SEMI-EMPIRICAL LIKELIHOOD RATIO CONFIDENCE INTERVALS FOR THE DIFFERENCE OF TWO SAMPLE MEANS

Transcription

1 Ann. Inst. Statst. Math. Vol. 46, No. 1, (1994) SEMI-EMPIRICAL LIKELIHOOD RATIO CONFIDENCE INTERVALS FOR THE DIFFERENCE OF TWO SAMPLE MEANS JING QIN Departent of Statstcs and Actuaral Scence, Unversty of Waterloo, Waterloo, Ontaro, Canada N2L 3G1 (Receved February 18, 1993; revsed June 2, 1993) Abstract. We all know that we can use the lkelhood rato statstc to test hypotheses and construct confdence ntervals n full paraetrc odels. Recently, Owen (1988, Boetrka, 75, ; 1990, Ann. Statst., 18, ) has ntroduced the eprcal lkelhood ethod n nonparaetrc odels. In ths paper, we cobne these two lkelhoods together and use the lkelhood rato to construct confdence ntervals n a separaetrc proble, n whch one odel s paraetrc, and the other s nonparaetrc. A verson of Wlks's theore s developed. Key words and phrases: Eprcal lkelhood, hypotheses tests, se-eprcal lkelhood, Wlks's theore. I. Introducton A proble arsng n any dfferent contexts s the coparson of two treatents or of one treatent wth a control stuaton n whch no treatent s appled. If the observatons consst of the nuber of successes n a sequence of trals for each treatent, for exaple the nuber of cures of a certan dsease, the proble becoes that of testng the equalty of two bnoal probabltes. In soe cases, however, we don't know or perhaps only partally know the underlyng dstrbuton, but we stll want to copare the two treatents. Consder N = n + ndependent easureents n two saples. The frst saple conssts of n easureents xl, x2,..., Xn recorded under one set of condtons, and the second saple conssts of easureents yl, y2,.., y,~ recorded under a dfferent set of condtons. For nstance, the x's ght be blood pressure ncreases for n subjects who receved drug A, whle the y's are ncreases for dfferent subjects who receved drug B. The proble s to copare the two populaton eans,.e. test #A = #B, or gve a confdence nterval for the dfference of the two eans A = PA -- #B. Suppose that based on our experence, we are qute sure of y's dstrbutonal for up to one paraeter, say Go(y) (aybe the drug B has been used a long te), but for new drug A, t s hard to say x's dstrbutonal for, so we have no knowledge about x's dstrbuton F(x). How can we test 117

2 - I )} 118 JING QIN Ho : #A = #B, or gve a confdence nterval for A = #A -- #B? In other words, we need to test the equalty of populaton eans based on one paraetrc odel and one nonparaetrc odel. We all know that we can use the lkelhood rato statstc to test hypotheses and construct confdence ntervals n full paraetrc odels. Recently, Owen (1988, 1990) has ntroduced the eprcal lkelhood ethod n nonparaetrc odels. In ths paper, we wll cobne these two lkelhoods together and develop a lkelhood rato test and confdence ntervals for ths separaetrc two saple proble. In Secton 2 we brefly descrbe the eprcal lkelhood developed by Owen (1988, 1990). In Secton 3, we gve our an results. Secton 4 gves soe proofs. Secton 5 presents soe lted sulaton results. 2. Eprcal lkelhood The eprcal lkelhood ethod for constructng confdence regons was ntroduced by Owen (1988, 1990). It s a nonparaetrc ethod of nference. It has saplng propertes slar to the bootstrap, but where the bootstrap uses resaplng, t aounts to coputng the profle lkelhood of a general ultnoal dstrbuton whch has ts atos at data ponts. Propertes of eprcal lkelhood are descrbed by Owen (1990) and others. Consder a rando saple Xl,X2,...,x~ of sze n drawn fro an unknown r-varate dstrbuton F0 havng ean #0 and nonsngular covarance atrx E0. Denote the j-th coponent as x J (j = 1, r), so that x = (x~, xr~ ~- Let..,..., ~2 L be the eprcal lkelhood functon for the ean. For a specfc vector # = (#1,...,/),, L(p) s defned to be the axu value of 1-IP~ over all vectors P = (Pl,-..,P~) that satsfy the constrants (2.1) Ep =l, xp = #, P >- O, = l,...,n. L =1 =1 An explct expresson for L(#) can be derved by a Lagrange ultpler arguent. The axu of I-[n=l P subject to (2.1) s attaned when (2.2) P z p(~t) = 7%--1{1 -~- ~'r (X - -1 where t = t(#) s an r-densonal colun vector gven by n (2.3) Z{ = 0. =1 n Snce l-i=~ P attans ts largest value over all vectors p = (Pl,...,Pn) satsfyng }-~-=1P n z 1 when p = Tt -1 ( = 1,..., n), t follows that the eprcal lkelhood functon L(#) s axzed at /2 = 2 = n -1 }-~=1 x and L(/2) = n -~. The eprcal lkelhood rato at the pont # s _ -[{ 1 + _ (2.4) L(/2) =1 Tt

