AND THE REST CATEGORICAL. by M. D. Moustafa

Transcription

1 TESTING OF HYPOTHESES ON A MULTIVARIATE POPULATION SOME OF THE VARIATES BEING CONTINUOUS AND THE REST CATEGORICAL by M. D. Mustafa This eseach was suppted by the United States Ai Fce '~hugh the Ai Fce Office f Scientific Reseach f the Ai Reseach and Develpment Cmmand unde Cntact N. AF 18(600) Repductin in whle in pat is pemitted f any pupse Qf the United States Gvenment. Institute f Statistics Mimegaph Seies N. 179 July ;.

2 3. AFOSR-TN ASTIA-AD UNCLASSIFIED SECURITY INFORJvIATION BIBLIOGRAPHICAL CON/fROL SHEET / 1. Univesity f Nth Calina Chapel Rill N. O. 2. Mathematics Divisin Ai Fce Office f Sientific Reseah 5. TESTING OF HYPOTHESES ON A MULTIVARIATE POPULATION SOME OF THE VARIATES BEING CONTINUOUS AND THE REST CATEGORICAL 6. M. D. Mustafa 7. July AF 18(600) File 3.3

3 TESTING OF HYPOTHESES ON A i1ultivariate POPULATION; SOME OF THE VARIATES BEING CONTINUOUS AND THE REST CATEGORICAL * by f'lj. D. Mustafa Summay: We cnside a (k+:).)-vaiate distibutin in which k vaiates ae cntinuus and i vaiates ae categical. The k vaiates ae assumed t have a cnditinal multivaiate nmal distibutin with espect t the i categical vaiates which ae assumed t have a multinmial distibutin. Apppiate hyptheses ae famed in this situatin analgus t the custmay hyptheses n a single ltdlhivaiate nmal distibutin lage sample tests f such hyptheses ae develped and sme f thei ppeties studied. Next instead f assuming a single multinmial distibutin n the i categical vaiates a pduct f multinmial distibutins is assumed (in case sme f the i-categical ae ways f classificatin) and hyptheses ae famed in this situatin analgus t the custmay nes f seveal multivaiate nmal distibutins and lage sample tests f such hyptheses and sme f thei ppeties ae studied. * This eseach was suppted by the United States Ai Fce thugh the Ai Fce Office f Scientific Reseach f the Ai Reseach and Develpment Cmmand.

4 2 Intductin: Many cntibutins have been made t the multivaiate analysis f vaiance and cvaiance f lage masses f data n eithe cntinuus categical vaiates. In this pape we ae applying the technique f the multivaiate analysis f vaiance and cvaiance t a multi-way table such that cetain ways efe t cntinuus vaiates and the the ways ae categical. F cetain pblems all the categical ways efe t vaiates; f cetain the pblems all f them efe t ways f classificatin; and f sme pblems sme f the ways efe t vaiates and the est t ways f classificatin. We use the fact that unde cetain bad cnditins the -2 lg A statistic f lage samples and n the null hypthesis has asympttically the1l 2 -distibutin with cetain degees f feedm (Wilkes Wald). Als we use the fact that f categical data the -2 lg A and the statistics ae bth equivalent (Andesn Ogawa) In sectin 2 we ncside the tw way (XZ) table in which we take X cntinuus and Z categical. Z may be a andm vaiate a way f classificatin. In sectin 3 we cnside the case f a thee way (XYZ) table in which X is a cntinuus vaiate Y and Z ae bth categical vaiates ne f them a categical vaiate and the the a way f classificatin bth ae ways f classificatin. In sectin 4 we cnside the case f a thee way (XYZ) table in which X and Yae cntinuus vaiates and Z a categical vaiate a way f classificatin.

