EMPIRICAL LIKELIHOOD INFERENCE UNDER STRATIFIED RANDOM SAMPLING USING AUXILIARY INFORMATION

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1 EMPIRICAL LIKELIHOOD INFERENCE UNDER STRATIFIED RANDOM SAMPLING USING AUXILIARY INFORMATION C.X.Bob Zong and J.N.K. Rao, Carleton Unversty Dept. of Mat. &: Stats. Carleton Unversty, Ottawa, ON. K1S 5B6 Key words: Dstrbuton functon; Jackknfe; Lkelood rato; Estmatng equaton; Confdence nterval. Introducton Suppose we are requred to make some nference about populaton parameters. In many practcal problems, we ave some nformaton about te populaton. For example, some populaton values of te auxlary varable x are known, suc as te populaton mean or medan. Anoter example s tat we may know tat te populaton varance s a functon of te populaton mean. By usng ts knd of knowledge, we would lke to provde mproved nferences on te populaton parameters suc as te populaton mean and te populaton dstrbuton functon. Emprcal lkelood metods, recently ntroduced by Owen (1988, 90, 91) n te context of d random varables, provde a systematc nonparametrc approac to utlzng auxlary nformaton n makng nference on te parameters of nterest. Hartley and Rao (1968) gave te orgnal dea of emprcal lkelood n sample survey context, usng ter "scale-based" approac. Cen and Qn (1993) extended ter results to cover dstrbuton functon and medan. Qn and Lawless ( 1994, 95) used emprcal lkelood and estmatng equatons n te d case to deal wt nterval estmaton or ypotess testng. Tey obtaned an emprcal lkelood rato test statstc (ELRS) and ts asymptotc dstrbuton under null ypotess. Te am of ts artcle s to study te case of stratfed random samplng. In secton 2, we gve te maxmum emprcal lkelood estmator (MELE) for te parameter of nterest, were we know te mean of X of te auxlary nformaton x and samples are taken wtout replacement n eac stratum and ndependently across strata. Some large sample propertes of MELE are also dscussed. Te emprcal lkelood rato est- mator (ELRE) s dscussed. Te results sow tat MELE s equal to ELRE. Emprcal lkelood rato tests (ELRT) are also dscussed. In secton 3, we use emprcal lkelood and estmatng equatons to deal wt nterval estmaton and ypotess testng. 2 Emprcal Lkelood Inference Usng Stratfed Random Samplng wtout Replacement 2.1 Introducton Suppose tat a target populaton s dvded nto H strata wt known wegt W for all stratum, ~ W In stratum tere are N unts wt values Z( -- 1,...,N; = 1,..., H), were Z s a d-dmenson varable, Z -- (x~,,y~,) r, x and Y are d-p and p dmensons respectvely and r denotes te transposton. Denote te t stratum populaton mean, medan,, T )7" and dstrbuton functon by (X~, y~)r, (m~ my and Fg(S) -- gl{6(z~l) (ZN~)} respectvely, were 5z, s te pont measure at z. Also, let (X r, Yr)r, (mrx, m~.) and FN(S) -- H ~=l WFg(s) be te mean, medan and dstrbuton functon of te target populaton respectvely, were N - E g =l N. Obvously, X = H E=l WX, Y - EH=I WY. We want to make nference about target populaton parameters suc as ]7" or my. 2.2 MELE wt X known Suppose tat Zl,''', Zn s a smple random sample wtout replacement from stratum wt dstrbuton functon Fg for all, and tat te samples are selected ndependently across te strata. Te 798