3 SEMI-EMPIRICAL LIKELIHOOD CONFIDENCE INTERVALS 119 and nus twce the logarth of ths rato s (2.5) = 2 log{1 + <(x - =1 Under approprate regularty condtons, Owen (1988, 1990) has proved that a verson of Wlks's theore holds,.e. under H0 : # = #0, W(#0) ~ X~. There s an obvous extenson of ths to construct confdence nterval for the dfference A of two saple eans. Let Xl, x2,.., x~ be ndependently and dentcally dstrbuted rando varables wth dstrbuton functon F(x) and Yl,Y2,..-,Y,~ be ndependently and dentcally dstrbuted rando varables wth dstrbuton functon G(y), where both F(x) and G(y) are unknown. We defne EL(A) as the axu value of Iq[~l P 1-Ij=l qj subject to constrants 7t P >_0, qj >_0, E P= I' E qj = 1 =1 j=l and px -- qjyj = A. =1 j=l The eprcal lkelhood rato statstc for A s (2.6) zw(a) = -2log {ZL( X)/2xZL(A) } Easly we can show that under H0 : A = Ao, the true dfference, EW(A0) --4 X~)- We do not gve the detals here. 3. Se-eprcal lkelhood and an results It s well known that lkelhood based confdence ntervals and tests perfor well n paraetrc odels. Owen's eprcal lkelhood rato confdence nterval can be used n nonparaetrc odels. In ths secton, we consder a seeprcal lkelhood based confdence ntervals for the dfference of two eans. Let xl,x2,..., x~; Yl,Y2,..., Y,~ be ndependent and suppose the x~ are dentcally dstrbuted as unknown F(x) wth ean #1 = f xdf(x) and the yj are dentcally dstrbuted as G0(y) wth ean #2 = f ydgo(y) = #(0), where G0(y) s of known for dependng on paraeter 0. We assue that Go(y) has densty functon go(y). The proble s to test H0 : #1 = #2 = #(0), or gve a confdence nterval for Ao = #1 -- #(0). The se-eprcal lkelhood functon s n H dr(x ) H go(y ) j=l

4 120 JING QIN It has axu value n -~ Ej=I go(yj), where 0 s the MLE based on the second saple. Let f~(f, O) ~- E%l de(x) 1-[j=l go(yj) ~-'~ [Ij<196(vj) Cr,n= { f xaf- ~(O) l F << F,~,r, O) >_,'}, n(a)=sup{r(f,o) l f xdf-,~(o)= Lr << r,~}, F,O where F << Fn denotes that F s absolutely contnuous w.r.t F~,.e. the support of F s contaned n the support of eprcal dstrbuton F~. Then A E C~,,~ f and only f 7~(A) > r. We want to show that -2 log ~(A0) ~ X~I). Wthout loss of generalty, we assue that A0 = 0, and T4(O) : sup E(np) E go(yj) p,,,o =1 /=1 j=l 9~(Yj) -1 We frst axze the jont lkelhood wth restrcton,1 =,2,.e. I n (3.1) ax Z logp + Eloggo(Yj) pl,...,pn,o =1 j=l Let then H=~-~logp-F~logge(Yj)-H~/(1-~p)--nA(l~(O)-~px), =1 j=] OH -1 1 Op -- P - "7 - nlx{ = O,=~ P nlx OH o: : or Also, P ~-- 1 1,~ + A(x{ -.(o)) OH00 -- ~ Ologgo(yj)00 + na#'(o) = O, J.e. (3.2) ~(e) = - ~_~.j Ologge(yj) 012(0) 00 00,~.,(e).,(0)