5 3 Sectin 5 pints ut the pssibility f an extensin t a (k+l)-way 'table whee thek's ae cntinuus and the l's categical. In all these cases we fmulate all pssible hyptheses t be tested cncening cnditinal independence jint independence and ttal independence; and we give the -2 lg A statistic used tgethe with its distibutin. as: max P(sample H ) A '"' ~-.;... max P(sample i H) The likelihd ati A is defined P(x l. x Pl - p '"'-"._._..... '-'1 p(x~... x a PI' -.L. -n I whee if the null hypthesis is such that Pi '"' p~ f i = 1 2 v and H H then the numeat f A is the likelihd functin afte subs'ljituting f Pi' i = v+l. m the maximum '" 0. likelihd estimates Pi subject t the null hypthesis H ; and the denminat f A is the likelihd filllctin afte substituting the maximum likelihd estimates Pi' i = 12. m f Pi subject t the altenative H. In all these cases we assume that the cnditinal distibutin f the cntinuus vaiates given\the categical vaiates is a multivaiate nmal whse paametes might depend upn the values f the categical vaiates (at which the cnditinal pbability is being cnsideed). In case sme f the diectins alng the categical pat f the table efe t "ways f classificatin" the paametes f the cnditinal nmal dlstibutin might als depend upn the~pai'ticula cell f the pat f the table cnstituted.".

6 by the ways f classificatin. In evey case it can be shwn 4 (and has in fact been shwn in anthe pape) that the cnditinal distibutin f the cntinuus vaiates given the categical nes satisfy Db's cnditins fm which by using cetain the theems it has been pved in the the pape that -2 lg A f the jint distibutin f the cntinuus and categical vaiates has asympttically a1l 2 -distibtitin with ppe degees f feedm. Actually hweve in each case it is nt -2 lg A but anthe algebaically simple and me cnvenient statistic pved t be equivalent in pbability t -2 lg ~ that is used. A methd f btaining such a statistic is utlined in sectin 6) the mathematical pf being eseved f anthe pape. 2 The case f the tw way (XZ) table X cntinuus and Z categical: Z may be a andm vaiate away f classificatin which may belng t exhaustive and exclusive categies i 12. say. We assume that the distibutin functin f X is nmal f the diffeent categies f Z but that the paametes viz. the mean and the vaiance f the distibutin funtin f X depend n the categy. Suppse we have a sample f n inidviduals; eveyne belngs t ne anthe f the categies and als has a measuement x. Let n i i = 12 be the numbe f individuals belnging t the th ~ i categy such that Z n. a n is fixed fm sample t sample. i=l J. If Z is a way f classificatin then n. i 12. is fixed J. I. fm sample t sample; but if Z is a andm vaiate then suppse that Pi is the pbability that an individual belngs t the i th categy.

7 2.1 ~ ~ ~ ~ ~ ~ vaiate: Accding t u assumptin the mdel f the cnditinal likelihd functin is 5 (2.1.1) x1 xnl ~} =!c ( J.= 1 2n~i) x exp - Z i=l n i )'2 z _...;J.;;wJ n i (x.. -1J.(i» j=l 2V{ i) _ 2 whee x.. is the jth measuement n X f an bsevatin belnging J.J ' t the ith categy and ~I = (n n n ). Theefe the 1 2 likelihd functin is: (2.1.2) ~ n. 1\ p. J. i=l J. We ae intees'ted in testing the cmpsite hypthesis that the distibutin f X is the same in the -diffeent categies i.e. H : and lj.() =!-L(-l) = V<)'= v(-l) =... =!-L(l) =!-L = v (1) = v against the altenative H" H ' whee!-l and v ae nuisance paametes. n. F n vey lage and 1: = a cnstant the -2 lg A s'tatistic is (2.1.3) h v -2 lg A = Z n. lg ~ i=l J. V( i) We shall pve in anthe pape that this st<:ltistic is asympttically equivalent in pbability t (2.1.4) n. (~(i) _ ~)2 Z J. i=l 1\ v + ( ~(. ) ")2 n i v J... V Z -.; i=1 2.02