2 emprcal lkelood for te above samplng sceme can be approxmated by L- IIH=lII'~=lP (2.1) f n << N and large N, were P -- Pr(z -- z) and z -- (Z~, y~)r as te dstrbuton functon FN We consder te case of known vector of populaton means, X, of te varable x - (xl,..., Xd-p) ~" H H Y f( -- E WE(X) - E WNl E X. (2.2) =l =l =1 Clearly, te maxmum lkelood estmator sould be sougt among dstrbuton functons satsfyng (2.2). Usng te same argument as n Owen (1990), we need consder only estmators of FN wose support s contaned n te set of observatons. Te problem terefore reduces to maxmzng 2.3 Asymptotc Results We assume tat bot te sample sze n and te stratum sze N go to nfnty as a certan ndex u attaced to nv and N~ goes to nfnty for all,.e., nu and Nu go to nfnty as u ~ exp. However, for convenence, we wll suppress te ndex u n te followngwenever possble. And we wll denote te soluton by ~ snce we are gong to deal wt large sample problems. Teorem 2.1. Suppose tat as u ---. c~, N, n, N- n go to nfnty, n/n ---+ k, k > O, n/n ~ O. ~.d bot E Wg E. ~'r"' II x~ II 3,.rd E~ w~e, ~' II Y, I ~v~ an upper bound ndependent of v, and (r,z = CoV(X,X) > O" 1 ~> 0 for a11 and u, were gl s a fxed postve defnte matrx. Ten ~;~( -?)& N,(O, I,), subject to and l - E E log P (2.3) E P -- 1 ( - 1,..., H; P >_ O) (2.4) were E - ~=l H and ~ - E=I n. A unque soluton for te above problem exsts, provded tat )( s wtn te convex ull of te ponts Xll~''',Xlnl;''';XHI,''',XHnH. Te estmators of P are te solutons of te followng system of equatons P -- 1/n[1 + mcr(z - X)], (2.6) were Z -- E PX, m -- nwn 1 and P subject to 0 < P ~ 1. We denote te solutons of (2.6), (2.7) and (2.8) as ~~. Te estmator we get by usng ts metod s called te Maxmum Emprcal Lkelood Estmator (MELE). An effcent metod for gettng P s gven n an unpublsed report. We wll dscuss te propertes of P and oter related topcs n te followng sectons. Proofs are omtted. A-½ U ~ L Nd-v(0, Id-p), A-½ (~',~(s) - FN(S)) L Na(O, V), as v --. ~, were Fn(s) - ~~~l(z < S), (7" v -- E W~(n 1 -- Nl)[Var(y ) - 2BCov(y,X) + BYar(z)Br], B - ~ kw~cov(y,x). [E kwgcov(~, ~)]-~ ~,d U ~ - C~, ~, - E~ k~w:v v(~,~), C - E~ k~w:, V = C -~ E~ Wgk~F~(~)(1- F~(~))- C-'a~2a ", a = E W~kCov(l(z < s), x~). From te results of Teorem 2.1, we know/~,~(s) s asymptotcally more effcent tan te emprcal c.d.f. F,~(s). Also, Y as te same asymptotcally varance as te optonal regresson estmator ~3,t + B(X - ~,,). In above dscusson we consder auxlary nformaton of te form of (2.2), but we can easly generalze t to oter form of auxlary nformaton suc as g w~ N; ~ E ~(~,) - o -1 Te coce W(X) -- X- X gves (2.2). Wen te populaton medan m~ s known and x s a scalar, we let W(Z)- I[~,_<mx] Varance Estmaton In ts secton, we wll consder ow to estmate ~r, te varance of Y and A½V, te varance of/~,~(s). 799

3 Under te condtons of Teorem 2.1, we can see and -1 _ N 1 ~ w~ nn _ l rl =1 A½f/ - A-½[EW~IF,(s)[1- F.