bayes tests als breaks dw.

7 I view f this it seems atural t itrduce a prir G 8 2 ad the wrk with the itegrated likelihd rati statistic f(x i ) G(d8 2 ) /s8up ~ g(yi,8 1 ) 1 r sme ther fuctial the itegrated likelihd. Oe shuld prbably chse G s that the assciated test is asympttically lcally miimax. It is plausible that such a G ad a assciated test always exists uder reasable cditis. As a first step. it is prved i Secti 4 that fr Example 1 the prir degeerate at b ad the crrespdig likelihd rati test assumig 8 2 = b wrks i this sese. A similar result is prved fr geeral expetials. O the ther had it is t hard t shw (thugh we d t prve it here) that the likelihd rati test is t asympttically lcally miimax fr these examples -- these are thus ew istaces f the failure f the priciple f maximisig the likelihd. Fr the case f (t strgly) idetified mixtures. a result aalgus t Therem 2.1 is btaied fr the likelihd rati test f the hypthesis i (1.2). T d this we assume a separati cditi 118(1)_8(2) II > E where E > 0 is a fixed quatity. I a subsequet cmmuicati we shall try t remve this cditi. A review f the literature this tpic is available i Bck (1981) ad Gupta ad Huag (1981) ad. the fial chapter f Everitt ad Had (1981). Eve thugh there is verlap with ur results, a paper f Mra (1973) deserves t be metied. Mra derives the asympttic distributi f the likelihd rati test f hmgeeity agaist special mixture alteratives i tw cases. amely. Pissa ad Gamma. Fr his alterative Mra csiders -5-

8 mixig distributis k G{(8-A )/a 2 } where G is a fixed kw distributi f which e eeds ly that the third mmet abut mea is zer. I ur set-up f tw pit mixtures, this wuld crrespd t assumig 8 0 = IT = k 2, s that H is equivalet t the scale parameter 18(1)_8(0)\ = O. O 2 We cclude by makig a few remarks abut mixtures f N(~, ). The therem i secti 5 applies directly ly the case f mixtures f meas with kw a ad ~ i a cmpact set. Hwever, we feel a similar result wuld hld if a is ukw but (~,) lies i a cmpact set, a # 0 ad ly mixtures f mea are allwed. If e als allws scale mixtures, substatial chages are eeded i the treatmet sice the derivative f the likelihd with respect t TI ceases t be square itegrable. Nte that a similar pheme ccurs i Mra's case if G has -zer third mmet abut mea. It shu1q be pssible t hadle such cases by suitable trucati f the derivatives. Fially, it may be ted that thugh we have cfied urselves t the case f mixtures ur mai cclusis hld fr ther cases f -idetifiable parameters. 2. The Mai Result fr Strgly Idetifiable Mixtures. Let 8 = (8,8,8 ) ad f(x,8) be as i the itrducti ad L (8) = be the lg likelihd based i.i.d. bservatis. Suppse H f (1.3) is true ad the true desity is All expectatis ad prbabilities will be cmputed i this ad the ext secti uder this assumpti but this will t be displayed i the tati. We w sketch a argumet leadig t Therem 2.1, itrducig tatis -6-

9 as we g alg. The details fr hadlig several remaider terms as well as the ecessary assumptis (AI thrugh AS) are cllected i the ext secti. Here we ly remark briefly the ature f the assumptis... The assumptis are similar t thse i the classical case but have t be stregtheed suitably t esure uifrmity i 8 2 at varius places. The latter as well as tightess f the Gaussia prcess T (-) itrduced belw makes it cveiet t wrk with 2 as a clsed buded iterval. Hwever all we really eed is cmpactess f 8 2 r its clsure. The restricti t dimesi e fr 8 2 is made s that the tightess Assumpti A (vide Secti 3) is easy t write dw; it may be exteded by makig use f aalgus cditis i Bickel ad Wichura (1972). As usual we take 8 as a pe rectagle i R P. 1 Amg ther thigs the assumptis f the ext secti guaratee that all quatities itrduced belw are well-defied. We w begi by rescalig the parameters thrugh 8 = eo + 11 / I where eo = eo + l1l / l I e 2 = 11 2 Let L (e) be deted as V (11) whe regarded as a fucti f 11 = (11 0, 1,11 2 ) Nte that V (O,0,112) is free f 11 2 ad hece may be writte simply as V (0). Let -7-