8 1 n. /I 1 1\ whee ~(i) =- ZJ. x.. I.l. = - n Z n. lj.(i) i J.J n j=l i=l J. 6 1 n. ~ (i) = - Zl. (x.. _ ~(i»2 n. J.J J. j=l A 1 n and Z i (x "")2 ij. v=... Z - IJ. n i=l j=l and that it has the~2-distibutinwith 2(-l) degees f feedm. 2.2 ~ ~ ~ ~ ~! wa~ ~ classificatin: Hee we have independent nmal ppulatins; and the hypthesis t be tested is that X(i) i = 12.. have identical distibutins. If n. is vey lage f i = 12. " then the statistics J.... (2.1.) and (2.1.4) hld and have the same 112-distibutin with 2(-l) degees f feedm thugh the asyn~ttic test is diffeent fm that f the pevius sectin. pwe f this ). The case f a thee way (XYZ) table X cntinuus and I Z categical: Suppse "~"hat Y can belng t the categies i = 12. and Z can belng t the categies k = 12.. s. We assume "that f given cell (ik) the pbability density functin f X is nmal with mean lj.(ik) and vaiance v(ik). Suppse we have a andm sample f size n whee n is fixed fm sample t sample such that n ik individuals belng t the th s (ik) cell and E Z n ik =n. Als evey individual bsevatin i=l k=l has the measuement x ikj j = 12...n ik f i = and k = We have s E n' k =: k=l J. n i and l: n. k = n k' We 1=1 J. 0

9 7 culd have n i ' i = 12.. n k ' k = n ik fixed in advance in which case eithe Y Z bth wuld be ways f classific3tin B'th Y and Z ae andm vaiates : ;;;;;;;;;;.;~ ~.;.-.;;;;;;.;..;;.;;..;. Let E' = (nil n 12 II' n ls ' n 21 n 22 n?s'. n l n 2.. U s )' and Pik is the pbability that an individual bsevatin belngs t the (ik)th cell whee s E P'k = p. k s and ~ E p.. 1 i"'l k=l 1.k The cnditinal likelihd functin will be (3.1.1) P { xl xnl B}... -IT ( 1 ) ~ i k. 2nv(ik) s n ik (x ik. - IJ.( ik»2.'. x exp E E E J i=l k=l j=l 2v(ik) and theefe the likelihd functin in this case is (J.1.2) P { xl'''' Xn'!!} = P { xl"" xn \!!} J\ ::k! ZVi~ik ik We cnside the fllwing kinds f cmpsite hyptheses: (3. 1a) Cnditinal independence between! ~ 1 siven~: Given that Z belngs t the k th categy then f i agains't H :f H and v(ik)'" v(k) and k = whee lj.(k) and v(k) ae abitay nuisance paametes.

10 8 The -2 lg ~ (J.1.3) statistic is s " E ~ n lg v(k) k=l i=1 ~k ~(ik). and this except f a quantity that cnveges t ze in pbability is equal t Futhe it has the J(2-distibutin with 2s(-l) degees We nte that f feedm. 1 n ik ~(ik) '" ~ ~ x'k' nik j=l ~ J ~(k) ~ n'k ~(ik) i"'l ~ ~'k) ~(ik) Likewise we can test the cnditinal independence between X and Z giv~n Y in which case the -2 lg X statistic has a ~-distibutin with 2(s-1) degees f feedm. (3.1b) Independence between ~ jintll ~ ~: The likelihd functin in this case can be witten in the fm

