On the Efficiency of Certain Nonparametric Tests
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(2) The Journal of National Chengchi University Vol. 58, 1988. where {( = (Xl, X 2 ), Y = (Y I, Y 2 ) and.!' = (VI ,IJ 2 ) is the common median. We would like to detect differences in variability or dispersion for the two populations. Two nonparametric tests Rand R* are suggested in Liu (1982). If the common median l: = (VI' V2) is known, we define Rm,n to be the Mann Whitney (1947) test statistic for the two independent random samples. m. Dij. = 1 if Ui > Vj for all i = 1, 2, ... ro, = 0 otherwise. j. = 1,2, ... n,. Ui = [(Xu-Vdl + (Xli -Vl)2] Jh. for i = 1,2, ... , ro, and. Vj =[(Ylj-vd2+(Y2j-V2)l]Jh. forj=1,2, ... ,n.. If the common median E = (v I, v 2 ) is unknown, we define R;l; ,n to be the Mann-Whitney test statistic for the two samples, * U2N, * ... , U*mN and V*IN, V*2N,···, VnN * UIN, m. i.e.,. where. > VtN. =0. viN = [(Xli-MIN)2. statis. norm nonp. two i b (P2 b. n. R~ n = .~ ~ Dij, ., i= 1 j= 1. Dj1 = 1 if viN. for t. is G, of id nativ. n ~. DiJ·, i.e., Rm,n = ~ i= 1 j= 1. where. 2.. /.l=(. for all i = 1, 2, ... , ro, j. = 1, 2, ... , n,. + (X2i-M2N)2] Jh,. VjN= [(Ylj-MIN)2 + (Y2j- M2N)2]0, N=m+n, and ~N = (MIN, M2N)is the combined sample median.. In this paper, we would like to seek appropriate par.ametric tests for bivariate. normal and bivariate uniform distributions and investigate the asymptotic relative. efficiencies (ARE) of the nonparametric tests Rand R* with respect to the. parametric competitors.. - 114. paran The 1 H an( 111,112. distri.
(3) On the Efficiency of Certain Nonparametric Tests. 2.. The Test Statistic F~ ,n Under Normal Theory. In the univariate case, the general distribution model of the scale problem for two independent random samples. is Gy _ J.J. (t) = Fx _ J.J. (9t) where J.J. is the common location. The null hypothesis of identical distribution then is H: 9 = 1 against either one- or two-sided alter natives. Under normal theory, the parametric test for the scale problem is the m. statistic Fm ,n. =. _. l; (Xi - X)2/(m - 1) i= I n. l; (Yj ~ y)2/(n - 1) j= I. We are now interested in seeking an appropriate parametric test for a bivariate normal two-sample scale-model so that later we can compare the efficiency of the nonparametric tests with it. Let us consider the bivariate normal two-sample scale-model as follows: Suppose (Xll,X21),···, (X Im , X2m) and (Y l l , Y2d,· .. ,(YIn, Y2n) are two independent random samples from N2 ((PI, J.J.2), (p a a PIa)) and N2 «J.J.I, J.J.2), ( b b P b2 b))" respectivelY,L where PI and P2 are the kno~n c3rrelation coefficients, P2 ~ ~ = (J.J.l, J.J.2) and!? = ('111' '112) are unknown means, a and b are unknown scale parameters. Then n = [(a,b,J.J.l ,J.l2 ,'111,'112) I 0 < a,b < 00, _00 < J.lI ,J.l2,'111 ,'112 < 00]. The hypothesis H': a = b, ~ and !l unspecified, is to be tested against A': a =f= b, ~ and!l unspecified. Then w = [(a,b,J.ll,J.l2,'111,'112)1 0 < a = b < 00, _00 <J.ll,/-l2, ''It ,'112 < 00]. We are going to derive the likelihood ratio test and study its distribution. The likelihood functions are. .~.. .'.. 'I ,:1. ,. ~I:!I. '. ~. ,. ..',.'. ,. ,. ':. ..• .•. '.. .... .~. ,': '.~ ';1. e. - 115 -.