10 U ( Z ) O be the rmalised derivative with respect t e ad let be the lxp (rw) vectr f rmalised derivatives with respect t the (T1 Z ) cmpets f 8 1. The presece r absece f Z i U U l lwed belw. Let ad (p+l)x(p+l) I(Z) -ro ( z ) r01(z)t idicates depedece Z r lack f it. The same cveti is f1- -8-

11 By Assumpti AI. I.. 's are related t the secd rder derivatives f 1J lg f(x 1,8) i the usual way. Expadig v () with respect t the first tw c-rdiates, by Al (2.1) where the remaider term R l is (1) p buded sets f uifrmly i 2 ad We prve i the ext secti that uifrmly i 2, Sup 0<8 <1 - er- GlEe l L (8) v (0) + Sup A () >0 er ER P (1) p (2.2) (The prf is similar t the classical case but e has t esure uifrmity i 2 ). By the well-kw Kuh-Tucker-Lagrage therem (viz., McCrmick (1967)] the supremum f A () is (2.3) if (2.4) ad the supremum is 1 U I-1U T ~f ~ l 11 l... (2.5) -9-

12 Similarly L (H ) 0 = Sup L (8) def e =0 8 l Ee l (Z.6) Hece, A ( z ) = Z{L ( z ) - L (H )} = 0 (1) def 0 p (2.7) (Z.8) S the likelihd rati statistic is by defiiti A = Sup L ( z ) - L (H ) 0 2 (Z.9) where W = Assume Sup Zez = [b,c]. By A4 ad A5 f Secti 3, the stchastic prcess T (e) takig values i C[b,c] cverges weakly t a Gaussia prcess T(e) C[b,c] whse mea is zer ad the cvariace kerel K is the same as that f T 1 (e) ad easy t write dw. The cvariace (uder ad T ( ZZ ) is give belw assumig scalar: where J(Zl, ZZ ) is the cvariace f u lo ( Zl ) ad U lo ( ZZ ) uder 8~. Nte that Var(Tl( Z» z 1 V Z. Sice A 11 Y (e) where is a ctiuus fuctial, we have -10-

13 Therem 2.1. Uder the assumptis A1 thrugh AS f Secti 3) A cverges i distributi t T(e). Remark 2.1 (a) The limitig distributi f A simplifies a little whe.. Xi assumes ly k distict values. Idetifiability f the mixtures wuld require that k-l ~ dim dim (b) The limitig distributis f T (8 2 ) ad A (8 2 ) are give explicitly ad applied t Example 1 i Secti 4. (c) T get alterative test statistics whse limitig distributis 1 m are easier t cmpute, e may apprximate 0 2 by a fiite set 8 )".)8 2 2 i ad the csider the statistics T (8 ), i = 1,..,m which are asymp- 2 ttica1ly multivariate rmal with zer mea uder H. 0' the dispersi matrix ca be csistetly estimated sice the m1e is csistet fr 8 1 whe H is true. Oe ca use as a test statistic ay suitable fucti i f T (8 )'s; fr example) chsig a suitably psitive defiite quadratic 2 frm i T 's ad the estimatig the cefficiets, e wuld get a limitig X2-distributi. 3. Assumptis ad Details f Prf. As i Secti 2 all expectatis are cmputed uder a fixed The assumptis are marked Al thrugh AS. Istead f cllectig them at e place we shall preset them as the eed arises, i curse f supplyig sme f the details left ut i Secti 2. fr Let 8 = (8,8 ), = (O,e ) A similar cveti is fllwed l j = l,.,p, uless ther d d Let D l ' l = -- D. =-- 0 ae, J ae lj, wise stated the derivatis are evaluated at (e l,8 2 ) 0 AI. (i) 0 is a pe set f R P, ad 0 a clsed buded 1 2 iterval [b, c] f R