11 (J.1.6) s x exp - 2: k=l p i Xl".' xne I E2] :It 2: =l n' k ZJ. j=l 1T (; 1 ) 2 i k 2nv(ik) (x ikj - lj.(ik» 2v( ik) 2 n ik s { n k 1 * n'kj "IT.JL 'JT p J. k=l i=l J.k 11" n'k 1 i=l J. 9 and Pf!!2j :It s n n J "'IT p k 6 k-l k "IT n k l k=l 0 Fm (3.1.8) we can wite the null hypthesis as: H : lj.(ik) = lj.(i) v(ik) = v(i) and P'k = p P k ' f i = l. k = 1. 2 J. J.O 0 against H H The -2 lg A statistic in this case is ~ ~ n. lg V(i?)... 2 ~ i=l k=l J.k v(ik) i=l and it is (3.1.8) asympttically equivalent in pbability t: s n'k(a(ik) - ~(i0»2 n'k(~(ik) - ~(i0»2 2: Z L J. -t -:;.;J.:--.._...~---- i=l k=l v(i) 2v 2 (i0) G n. n k)2 J.O. n ik - + n _7. nink n futhe this statistic has the "'X. 2 -distibutin with 2(s-1) + (-l)(s-l) = (s-1)(3-l) degees f feedm.

12 10 Likewise we can test the independence between (XZ) jintly and Y. Hee the -2 lg ~ statistic has asympttically the)l2-distibutin with (-l)(3s-1) degees f feedm; and it will be asympttically equivalent in pbability t: (J.1.9) s n'k(~(ik) - ~(Ok»2 1: 1:[":1. k=l i=l. ;(k) + ( n _ :1;0 0 ik n n. n k 10 0 n n. n k) 2 (3.1 ) Inddpendence betwe~~! ~ (YZ) jintly: Th6 hypthesis t be tested is H: lj.(ik) = IJ. v( ik) = v f all i k whee IJ. v ae nuisance pamnetes against H ~ H The -2 lg X statistic is s (J.1.10) 1: 1: n lg v '" ik i=l k=l ~(ik)... 1 s n' whee v=- 1: 1: 1: 1 k (X k - "l IJ. n i=l k=l j=l :1 J s 1 n' k 1 s 1:J. Z n.;(ik) n i=1 k=-l j=l n i=l k=l J. A and I-L = - 1: 1: x ikj.. - 1: This stntistic is asympttically equivalent in pbability t 1\ ")2 (.) -"\)2 ~ ~- nik(lj.(i~k) - I-L + :J~k(~::~ - v - (31.11) J... A 2 _7 ; ~l ~1 v ~. and it has the "X2-distibutin with 2(s-l) dei5ees f feedm.

13 11 (3;1d' Ttal independence between ~ ~ vaiates: In view f (3.1.2) the hypthesis t be tested is nuisance paametes. H: lj.(ik) = IJ. v(ik) :It v and Pik.. Pi P k f all i k against H ~ H ' whee IJ. v Pi and P k The -2 lg A statistic is ae all S A (J.1.12) i: i: " n' k lg _ v i=l k=l ~ v(ik) and this is asympttically equivalent in pbability t s (3.1.13) i: ~ ". i=l k=l nik(~(ik) - ~)2 nik(;(ik) _ ~)2 + A V 2v 2 + n n (n _ 10 k )2 ik n It has the~2-distibutin with 2(s-l) + (-l)(s-l) = 3s - - s - 1 degees f feedm. 3.2! ~ ~ andm vaia~~ ~ ~ ~ way! ~~assificatin: In this case the hypthesis f (3.la) wuld be eplaed by the hypthesis that X and Yae independent f each Z sepaately that f (3.lb) wuld be eplaced by the hypthesis that the jint distibutin f (XI) is the same f all Z and that f (3.10) des nt seem t have an immediate analgue. The hypthesis f (3.ld) wuld be eplaced by the hypthesis that X and Yae independent each having a distibutin which is the same f all Z.