~(s)] n ' n are consstent estmators of cr and A½ V respectvely, were [? -[~_, nw~n; 2 E(y, - ]~)(z,- 2)r]s;: We can use ts result to get confdence ntervals for Y and F(s), wc are asymptotcally correct. Anoter consstent estmator of cr s te jackknfe varanceestmator. We omt te teorem ere. 2.5 ELRE estmator Now we turn to te emprcal lkelood rato estmator (ELRE) of f'. Here we denote E(z) = E W E(z ) -- 0, were 0 - (0;, or) r, Ok -- X, Ou - ] 7", and k, u mean "known", "unknown" respectvely. By (2.1) we know te emprcal lkelood functon s L(F) - IIH=I =l (Z) - HH=I I-[n_lP, (2.9) were p - df,n~ (z)- Pr(z - z) and F denote te populaton dstrbuton. Notng tat p > 0, P - 1 E WF,N, we know (2.9) s maxmzed by F~(s) - ~ WF,~(s), were F,~ = ~ ~ I(z < s). Te emprcal lkelood rato s ten defned as R(F)- L(F)/L(F,~) wc reduces to R(F) IIH=I n. (2.10) -- II=l np Snce we are nterested n te parameter 0~ - 12 and we ave auxlary nformaton (2.2), we defne te profle emprcal lkelood rato functon RE(O) -- sup{r(f) l p > O, EP -- 1, W ~PZ - O}. (2.11) As dscussed n secton 2.2, for a gven Ou, a unque value for te rgt-and sde of (2.11) exsts, provded tat 0 s nsde te convex ull of ponts z11,"', z1~1 ; " " " ; ZH1, " " ", ZHnH. Usng te Lagrange multpler metod, let E E log P -- E ~ (E P -- 1) - o) were t - (tl,t2,''.,td) r are Lagrange multplers. Takng dervatves wt respect to P, we ave 01 OP E Hence, -- P A -- nw trz -- O, -- 1,..., n; Ol P OP - 1,.-., H, = n -- A -- nw tr EPZ -- 0 ::~ /~ -- n -- nwt r EPZ. P -- 1/n[1 + mtr(z -- Z)], (2.12) were Z - ~ PZ, m -- nwn~ 1 and P wt te restrcton E W E P Z W 1 = E-~ l+mt'(z--z) Z. We know tat t can be determned n terms of 0, and t - t(0) s actually a contnuous dfferentable functon of 0. Te emprcal lkelood functon for 0 s now defned as L~(e) n~=, n H=l(n mtr(z - Z)' wc leads to te emprcal log-lkelood rato statstc" le(o) - E E log[1 + rn tr (Z -- Z)]. (2.13) We mnmze le(o) to obtan an estmator 0~ of te parameter Ou, 0 -- ( f[r,0~,) ~r r, called emprcal lkelood rato estmator. In addton, ts yelds es- tmators P, and an estmator for te dstrbuton functon F: F, (s) - E W E P, l(z, < S). (2.14) 800

4 - 0 - O. Now we turn to te calculaton of 0,.e., mnmzaton of IE(O). From te context above we know tat t s equvalent to maxmzng L- IIIIP (2.15) Teorem 2.3. Under te ypotess Ho and te condtons of Teorem 2.1 and Var(z ) > (71 > 0 for any and u. WE(O0 ) L_~ X2 (p) as u ---, cx:), subject to: E P - 1, P ~_ O, (2.16) E W E PX -- Ok, (2.17) E W EPY - O~ (2.18) were 0~ s also unknown. Snce 0~ s unknown, (2.18) does not add any nformaton nto te estmaton problem. Hence, L wll be maxmzed droppng (2.18). We can sow ts as follows. Let E E ]Ogp -- E "~(EP - 1) - ) - ), were tk,t~ are multplers wt te same dmenson as Ok) 0~, respectvely. Ten" Ol~ = ntu - 00~ = > tu - Ts sows tat we can drop (2.18). Terefore, P'S are te same as P'S n secton 2.2, and 0~- Y - ~ W ~ P Y. We note ere tat under te ypotess H0 " 0~ - 0~,o, te estmator for P wll be canged to P - 1 ) n[1 + ~rlr(oo)(z - Z)] ) )T. were Z - ~ PZ O0 (X ~, 0~o Te emprcal lkelood statstc for testng Ho s gven by we(oo) 21E(0o) -- 21E(0) were Z-- ~-, P Z. 2 E E l g[1 + rnff(oo)(z - Z)]- -2 E E log[1 + rn 7" (O)(Z-- 2)], were00-(0~ O r )r ) It o 3 Emprcal Lkelood and General Estmatng Equatons 3.1 Introducton Lkelood and estmatng equatons provde te most common approaces to parametrc nference. Our purpose s to combne emprcal lkelood, estmatng equatons and stratfed samplng tecnology togeter. In order to smplfy our dscusson, we assume..d, samplng n eac stratum, but te dscusson can be carred over to smple random samplng wtout replacement. Suppose tat a target populaton wt d- dmensonal caracterstc x as te unknown dstrbuton functon F and a p-dmensonal parameter 0 assocated wt F. We are nterested n makng nference on 0. Te samplng sceme s as follows. Suppose tat te target populaton s dvded nto H strata wt known wegt W's for all strata. We get an..d, sample Xl,"', Xn from X, te straturn wc as dstrbuton F, were X ~S are d-dmensonal and - 1,...,H, - 1,.