(4) ThP, Journal of National Chengchi University Vol. 58, 1988. The. 3 Ifand. L(w). then. .~. 1 1=1. «xli-Pl)2 - 2Pl (xli-Pl)(x2i-P2) + (x 2i - P2)2). exp { - -2-a [ - - - - - - - - - - - - - - - (l-p/). 310gL(U) If - - . oa. 310gL(U). 310gL(U). 3b. OP l. --:-:--- ,. ,. 310gL(U) OP2. ,. 310gL(U) <ml. and. ologL(U). 317 2. are equated to. maxiJ. Hence. - 116.
(5) On the Efficiency of Centrain Nonparametric Tests. 1. •. . ' .. exp(-m-n). 2 (21rf1+n(anf1<bnf(1-P12f1/ (1-P2 2f/2. If. alogL(w). o1OgL(w). •. aa. then. •. alog(w ). aJ.Ll. il 1W. alog(w). •. in/ 1. aJ.L2. in/ 2. are equated to zero.. m. -2 -2 ..E [(Xli-Xl) - 2Pl (Xli-Xl XX2i-X2) + (X 2i -X2) ]. 1. -. W. alog(w ). = Xl' il 2w = X 2 • ~lW = Y l' ~2W = Y2 and. ~. a. •. 1=1. ~m~. {-------------------------1 -PI. 2. ,. ;j. ,. 1. n -2 -2 .~ [(Ylj-Y l ) -2plYlj-YI)(Y2j-Y2)+(Y2j-Y2)]. + J-l. ~. .1. } I-P2. q. 2. n. ed to. ~. maximize L(w). The maximum is. ri. ~. 1. L(w) =. il. ,:,. exp(-m-n). (21rf1+n(aw)m+n(1-P12f1/2(1-P22f/2 (anf1(i'n f. L(w). Hence. ij. A=-~-=----. L(n). (awyn+n. m. -2. -. -. m. -2. .E [(Xli-Xl) - 2P l (Xli-X1)(X2i-X2) + (X2i-X2) ] { rl. }. 2in(1-p/) = m. 1. -. 2. -. -. -2. ..E «Xli-Xl) - 2PI (Xli-Xl )(X 2i-X2) + (X2i-X 2) ). 1=1. {,.,,--,--\ [. 2. I-PI. - 117. ~ .~. "1,1..,.
(6) The Journal of National Chengchi University Vol. 58, 1988. =. Thus, the the likelihl p. and 11 un. - In- th. en. = (0,(. of freedom. Lemma 2.] according the X2 dist Proof: See. Lemma 2.2 according N. ';1. 1. c~{3. -. Yi"'" Y. -. Proof: See. Theorem 2. [(. (m+n)m. =. mmnn. m-l. )F*. n = (0,00), '. ]m. n-l· m,n -m---;I;--.----, where +1] m+n [( )F n-l m,n. - 118. •. (1) Fm,nil. (2) E(F*m,.