14 12 F each f these pblems the statistic and the distibutin (n the null hypthesis) ae the same as f the cespnding pblem f the pevius cases althugh the asympttic pwe f the test f any pblem in this case wuld be quite diffeent fm that f the cespnding pblem f the pevius case. 3.3 ~! ~ ~ ~ ways! classificatin: Hee n f each ik culd be fixed the maginals ik.21 =: (n l n 20 be fixed. ns) culd (i) If n ik i =: l k =: l s ae fixed fm sample t sample we shuld have s independent vaiates A(ik); n each we have a sample f n bsevatins. ik The hypthesis (3.la) shuld be eplaced by the hypthesis that the distibutin f X(ik) will be independent f Z f each Z. The hypthesis (3.Ib) and (].ld) have n analgues. But the hypthesis (3.lc) shuld be eplaced by the hypthesis that X(ik) will have the same distibutin f eachy and Z. F each f these tw pblems the statistic and the distibutin (n the null hypthesis) ae the same as f the cespnding pblems in sectin 3.1 except that the asympttic pwe f the test f any pblem hee wuld be quite diffeent fm that f the cespnding pblem in sectin 3.1. (ii)... If the maginals ni'.. (n l n 2 n ) and n 2 ' => (n'l n" ns) ae fixed the likelihd functin f the sample will be.. (J.].l) P {X l x 2.. x n!!1!!l andl!2 fixed] = P Lxl'x 2.. x n ll! such thdt ~l and.22 fixedj x P 1.2 )!!l and l!2 fixedj

15 whee 13 in which case p{!!i!!l and!!2 fixedj culd have been btained fm nj nik =-~_. 1T Pik j'f n'kj ik k J. J. by putting P ik = PiP k and unde this cnditin f independence finding P {!! J subject t!!l and!!2 being fixed. This makes it difficult t wite dwn P {!! \ 121 and!!2 fixed \ unde a distinct assumptin the than Pik" PiP k (Ry and Mita). If the nik's ae vey lage the hyptheses (3.1a) and (3.1c) wuld be the same in this case and each will have the same statistic with the same 1l 2 -distibutin unde the null hypthesis. F the hyptheses cespnding t (3.1b) and (3.1d) the statistics culd nt be btained diectly due t the fm f P {!! \!!l and!!2 fixed 1; but may be taken ve fm (3.1b) and (3.1d) by analgy. each f these tw tests cannt be btained. The pwe f 4. The case f a thee way (KYZ) table X and Y cntinuus and Z categical: Suppse that Z belngs t the i th categy whee i = 12..; and f given i we assume X and Y t have bivaiate nmal distibutin with paametes given by the vect ~'(i).. (~l(i) ~2(i» f means and v(1) f vaiance-cvaiance matix.

16 Suppse we have a sample f n bsevatins whee n is fixed 14 fm sample t sample such that n. individual bsevatins belng l. t th:.; i th categy i 1~2... and ~ n i = n. Als evey iel individual bsevatin has tw measuements (x.. Yi') j = 12.. n.. l.j J l. We culd have n: as andm numbes subject t l. in which case Z wuld be a andm vaiate in which case Z wuld be a way f classificatin. 4.1 ~ ~ categical vaiate: n.2: n i = n being fixed l.=l n. fixed f i = 12. l. Let Pi be the pbability that an bsevatin belngs t the i th categy P. = 1; and put Xl (x y). i=l 1. - The likelihd functin in this cas is ( 1 )ni ni 1( I = "IT I exp - 2: 2: -2 x.. i ""1 \ 2n 1veil t. i=l j=1 -l.j nl x--- "IT n.! i=l l. n. T[ p.l. i=1 l. We cnside the fllwing pblems in testing f hyptheses. (4.1a).Q.nd!~~ ledependence between! and 1 given~: th If Z belngs t the i categy we set H : v 12 (i) = 0 f i = 12. against H H. The -2 lg A statistic is (4.1.2) 1 2: n. lg... 2 i=l l. (1 - (i»

17 A whee ei) is the sstimate f the dinay celatin cefficient calculated fm the i th categy; and except f a quantity that cnveges t ze in pbability is asympttically equivalent t.2 ~ n. (i) i=l l. 1$ Futhe it has the 1l 2 -distibutin with degees f feedm. (4.lb) Independenc~ between (XY) jintly ~!' The hypthesis t be tested is H I 1: 1 (i) v(i) = 1: ' '" V f i '" 12 (4.1.4) against H H whee i!: and v cnsist f five abitay paametes. The -2 lg A statistic is ~ n. l' g --- V i=l ~ I V( i)f lg. (.) '" (. )(... 2(.) ; vii ~ v l- i) and this is asympttically equivalent in pbability t whee VI = (vii v 22 v 12 ) -1 ( ~-l ) 11 v12 U = 1 A-I.. -1 v l2 v 22 ("() ") ("( ")1 Uli!: i - 1!: + l: n. v i) - v x i=l 1.- u 2(Y( i) - y)