-.,n and F's are unknown, and te samples across strata are also ndependent. We ave F -- E WF. (3.1) We also assume tat nformaton about 0 and F s avalable n te form of r >_ p functonally ndependent unbased estmaton functons, tat s functons gj(x,o),j- 1,...,r suc tat EF{gj(x,O)}- O. In vector form, g(x, 0) -- (g~ (x, 0),..-, g~(x, 0))', were g(z, O) satsfes Er{g(~, 0)} - ~ W~E~ {9(~, 0)) (3.2) In te followng we wll sow ow to use suc nformaton to estmate 0 and F, n conjuncton wt emprcal lkelood. 801

5 - E 3.2 MELR estmators We defne emprcal lkelood as L(F) - HH=I II~lP, (3.3) were P -- Pr(x -- X). Only tose F dstrbutons wt F's wc ave an atom of probablty on eac X ave nonzero lkelood. Notng (3.2), we know (3.3)s maxmzed by te emprcal dstrbuton functon F~(s) - ~ F,~(s), were n - E n, F,~, (S) -- A~ E l(x < s). Te emprcal lkelood rato s ten defned as R(F) - L(F)/L(F,~), and after a lttle calculaton we get R(F) - HH=I II~ 1 np. (3.4) Snce we are nterested n estmatng te parameter 0, and we know te estmatng equaton (3.2), we defne te emprcal lkelood rato functon R~(o) - =~p{r(f) I P~, >_ O, ~ P -- 1, E W E Pg(O) - 0}, (3.5) were g(o) -- g(x,o) for all and. For any gven 0, a unque value for te rgt sde of (3.5) exsts, provded 0 s nsde te convex ull of te ponts gll(0),""", gln,(o); " ";gll(o),'" ", ghnh(o). Te maxmum may be found va Lagrange multpler metod. Te estmators for [9 are solutons of followng equatons 1 P -- rg[1 + mtr(g(o) - g(0))]' (3.6) g(o) (3 7) O(O) - E n[1 + rnt'(g(o)- ~(0))] ' " }~ w~o~(o) -o, (3.s) from wc t can be determned n terms of 0, and t - t(o) s actually a contnuous dfferentable functon of 0. Terefore, te emprcal negatve loglkelood rato statstc IE(O) -- E E log[1 +,nt" (O)(g(O) -- t~ (0))]. (3.9) We may mnnaze IE(O) to obtan an estmate 0 of te parameter 0 (called MELR ). In addton, ts yelds estmates P ~S from (3.6), and an estmate for dstrbuton F as F, (s)- ~ W' ~ P l(z, < s)) (3.10) { Here are a few condtons for te followng teorems to be true. (1) As n ~ oo, n/n ~ k > 0 for all. And suppose tat n a negborood of te true value 00, E[(g(X, 00) -- Eg(X, O0))(g(X, 00) -- Zg(X, 00))'] = 0.(O0) > 0.1 > 0 for all, were O" 1 s postve defnte, II g(~, 0)II 3 s bounded by some ntegrable functon G(z) n ts negbourood, ten for gven 0. (2) Og(x, 0)/00 s contnuous n a negborood of te true value 00; tat II Og(~,o)/oo II s bounded by some ntegrable functon G(x) n ts negbourood; and tat te rank of ~ WE[Og(x, 00)/00] s p. (3) 2g(=' ) 0000 ~ s contnuous n 0 n a negbourood o~g(=,o) of te true value 00, ten f 1[ oooo, II s bounded by some ntegrable functon G(x) n ts negbourood. Teorem 3.1. Under condtons (1), (2) and (3) above, we ave x/~(0-00)--* N(0, V), x/%(7-0)---, N(0, U),,o v~(r. (~) - F(~)) -~ N(0, W), were Fn (x)- EWE P l(x < S), P -- ~T ~[1 + m~ t (g~(o)- 0~(0))] -~, ~d u, v, w ~ defned below: U - M~I(I + M12M~!lM21M~l), were -- ~ W ~-, Og~'o(O ) ~ M12, M21 -- M[2, --E ~ E(g(Oo)--g(Oo))(g(Oo)--~l(O0)) r p Mll, M M21M~ll M12, V- (M21M~ll M12) -1, ~nd W - E W:k~F~(~)[1- r~(~)]- BUB ~, ~~ B - E W~kZ[(g(z, 0o) - Zg(X, 0o)) ~ l(x < ~)]. Teorem 3.2 In te semparametrc model (3.2), for testng Ho 0 - Oo te emprcai Ikelood rato test statstc R 2{~ ~ log[1 + m~'(oo)(g~,(oo)- M~(00)] E l g[1 + rn 7" (O)(g(O)t - O(O)t]} s asymptotcally Xp 2 under Ho, assumng (1) (2) and (3). 3.3 MELR estmators and testng wt constrants In ts secton we extend te emprcal lkelood metods to deal wt te case n wc tere are 802