5 SEMI-EMPIRICAL LIKELIHOOD CONFIDENCE INTERVALS 121 7% where, 12 = (l/n)}-~j=l loggo(yj). By the sde condton E:lP(X- #(0)) = O, we have, (3.3) 1 ~ x - #(0) :1 1 + l~---p(0)) = 0. We wll prove that there exsts a root 0 of ths equaton such that the root les wthn an Op(n -1/2) neghborhood of the true value 00 when n s large enough. In the followng, we wll ake assuptons on the dstrbuton Ge(y) whch concde wth the condtons of noralty of the MLE n full paraetrc odels, gven n Lehann (1983). ASSUMPTIONS. () The paraeter space f~ s an open nterval. () The dstrbutons Go(y) of yj have coon support, so that the set A = {y : go(y) > 0} s ndependent of 0. () For every y E A, the densty go (Y) s dfferentable three tes wth respect to 0. (v) The ntegral f 9o (y)dy can be twce dfferentated under the ntegral sgn. (v) The Fsher nforaton I(O) = E[Ologgo/O0] 2 satsfes 0 < I(O) < oo. (v) I(oa/oO3)loggo(y)l < M(y), for all y E A, 00 - c < 0 < 0o + c, wth E0o [M(y)I < oo THEOREM 3.1. If F(x) s a nondegenerate dstrbuton functon wth f ]x[sdf < oo, >(0) s contnuously dfferentable at 00 wth #'(0o) # O, 9o satsfes the above assuptons () through (v), and?%/ ---* 7 > 0 as n, ---* oo, then the log se-eprcal lkelhood rato statstc under the null hypothess Ao = 0, log~(o) : ~ log~p{(~)+ ~ log[a(yj)/a(y~)] =1 j=l satsfes -2 log 7~(0) --+ ~1) and l7%--.oo P(Ao E Or,n) = P(X~I) ~ -2 log r). 4. Proofs Frst we gve a lea. LEMMA 4.1. Under the condtons of Theore 3.1, there exsts a root 0 of (3.3), such that 0-0o = 0p(7%-1/2). PROOF. Let x - ~(0) h(o) =! Z 1 + ATK( 7,(o)) n?% 1 + (x~ -.(0)) 2 V(o)7~ :;(o))