18 2.. 1("-1)2 1(-1)2 ~ -1-1 '2 vn 2 12 v v ll 12 1(" 1)2 1('-1)2 '* v12 2 v 22 v 22 v hl "-1 " (-1)2 v ll v 12 v 2 2 v 12 v ll v 22 + v 12 A l 1 '" v12 -v ~-l v =- v Iv t tv 1 16 '" and is the celatin cefficient estimated fm all the categies pled. This statistic has asympttically the1l 2 -distibutin with 5(-l) degees f feedm. (4.1c) Independence between (XZ) jintly ~ 1: A necessay and sufficient set f cnditins f this is that v 22 (i).. v 22 ' v 12 (i)" 0 f i.. 12 against H " H That this set f cnditins is sufficient is bvius; but the necessity we pve in the appendix. The -2 lg ~ statistic is (4.1.6) v 22 2: n. lg A. "2 -. i=l ~ v 22 (i)(1- (i» This statistic except f a quantity that cnveges t ze in pbability is equivalent t n.(a 2 (i) - ~2)2 2: L...;~-...; It ~= v 22

19 2 and has the X -distibutin with (J-2) degees f feedm Id ~ independence: The hypthesis we ae inteested in is H: j;!:( i) = u J;;; V(i) = V and v 12 " 0 f i = 12. against H H Hee we shall have fu abitay nuisance paametes viz. ~l' ~2' vll' v22 The -2 lg X statistic is given by (4.1.8) l: n i lg i=l and this is asympttically equivalent in pbability t 2 { ni(~k(i) - ~k)2 + ni(vkk(i).. V kk ) 2 } EL l: #0. _7 k=l ial V kk 2v kk + l: i=l n " 2 i (v 12 (i» Futhe it has the~2..distibutin with 4(-l) + = 5-4 degees f feedm. Thee is a case which we culd nt test viz. 4.le Cnditinal independence between! ~~ given 1: Owing t sme difficulties abut establishing f this situatin the asympttic~-distibutinthat has been established f the the situatins by using a line f pf discussed in anthe pape we culd nt test this case

20 ~~ way ~ classificatin: If Z may belng t categies then we independent bivaiate naml ppulatins. shall be dealing with The hypthesis in (4.la) will be that X and Yae independent in all the diffeent bivaiate ppulatins that in (4.1b) will be that (XY) will have the same distibutin in all diffeent ppulatins that in (4.1c) will have n analgue hee and finally that f (4.ld) will be that X and Yae independent and have the same distibutin in all the diffeent ppulatins. F each f these pblans the statistic and the distibutin (n the null hyptheses) ae the same f the cespnding pblems in sectin 4.1 althugh the asympttic pwe f the test f any pblem hee wuld diffe fm the cespnding pblem in sectin Fm the line f pf f these esults which will be given in anthe place simila test pblems t these culd be cnsideed geneally f a (k+})-way table in which k ways efe t cntinuus vai3tes and the ~ ways ae categical. The i categicnl ways may efe t all andm vaiates all ways f classificatin sme f them efe t andm vaiates and the emaining t ways f classificatin; and the hyptheses have t be phased accdingly. 6. In evey case we cnside we give besides the -2 lg A statistic which we calculate diectly fm (1.1) accding t the null hypthesis H and the altenative H a statistic that diffes fm the -2 lg A by a quantity that cnveges t ze in pbability. Such a statistic will be called ne which is equivalent in pbability t