6 constrants on parameters. Suppose tere are q- dmensonal constrants on O" r(o) -0, (3.11) were r(o) s a q 1 (q < p) and te q p matrx R(0) - o --~ s of full rank q. To mnmze IE(O) defned by (3.9) subject to r(o) - O, we consder a: -!l~(o)+.~(o), n were u s a q x 1 vector of Lagrange multplers. Dfferentatng G2 wt respect to 0 and u, we ave 1 ale(o) n 00 ~,~R(o) - o, r(o) - o. (3.12) Consequently, to mnmze le(o) subject to r(o) - 0, we consder te soluton of Q~..(o. t. ~) - o. q2..(o, t. ~,) - o. q~..(o, t. ~,) - 0. (3.13) were Now we turn to te problem of testng H0 : r(0) = 0. Tere are tree popular metods based on lkelood: lkelood-rato test, Lagrange-multpler test and Wald test. Of course ere tey sould be based on emprcal lkelood. Tese statstcs are defned respectvely as: e~ ~T e~ ELR- 21E(Or), LA - n u r HZ 1 ur, (3.16) 6,. WA - nr T (O)H~r(O), (3.17) were 0 s a soluton of te estmatng equaton E Wn 1 E g(o) -- O, and Ho defned n (3.15), s te asymptotc covarance matrx of x/~ ur. Te followng teorem gves te asymptotc beavor and relatonsp between te test statstcs defned n (3.16) and (3.17). Teorem 3.4. Under te assumptons of Teorem 3.3, te tree test statstcs n (3.16) and (3.17) are asymptotcally equvalent, and eac of tem s asymptotcally dstrbuted as Xq 2 under Ho - Tl I "~ m r[g(o)- g(0)]' References Atcson, J., ~ Slvey, S.D. (1958). Maxmumlkelood estmaton of parameters subject to re- Q2,~(o,t,u) [E W E 1 strants. Ann. Mat. Statst., 29, n 1+ mtt[g(o)- g(0)]bckel, P.J. & Freedman, D.A. (1984). Asymptotc normalty and te bootstrap n stratfed samog~,(o~]t~ O0 + R ~ (O)u, (3.14) png. Ann. Statst. 12, Cen, J. & Qn, J. (1993). Emprcal lkelood est- Q~. (o, t..) - ~(o), maton for fnte populaton and te effectve usage of auxlary nformaton. Bometrka were O(O) defned by (3.7). We denote te soluton Cocran, W.G. (1977). Samplng Tecnques, 3rd to (3.13) as (0r, tr, ur). ed. New York: Wly. Hartley, H.O. & Rao, J.N.K. (1968). A new est- Teorem 3.3. Under condtons (1) (2) (3) and maton teory for sample surveys. Bometrka 55, (4) g(x, O) s contnuous and dfferentable n te negbourood of Oo and E II g(~, O)113< ~. And Owen, A. B. (1988). Emprcal lkelood rato conf- Cr(Oo) > 0 for all and tat E E{ Og(z' ) oe } s dence ntervals for a sngle functonal. Bometrka full rank (5) n a negbourood of 0o, (O) s a contnuous Owen, A. B. (1990). Emprcal lkelood confdence dfferentable functon; tat te q p matrx R(O) s regons. Ann. Statst of rank q; and tat Owen, A. B. (1991). Emprcal lkelood for lnear O~(O) O: g(~, O) models. Ann. Statst Qn, J. & Lawless, J.F. (1994). Emprcal lkelood 0o0o ~, 0o0o ~ and general estmatng equatons. Ann. Statst exst and are bounded by some constant and some ntegrable functon respectvely, we ave (3.15) Qn, J. L: Lawless, J.F. (1995). Estmatng equatons, emprcal lkelood and constrants on parameters. Te Canadan Journal of Statstcs