14 «ii) f(x,8) is ctiuus i 8 ad twice ctiuusly differetiable with respect t 8 01, (iii) E(D j lg f) = 0, E(D.D., lg f) = -E(D. lg f x D., lg f) J J J J (iv) E{ Sup IDjD j, lg f(x,8 0l,8 2 ) - DjD j, lg f(x,8~1,82)1} ~ 0 II 8l-8~111 <0 8 2 E8 2 as 0 ~ 0, j, j' = 0,1,., p T hadle the remaider i (2.2) we prceed i three stages, Assumpti A2 culatig the supremum f L (8). arbitrary but fixed eighburhds f The with the help f A3 we wrk with which is replaced at the third ad fial stage by eighburhds that shrik like -~, i. e., buded eighburhds i the l-plae. The fact that R l i (2.1) is (1) p uifrmly buded l-sets w cmpletes the prf. A2. There exists a cmpact eighburhd N f 8 01 such that Mrever, W(X1,) is ctiuus 8 2, IW(X l,8 2 )1 ~ H(X l ) V 8 2 ad E(H(X l» < 00. By the uifrm strg law f large umbers (USLLN) applied t we get ad the fact that Sup L (8) - V (0), 8 E[O,l]XN c 01 will allw us t restrict atteti t a cmpact subset f 8 1 while cal- -12-

15 L (8) - Sup 8 0l E:[0,1]XN L (e) = 0 (1) p (3.1) uifrmly i 8 2. A3. Fr each there exists a pe ball with cetre ad radius 0 such that if U = U(e 1,0) is its itersecti with 0 0 [0,1] x 01 ad the By A3 ad ctiuity f f Let U = {e l ; 0 < e U c [0,1] x N by sets Nw apply the USLLN t clude that < 0, I18l-8~1 I < J. Csider a pe cver f U(8 0l,01) ad chse a fiite subcver Ul',U m = 0 (1) P uifrmly i e 2 This cmpletes the secd stage f the prf... At the third ad fial stage, te that, by Taylr's therem ad AI, L (8) -1 L:t/J(X i,u j,8 2 ), j = l,,m, 8 2 E 02 t c- -13-

16 where IJij(T) + 1ij(T)Z)I = 0-0 (1) uifrmly i V x 8 Z We w use p (p+1)x (p+l) A4. 1(8 ) 2 is ctiuus i 8 2 ad its miimum eige value is greater tha 0 > 0 V 8 2 ; ad AS. EID lg f(xl,8~1,e2) - D lg f(xl,e~1,e2)la ~ KI82-8ill+Y fr sme a,y > O. AS esures tightess f U (.). f Dharmadhikari et a1. AI, U 1 K ad 0 is als (1). p (1968». Hece, Hece (by A4) such that fr >, (T see this e has t use the therem Sup U (T)Z) T)Z fr give, is 0 (1). Als by p we ca fid ' <, (Sup r P the U ( Z ) + Iu 11 < K, (p+1)x(p+l) smallest eige value f [Jij(T)] > ', " E: U0 x H 2 > 1 - The by first makig a suitable rthgal trasfrmati, e ca fid M such that fr > ' P{V(T) < V(O) if T) E U A() < A(O) if Iiill > M} > 1- x HZ ad I Ill I > M ad where M depeds ly K ad '. Thus with prbability > 1- E: fr >, a the supremum f L (8), by AI, i.e., A () (ver [0,1] x R P ) is attaied i R 1 f (2.1) is 0p(l) v () (ver II l ll ~M. U ) ad that f the prf f (2.2) is cmplete. Sice this set The prf f the similar result (2.6) fllws alg similar lies frm Al thrugh A4 which are f curse much strger tha what we eed fr (2.6). -14-

17 Remark 3.1 (a) The tightess assumpti A5 hlds if (3.2) ad E(~(X»8 < 00 fr sme 8 > 1. (b) The limitig distributi f T (e) depeds the true value f 8 1. It is weakly ctiuus i 8 1 prvided (i) the cvariace kerel f T( ) is ctiuus i 8 1 ; ad (ii) the Lipschitz cditi (3.2) is suitably stregtheed t be uifrm ver 8 l -eighburhds, (i) guaratees cvergece f fiite dimesial distributis f T(e) as ad (ii) guaratees tightess. Uder these cditis the limitig distributi f A is als weakly ctiuus i e l. Suppse this is s ad let t(ol,a) be such that lim P {A ~ t(ol,a)} = a. If fr each 8, t exists, is uique ad a pit f ctiuity f the limitig distributi f A uder 8 1, the t is ctiuus i 8 1 By Reder's (1981) result, '" 01 is csistet fr 8 1 uder H Hece, as pited ut t us by Peter Bickel, lim P 8 {A ~ t(8 l,a)} = a. Thus the test which rejects H " 1 if A ~ t(8 l,a) wuld be asympttically similar prvided the cditis assumed here hld. They are easy t check uder Al t A5 if e 2 is a fiite set. 4. Asympttically Lcally Miimax Tests i sme Examples. We shall calculate the asympttic prperties f tests based A (8 2 ), where A (02) is defied i (2.2) ad (2.7), ad shw that it is asympttically lcally miimax fr prblems like Example 1. Fix as i Secti 2 ad a sequece f alteratives K cr- respdig t a fixed = (, l, 2 ). We fix als a value b f 8 2 ad csider the limitig distributi f T (b) uder eo ad K 1-15-