6 122 JING QIN Frst we prove that h(o) = 0 has a root n an Op(?), -q) neghborhood of 0o, where 1/3 < q < 1/2. In fact, note that ol2(0) o12(0o) o212(0o) oo - O~ + (o - Oo) 0o ~(0 - )2 312( *)o03 where 0* les between 0o and 0. Snce 012(0o)/00 = Op(n-1/2), we have A(0) = Op(n -q) when 0 E (0o - n -q, Oo + n-q). By the assupton Elzl 3 < ~, we have axl<<n [x[ < n 1/3 for all but fntely any n, whch ples A(0)(x - #(0)) = Op(1) n the nterval (Oo-n -q, Oo+n-q). Note that h(o) s alost surely contnuous n ths nterval for n large enough, and consder the sgns of nqh(oo + n -q) and nqh(oo - n -q) for large n. h(oo + n -q) = 2 - #(0o + n -q) (x~ - ~(Oo + n-q)) 2 - (0o + n-q) n 1 Z 1 + A(Oo +n-q~-- ~o 2k n-q))" Note also that #(Oo + n-q) = #(Oo) + ~'(Oo)~ -q + o(~-q), "~(00 ~-?)'--q) ~- -- 0/2(0000 -~ n-q) / ]~t(00 -- n-q) ro /0o/ 1/ L ~ + oo2 - + ~(n-q) ~'(0o+s-q) = Op(n--1/2) _L [[2/#t(Oo)]T--q -~- Op(n q), (x -- p(o0 + n-q)) 2 = S 20p(1), where n Z. 1 + A(00-~ ~ P~o-+ n-q)) + _1 ~-~(z - #(0o)) 2 : S 2, /2 : E ( n / 0212(0 )) 002 > O, so that Slarly nqh(o0 + n-q) = nqop(n -1/2) - #'(00) - I2S2/j(O0) + 0~(1) = -(#'2(00) + I2S2)(#'(00)) -1 + Op(1). nqh(oo - n -q) = (#'2(0o) -~-/2S2)(#/(0o)) -I ~- op(1),.e. nqh(oo + n -q) and nqh(oo - n -q) have opposte sgn for large n. By the nteredate value theore, there exsts a root 0 n (0o - n-q, Oo + n-q). Slar to the above arguent, we have 0 = h(o) = Op(n -1/2) ) -}- Op(O -- 00),.e. 0 - Oo = Op(n-1/2). []

7 SEMI-EMPIRICAL LIKELIHOOD CONFIDENCE INTERVALS 123 PROOF OF THEOREM 3.1. Taylor expanson, we have 77~ Note that/2(0) = (/n) }-~-j=l loggo(yj). Usng a Fro (3.2) /2(0) -- /2(0) (0)-'~ \_(0-- O) Jr- 2--1(0-- 0)- 202/2(0)~ -~-Op(n--1), #(0) --.(0) = [.t' (O)(O -- O) -~- 0p(~--1/2). (4.1) o12(~) (~ _ 0) + op(~-l). A(g)(~(~) -,(0)) - oo Expandng 012(0)/00 at 0 = 0, and notng 012(0)/00 = 0, we have ol~(g) of 2(0) 2z2(0)(4 0)+op(~-l/~), 00 - O0 + ~ O0 -.e. O0 (4.2) g - ~ o212(0) (4.3) [o~12(0)] -'~-OP(n--1/2)=--A(O)#'(O) L 002 J 020.(4) -.(0) =/(o)(o - O) + o.(,~ -~/2) op(n 1/2), k j + op(n-1/2). Fro (3.3) we have 1 ~ -,(0) o= - E 1- n = 1 E[ 1 A(O)(X,(~))](Z.(4)) ~- Op(n--I/2),.e. (4.4) ~-.(0)= a(0) 1 Z(x~-.(0)) 2 + op(~-1/2). Fro (4.3), (4.4) we have 1 #t2 -.(0) = A(0) y~'(x -.(4)) 2 (4) o212(0) 002 -~- Op(n-1/2), or

8 124 JING QIN -1 (4.5) :,(o) = (~ -,(0)) I¼ Z(x~ -.(0)) 2 02z2(0) "'2( ) 002 -}- Op(n--1/2). So the eprcal log lkelhood rato statstc s T~ logt~(0) = ~lognp + ~-~log[go(yj)/gg(yj) ] =1 j=l Snce log(1 x) = x - (1/2)x 2 o(x2), = - E log[1 - A(0)(#(0) - x)] + n[12(o) - 12(0)]. - ~ log[1- A(O)(/~(O)- x~)] = -na(0)(:~- #(0))+ ~ A2(0) E(x- #(0))2 + %(1). / Expandng 12(0) at 0 and notng (4.1) and (4.2), we have ~[l~(~) - l~(0)] = -~ (~ - ~) + ~(0 - ~)~ oo~ + ~(~-~) = -~(0)(~(~) - ~(0)) - ~2(0) Tt~/2 (0) 0212(0) - - +op(1). \ 002 Hence by (4.3)-(4.5) ]og7~(o) = -~a(0)(~-u(0)) = - ~(~ - ~(O)) 2 ~(x~ - ~(0)) 2 - ~'2(0)/ ~O ~ -1 +o;(1). Under Ho : #1 = #2 = #(0o),,/Z(:~ - ~(0)) =,/~(~ - ~(Oo)) + ~(~(Oo) - ~(~)) + x(o, ~ +.~) where, ~1 z = var(x), and o-22 = #,2 (00)[- 1/E(0212 (00)/002)], hence -2 log 7~(0)---+ X~I)" r-

