A NOTE ON THE EQUIVAImCE OF SOME TEST CRITERIA. v. P. Bhapkar. University of Horth Carolina. and

Transcription

1 ~ A NOTE ON THE EQUVAmCE OF SOME TEST CRTERA by v. P. Bhapkar University f Hrth Carlina University f Pna nstitute f Statistics Mime Series N. 421 February 1965 This research was supprted by the Mathematics Divisin f the Air Frce Office f Scientific Research Cntract N. AF-AFOOR DEPARTMENT OF STATSTCS UNVERSM OF NORTH CAROLNA Chapel Hill N. C.

2 ~. A NOTE ON THE EQUVAENCE r SOME TEST CRTERA ntrductin. ~-. 1ll'w_~ by V. P. Bhapkar University f Nrth Carlina University f Pna ~. t is shwn in this nte that fr testing linear hyptheses in categrical data Wald' s methd [1943] i8 equivalent t the minimum -xi methd due t Neyman [1949] fr testing nnlinear hyptheses it is equivalent t using Neyman's linearizatin technique alng with the minimum -xi statistic. (2.1) Let (2.2.) Suppse s independent rm samples f experi- mental units are taken frm s ppulatins n j is the size f the sample frm the J. th ppulatin ~ij is the bserved frequency in the! th categry f the J. th sample i =12 r j 12 s. f we assume that Pij is the prbability that an experimental unit drawn at rm frm the J. th ppulatin belngs t the! th categry that either the sampling i8 with replacement r if withut replacement sampling tractins are negligible then the prbability distributin f the bserved frequencies is given by

3 Suppse nw we have t test the hypthesis k = 12. t where F's are t independent given functins f ps ' = [PllP2l.. P(r-l)1 ; P1SP2s P(r-l)s]' t is assumed that Fl s pssess cntinuus partial derivatives up t the secnd rder with respect t p's that the rank f the t x (rs-s) matrix [CFk( }/OP ij ] is t ~ (rs-a). t is then wellknwn (e.g. refer t [2]) that):j can be tested in varius asympttically equivalent ways by using either the minild.ull -r minimum -xi r the likelihd-rati statistic tgether with any set f BAN [2] estimates f p's btained subject t cnstraints (2.3). n general equatins giving maximum likelihd r minimum -r (r xf) estimates are -fairly cmplicated t slve iterative rethds have t be used; the minimum -xi estimates thugh can be btained fairly easily by slving nly linear equatims whenever the cnstraints (2.3) are linear in p's. f the functins F k are nt linear Neyman [2] has shwn that the prblem can still be reduced t the linear case by cnstructing the minimum -xi statistic using the minimum -xi estimate subject t the 'linearized' cnstraints (2.4) ( CF k (» ap (Pij-~j) = 0 ij =g k = 12. t. The purpse f this nte is t pint ut that the minimum -xi statistic in the linear als in the nnlinear case (using 'linearizatin') is equivalent t the statistic prpsed by Wald [3] fr a much mre general prblem but as adapted t the categrical set up. 2

4 We shall assume that the cnstraints (2.3) tgether with the basic cnstraints EiPij = 1 have at least ne set f slutins (Pij} fr which Pij > 0 fr all i = 1. r ~ j = 1. s. f s the event (n ij > 0 all ij}has prbability appraching ne as N -+00 withr j's remaining fixed (r mre generally j -+ a j with a j > 0 Eja j = 1). Fr asympttic purpses then we assume that all n ij are psitive. are linear (3.l) it is knwn (e. g. see [1]) that the mininmm -x.~ (3.2) where (3. 4) say -1 c'n. c N X N g = [gkj]' kj =12 t. f the cnstraints F k are nt linear let k = 12.t statistic is if i = 1 r-l 0 fr i = r. Then the 'linearized' cnstraint (2.4) is linear f the frm (3.l) with f's replaced by h's in (3.4) f F's 3

5 The minimum -xf ' subject t the linearized cnstraints (2.4) is then with ~. = [~h2'".h t ] ~ = Fk(sa) g given by (:3.3) with f's replaced by h's in (3.4). as fllws: Nw Wald' s general therem [3] (under sme regularity cnditins) is Let $("l"" ~; a1"'" au) be the jint prbability distributin f the rm variables x's invlving unknwn parameters a's which is assumed t be psitive definite. hd estimate f (3.6) where 2. the statistic n when the t independent cnditins 4 ] a ~ = 12 u. 1\ Then if e is the maximum likeli-. (k=l t; a=l u) has a'limiting chi-square distributin with t degrees f freedm as k = 12 t

6 are satisfied. Nw fr the categrical data with' given by (2.1) n =N u = rs-s. t is easy t verify that P where Then (:;.7) (3.8) with ~ ( ) = 1 N given as in (3.5) "" "" "" n ( p-1-1 ) 02 '"'2 + Pr2 k E j = diagnal (P1j.' Pr - 1 j) k = [l](r-l)x(r_l) l!-1( ) =N! ( ) where j = [Plj. Pr-1j] h "" "" S t test the hypthesis ~ given by (2.3) Wald's statistic in view f (3.6) is l!(i1) '" [:~(!V ] = ['J<ij ]. j = ~ tx(rs-s) Q 5

7 Hence where fit ~(!1) = [~l.' ~. ~s] nkj = [~lj' ' ~(r-1)j] Therefre = [1:jn~~ ~j (9.j -!1j!1j) Q.tj] txt where ~j = diagnal (q1j' ~-1 j)!1j = [q1j' ~-1 j) t is then easy t see that Wald' s statistic {:3. 8) thus reduces t Neyman's statistic (3.5) in view f (3.3) (3.4). s that f the hypthesis is linear given by (3.1) Q = x. ~ij = f kij - f krj ~j = ~j - ~j!' where ~j = [fklj f k (r_1)j]!' = [l]1x(r_1) The matrix (3.9) is then equal t the matrix!! in (3.3) the assert:t.n fllws. 6

8 tt REFERENCES [1] V. P. Bhapkar "Sme tests fr categrical data" Ann. Math. Stat. Vl. 32 (1961) pp [2] J. Neyman "Cntributins t the thery f r test" Prceedings f the Berkeley Sympsium n Mathematical Statistics Prbability; University f CaJ.ifrnia Press Berkeley 1949 pp [3] A. Wald "Tests f statistical hyptheses cncerning several parameters when the number f bservatins is large" Trans.Amer. Math.Sci. Vl. 54 (1943) pp