18 where T (b) is defied i (2.4), Let Z* = V () - V (0). N(-~OlI(2)~1,OlI(2)~1) The by (2.1), Z* is asympttically By a well-kw result f LeCam, amely his first lemma ctiguity [cf. Hajek ad Sidak (1967, p. 204)], this shws K is litiguus t Sice T (b) ad Z* are asympttically bivariate rmal, by ather well-kw result f LeCam, amely, his third lemma ctiguity (vide Hajek ad Sidak (1967)], T (b) is asympttically rmal uder K with same asympttic variace as uder 6~ ad mea uder K equal t mea uder 6. Mrever, A (b) 1 relati hlds uder 6 1 plus the asympttic cvariace uder 2 = T(b)I{T (b»o}+ 0 (1) uder - p 6 1 ad K is ctiguus. K, sice the same Uder is asympttically rmal with mea zer ad variace uity. is T (b) Als the asympttic cvariace f Z* ad T (b) uder 00 -~ 00 P 01 p= {I (b)} [ I ( 2 )Cv(U (b),u ( )) 2 + L I. ( 2 )Cv(U l"u ( ))] 0 1 J J 2 J= where 1 01 ad j U I' J are the j-th cmpets f ad Nte p depeds ly ad ad s may be writte By the remarks i the precedig paragraph, the fllwig result is true. Therem 4.1. Assume the cditis f Secti 3. The lim P {A (b»x} = 1- <P(/x) if x> = 1 if x = 0 lim P K {A(b)":::'x} = 1 - <P(v'x- p (,T)2)) if x> if x = 0-16-

19 where ~ is the stadard rmal distributi fucti. Csider w Example 1 give i the itrducti ad fix a <.5. Let the limitig pwer f a sequece f tests ~ f size a uder K be deted by S({~},e~,,l,2). Let us say lcally miimax if fr all sequeces if l ERl S({~},e~,,l,2) Z E 0 Z {~} is asympttically { ~} which have limitig pwer, 'fi fr every > 0 ad By the classical thery fr the likelihd rati test ~ based if if 1 ER1 S({~},e~.,l,b) ~ r ERl S({~~}.e~,l,2,b) fr every > 0 ad Hece asympttic lcal miimaxity f ~ will fllw frm Therem 4.1 if we shw p(, z ) ~ p(,b) V Z ~ b. Fr Example 1 this fllws easily by direct calculati. Hwever the fllwig lemma shws this prperty is true fr geeral expetials. Befpre statig the lemma we itrduce sme tatis. Let ge = g(x.8) = A(e)exp{8x}h(x), e E sme pe iterval J, be a family f prbability desities. Let a < b be fixed elemets f J. Let 1/!(8) ge gb I 'Cv(- --) 11 g' g a a -17-

20 where the cvariaces are cmputed uder Ie= a ' T relate t Example 1 (ad similar prblems) te that with e l = a. We assume ~(e) is fiite J. Lemma 4.1. ~(e) ~ ~(b) if e > b. Prf. Nte that ~(a) = 0 < ~(b) (4.1) Als ~(e) ca be expressed as g g' /{I ~ - I l(b) ~ - I11}gedlJ 11 gg a a say Sice is cvex, fr ay cstat K, (x) - K ca have at mst tw sig chages ad if there are tw, they must be frm psitive t egative ad egative t psitive. Hece by Karli's well-kw result sig dimiishig prperties f the expetial desities (see, e.g. Karli (1968», ~(e) - K has similar sig chage prperties. If there exists 8' > b such that ~(8') < ~(b) the this sig chage prperty wuld be ctradicted at the pits a,b,8' prvided we chse K such that Max{,~(e')} < K < ~(b). This prves the lemma. -18-