21 19-2 leg A. This statistic has the same 'patten f all thd cases and culd be witten dwn easily nce the null hypthesis f any p[jticulp.j case is laid dwn. We give hee an utline withut mathematical pfs f the methd we used t btain such a statistic (which is asympttically equivalent in pbability t the -2 lg ~ statistic). The mathematical pfs ae left t be given in anthe pape. Having satisfied uselves that the pbability density functin f the vaiates cnsideed in evey case cmply with Dbls equiements we the -2 lg ~ can use the fact that f lage samples has the 1l 2 -distibutin with cetain degees f feedm. If the null hypth(~sis H is n t a simple hypthesis as happens in 311 the nsaa W8 cnside we define a simple hypthesis H * and cnside a st3tistic equivalent in pbability t -2 lg ~. whee l' max P (samplei H:) A =------~- 1 max P (sample I H ) Alsu tgethe with the altenative hypthesis H we define a simple hypthesis H* =H: and cnside [l statistic equivalent in pbability t -2 lg A 2 whe8 max P (sample \ H*) A = 0 2 max P (sample th) Fm -2 lg ~l based n H* against Hand -2 lg A 2 based n H* 000 against H we g8t -2 lg A based n H fm against H by the fllwing (6.1) -2 lg A = -2 lg A lg Al

22 Nw if n the ight side f (6.1) we eplace as indicated abve -2 lg A 2 and -2 lg Al by statistics equivalent t them im pbability the ight side f (6.1) becmes nw a statistic equivalent in pbability t -2 lg A and invlving H*. It is pssible t eplace this by anthe statistic equivalent in pbability and t shw that this latte equivalent statistic des nt invlve the unknwn tue values.. H~~ fthe paamet es ccu1dg ~n 20 Futheme since this is equivalent in pbability t -2 lg A theefe this has asympttically the cental1l 2 -distibutin with apppiate degees f feedm if: (a) H: is tue and (b) themaximum likelihd estimates f the paametes ae in the neighbhd f H*. Ccmbining these tw esults we bseve that this equivalent statistic invlves nly H (and H) and has a limiting cental1l 2 -distibutin (a) f all H* cntained ~. in H and (b) f all maximum likelihd estimates f the paametes that lie in the neighbhd f the unknwn tue H* cntained in H 0 The fetue (b) thewise called the ppety f cnsistency is pved f each case sepaately in the the pape. W6 we h-nve: give us an example the pblem in sectin 2.1 in which and v(i) = v f i = 12. whee ~ and v ae 8bitay nuisance paametes against H f l We dafine H* as:..~ whe=- ~ H*: (i) 0 ~ =IJ.=IJ. and v cnside H~ a~ainst H ' We get: and v(i) = v v e the tue values f the paamdtes ij. ( " 0)2 (" 0)2-2 lg Al ~ n IJ. - IJ. _ + n v - v V 2v 2 i = v and we

23 Als. d. H~~ H*. t H t cns~ e~ng! 0 aga~ns we ge : n.(~(i) _ ~)2 ~ -2 lg :::: Z i=l + 21 Fm Al and ' we get: (6.2) -2 lg A a -2 lg lg Xl f ~ Z ~~ i=l v n.(~(i) - ~)2 _ n(p: " ~ l+ 0 ) 2 J v Nw if ~ and ~ ae in the neighbhd f the tue values ~ and V O (iespective f these values) and this fllws fm the ~ cnsistency f ~ and ; which is pved in the the pape -- then we can shw that except f a quantity that cnveges t ze in pbability the -2 lg A given in (6.2) is equal t ni(~(i) - ~)2 n.(~(i) _ v)2 Z E ~ i=l v i=l 2v? which is the statistic we suggest f this case; it des nt invlve 002 the tue values ~ and v and has a limiting"1c -distibutin with 2(-l) degees f feedm n matte what ~ V O might be.