9 SEMI-EMPIRICAL LIKELIHOOD CONFIDENCE INTERVALS 125 COROLLARY 4.1. Under the condtons of Theore 3.1, let?% j 002 and let 7 be any real constant. probablty, and Then -21og7 (~ - #(0) T~n -1/2) --+ T 2 n -2 log 7~(/zx - p(00) + 7-0?% -1/2) ~ )~1)(T2) PROOF. By a nor odfcaton of Theore 3.1, we note that -- 2log~-~(X -- ~t(0) -~- TO-n -1/2) = n{x -- ~(()) -- IX -- TO-~%--1/2]}2(Y -2 ~- op(1) = Op(1) and - 21og~(#1 - #(00) + T0-n -1/2) = n{:~ -- ~t(0) -- [~1 -- ~t(00) -I- T0-n--1/2]} ~- Op(1), ~v/~{ :~ -- /1,1 -- (~t(0) -- ~(00) ) -- T0-n -1/2} ~ N(--T0-, 02), ~2 ~ 0"2 n prob. [] 5. Sulaton results In ths secton, we gve soe lted sulaton results. We copared three ethods of obtanng confdence ntervals for the dfference of two saple eans. The frst one s based on the eprcal lkelhood rato statstc (ELR) n (2.6) wthout a dstrbuton for assupton on F(x) and G(y). The second one s based on the se-eprcal lkelhood rato statstc (SLR) wth paraetrc assupton only on G(y). The thrd ethod s based on the paraetrc lkelhood rato statstc (PLR) wth paraetrc assupton on both F(x) and G(y). We generated data by usng the S language. Fro each saple, 90% and 95% eprcal, se-eprcal and paraetrc lkelhood rato confdence ntervals were coputed. In Tables 1 and 2, we reported the estated true coverage, ean length and ean value of dpont of those three lkelhood confdence ntervals. Each value n those tables was the average of 1000 sulatons. We consdered the paraetrc odels wth dstrbuton F(x) fro ux~l) and G(y) fro log N(#, 1) and F(x) fro exp(01) and G(y) fro exp(02) n Tables 1 and 2 respectvely. Fro those tables we can see that the perforance of the se-eprcal lkelhood rato statstc les between eprcal and paraetrc lkelhood rato statstcs. All eprcal coverage levels are close to the nonal levels when the saple sze s oderately large.

10 126 JING QIN Table 1. x ~,, px~i), g ; y ~ logn(#, 1), # = 0, A0 = n==10 n----=20 n==40 90% CI 95% CI Cov. Av.length Av.dpt. Coy. Av.length Av.dpt. ELR SLR PLR ELR SLR PLR ELR SLR PLR Table 2. x ~ exp(01), 01 = 1; y ~ exp(02), , A 0 = 0.5. n==10 n==20 n==40 90%CI 95%CI Coy. Av.length Av.dpt. Cov. Av.length Av.dpt. ELR SLR PLR ELR , SLR PLR ELR SLR PLR Acknowledgeents The author wshes to thank Professors J. F. Lawless, A. B. Owen and the referee for any useful suggestons. REFERENCES Lehann, E. L. (1983). Theory of Pont Estaton, New York, Wley. Owen, A. B. (1988). Eprcal lkelhood rato confdence ntervals for a sngle functonal, Boetrka, 75, Owen, A. B. (1990). Eprcal lkelhood confdence regons, Ann. Statst., 18,