21 5. The Case f Idetifiable Mixtures. Let 0 be a pe buded real iterval ad 0 its clsure. Let g(x,6), 6 E 0 be a family f desities ad csider the mixtures (1-8 0 )g(x,6 l ) g(x,8 2 ), 8 i E 0. where < 8 0 < 1 ad :, beig a fixed 'psitive umber. Withut lss f geerality we may take < 80 ~~. We wish t test hmgeeity, i.e., H : 8 = agaist the abve mixture alteratives. I the sequel Suppse H is true ad the true desity is We make the fllwig blaket assumpti B1. Let 6 be ay pe set ctaiig 8~ ad 1 O ay clsed set such 2 that 0 O 1 2 = 0. The AI, A3, A4, AS hld with this 0,0 (Sice is cmpact by assumpti, A2 is drpped.) 2 Oe ca w imitate the argumets i Sectis 2 ad 3. As i Secti 3, we ca shw that i rder t maximise the lg likelihd L (8) we may restrict t < e < 0 ad I81-8~1 < 0 (0 < E). Hece 8 2 may be 0 restricted t < 8 - ( -0) ad 8 2 > 8 + ( -c)). Call this set Defie,V(),A() etc. as i Secti 2. The e ca prve as i 0(1) Secti 3, that with prbability tedig t e bth L (8) ad A (). attai their maximum i a buded eighburhd f 1 = (0,0). Hece uifrmly i 8 2, Sup L (8) v (0) + Sup A () + 0 (1) p 8 01 where the maximisati f A is ver the set -19-

22 Because f the ature f Va' fr give E' e ca fid K 2 such that with prbability > 1 - E, the maximum f A (ver attaied 0 0 if ~ E + 0 r at ' l 1 E + K E - < ~ E - K A easy calculati w shws ad Sup L (8) = V (0) + 8 A () + 0 (1) p 0(2) 0 where = {8 H: E r 2 ~ 8~ +d. The supremum f L (8) uder H has therefre the same expressi as a i Secti 2 with = 0(2) Sice the expressi fr the supremum f 2 2 L (8) uder H remais ualtered, the cclusi f Therem 2.1 is 0 valid, i.e., the fllwig is true. Therem 5.1. Assume Bl. The the limitig distributi f the.. likelihd rati test uder 8~ is the same as that i Therem 2.1 with = 0(2) 22 ACKNOWLEDGEMENT Thaks are due t the referee whse cmmets clarified may issues ad led t a better presetati. REFERENCES Bickel, P.J. ad Wichura, M.J. (1971). Cvergece criteria fr multiparameter stchastic prcesses ad sme applicatis. Statist., 42, A. Math. -20-

23 Bck, H.H. (1981). Statistical testig ad evaluati methds i cluster aalysis. Paper preseted at the lsi Glde Jubilee Cferece, t appear i the Prceedigs. Cherff, H. (1954). O the distributi f the likelihd rati. A. Math. Statist., 25, Dharmadhikari, S.W., Fabia, V. ad Jgde, K. (1968). Buds mmets f martigales. A. Math. Statist., 39, Everitt, B.S. ad Had, D.J. (1981). Fiite Mixture Distributis. Chapma ad Hall, Ld. Feder, P.I. (1968). O the distributi f the lg likelihd rati test statistic whe the true parameter is ear the budaries f the hypthesis regi. A. Math. Statist., 39, Gupta, S.S. ad Huag, We-Ta (1981). O mixtures f distributis: a survey ad sme ew results rakig ad selecti. Sakhya ~ 43, Hajek. J. ad Sidak. Z. (1967). Thery f Rak Tests. Academic Press, New Yrk. Karli. S. (1968). Ttal Psitivity, Vl. 1, Stafrd Uiversity Press, Stafrd, Califria. McCrmick. S.P. (1967). Nliear Prgrammig. McGraw Hill, New Yrk. Mra. P.A.P. (1973). Asympttic prperties f hmgeeity tests. Bimetrika, 60, Reder. R. (1981). Nte the csistecy f the maximum likelihd estimate fr -idetifiable distributis. A. Statist., ~, Wald, A. (1949). Nte the csistecy f the maximum likelihd estimate. A. Math. Statist., 20,