24 22 APPENDIX T pve that f the independence f (~ and~) jintly and 1 a set f necessay and sufficient cnditins is that ~2(i) = ~2 ' v 22 (i) = v 22 ' v 12 (i) = 0 f i = 12 We pceed as fllws x ns j'( n. t 1=. 1 J. n. T Pi~ i=l If (~ and ~) jintly ae t be independent f 1 then ~ and 1 shuld be independent and ~ and 1 shuld als be independent. F necessity theefe we pve that (i) 1 and E ae independent and next that (ii) ~ and 1 ae independent. (i) and we equie this t be independent f E' We have

25 n. 23 ~ 1 'IT ( )2" exp _ n. Z i=l 2nv 22 (i) i=l j=l 2v (i) 22 = 1:~ (Yi.(~2(i»2 x.j theefe 'IT ( i=l Z i-i ( Yij-IJ. 2( i ) ) 2 2v 22 (i) In de that this shuld be independent f categies i given n (a vect f andm nnmbessubject t 'Zl n i... n fixed) we - ~= equie that : a functin f 1 nly f any Suppse that (1) n... (n l + 1 n 2 n i - 1. n ) and let (0) I n then = 1 This is tue f any Yl n +1 Yin. 1 ~ and if Y 1 +1'" y then n 1 1.n i 1J.2(1)... 1J.2(1) This shws that v 22 (1)'" v 22 and 1J.2(i)'" 1J.2 ' f i = 12 ae necess~y cnditins f the independence

26 f (x and n) jintly and (ll) - F x Bnd 1 t be independent we have that P(~B ~ 3.) = p(~ \ 3.E) P(3. \!!) P<~) P(I) 24 v 22 n i.. )( ( )2' exp - ~ i=l 2n IV(iH i=l 2 IV(i) ~ nl x---- ' J nil i=l IT i=l n. P.l. ]. and P(~ \].) = i: P(~E 11.) ṉ this shuld be independent f 1. i.e. = (~) (say) whee x exp - i: i=l i.e. equals the uncnditinal distibutin f x If this is t be tue f all x.. and y.. (which ae andm vaiates) l.j l.j then v 12 (i) = 0 f i = l2... This theefe is anthe necessay cnditin f (x and n) t be jintly independent f n. That this set f cnditins is sufficient f the independence f (! and E)

27 25 jintly with 1 is evident. In cnclusin the auth expesses his thanks t Pfess S. N. Ruy f suggesting the pblem and the geneal line f appach and t Pfess J. Ogawa f his many valuable cmments and suggestins. REFERENCES 1. Andesn T. W. and Gdman L. A. (1957) "Statistical Infeence 2. abut Makv Chains" ~. ~'. Statist. 28 pp Came H. "MathematicJl Methds f Statistics" Pincetn Univ9sity Pess Db J. L. (1934) "Pbability and Statistics If Tans.~. Math. ~. 34 pp e 4. Mita S. K. (1955) "Cntiutins t the Statistical Analysis f Categical data" Nth Calina Institute f Statistics Mime. Seies Ogawa J. (1957) "A limit theem f Came and its genealizatinsii Nth Calina Institute f Statistics Mime Seies Ogawa J. (1957) "On the mathematical pinciples undelying the they f the x. 2 -test II Nth Calina Institute f Statistics Mime. Seies Ogawa J. lviustafa M. D. and 11.y S. N. (1957) "On the Asympttic Distibutin f the Likelihd Rati in Sme Mixed Vaiates Ppulatins" Nth Calina Institute f Statistics Mime Seies (t be published) B. Ry S. N. and lvlita S. K.(1956) "An intductin t sme nnpaametic genealizatins f analysis f vaiance and multivaiate

28 Gnalysis" Bimetika 43 pp W-31d A. (1943) "Tests f Statistical Hyptheses cncening seveal paametes when the n:umbe f bsevatins is lage" Tans. ~. ~" ~ Wilks S. S. (1938) "The lage-sample distibutin f the 26 likelihd ati f testing cmpsite hyptheses"~. ~. Statist. IX e.