(7) On the Efficiency of Centrain Nonparametric Tests. ( F*mn)m. (m+n)m+n(m_l)m(n_l)n. ,. =. [(m-l)P* + (n-l)]m+n m,n. mnnn. Thus, the critical region A EO;; AO is equivalent to F~ ,n EO;; c I or F~ ,n ~ c 2. Finally, the likelihood ratio test for testing H': a = b, 11 and 1'1 unspecified, against A': a =#= b,. - In- this. 11 and 1'1 unspecified, can be based on. n. l+n. F~ n'. -. ,. . nonna1-theory model, we can show that for every 0. = (a/b)I/2. the distribution of F~ ,n/02 is F with 2(m-1) and 2(n-1) degrees of freedom. We fIrst need the following lemmas. I. en. = (0,00),. Lemma 2.1: If a p--dimensional random vector. ~. = (Xl' .... , Xp) is distributed. according to Np (Q, 1.;) (nonsingular), then ~ - l~' is distributed according to the X2 distribution with p degrees of freedom. Proof: See {Anderson (1958), p. 54} .. Lemma 2.2: Suppose :>Sl' . . . , according to Np. ~a' 1.;).. Let C. N. ~N. are independent, where. = (c0cf3). ~a. is distributed. be an orthogonal matrix. Then Ya =. t1 ccx?--f3 is distributed according to Np (~a'. 1.;), where. N. Ea = ;'1 cOt(3~~ and. YI" .. , YN are independent.. -. -. Proof: See {Anderson (1958), p. 52 }.. Theorem 2.1: In the previous nonna1-theory model, for every 0. = (a/b)I/2. e. n = (0,00), we obtain (1) F~ n/02 has an F distribution with 2(m-l) and 2(n-1) degrees of freedom,. '* _ (n-1) 2 (2) E(Fm,n) - (n-2) 0 ,and. (3) Var(F* ) = (n-l)2(m+n-3) 0 4 . m,n (m-l )(n-2)2 (n-3) Proof: (1) Fix 0 = (a/b)112. en = (0,00).. - 119.
(8) The Journal of National Chengchi University Vol. 58, 1988. =----------------------------------. d. /[(I-P1 2)(m-l)J. In. /[(I-P 2 2)(n-l)J. Co. /[a(I-P 1 2 )2(m-l)]. l[b(I-P 22)2(n-l)J. m. = (J. _. :E (?Si-~l:x 2 i=l -. -1. _,. (~c?9 /2(m-l). n 1 :E (Y--Y)l:y - (Yj-Y)'/2(n-l) j=1 -J _ -. where. ~i = (X li,X 2i), ~. = (X l ,X2),. !j. r. Thu. = (Y lj'Y 2j) ,. =. (~\'Y2)' b. ~. = (P2b. P2b b)'. m 1 2 We may proceed to prove that i~l (&-~):E~ - C~i-~)' has a X distribution. with 2(m-l) degrees of freedom. Since (Xu ,X2l ),. ..• ,. sample from a bivariate normal population with mean matrix :Ex_. = (a pa. ~. (X lm ,X 2m ) is a random. = (PI ,1l2) and. covari~n~. pa), a there exists an m X m orthogonal matrix C .=(ck1.) with the .. - 120.
(9) Onthe Efficiency of Centrain N()nparametric Tests. m. last row (l/y'Iii, ... , Ifv'Iii). Let Zk =.1; ~i~i' Then. by Lemma 2.2, ~ is. -. 1=1. m. . distributed according to Np (~k' 1;~), where ~k =i~1 ~ie, and ~1'. .•.. ,~ are. independent. m. m. In particular ~m = 1; cmi~i = 1; (1/ v'ffi)~ = i=1 i=1 -. m. 1; ~k1;x k=1 . Consider. m. ~k'. m. m. . -1. = 1; (1; cki~i)1;x k=1 i=l m. ...;mX.. -1 m. .,. (1; Ck'~j) . j=1 J. m. = 1; 1; (1; ckick' )~~x i=1 j=1 k=1 J . -1. ~. ,. m m. = 1; 1; 6 ..X,~x -IX.', where 6 .. = 1, ifi=j i=1 j=1 1J-1- _ -~ 1J. 6..1J = 0 ' ifi±; '-J. and m -1' = ,1; ~~x ~. 1=1. Thus,. m. .Ii. . _. i~1 (~-~)~~. -1. _,. (~i-~). m. = 1; X,~x i=1 _1 _. -1' -1-' ~ -m~x X -. k~1 ~k~x. m. -1'. m.-l. -1'. = 1; ~k~X. ,istribution. -. k=1. _. ~k - ~m~x. . s a random covariance. i) with the. - 121 . ~k'. . -1'. ~m.
(10) The Journal of National Chengchi University Vol. 58, 1988. m. We note E(~) = .~ = ~ ckill. i=l. constants:. . Sl-w and'. of the sam Also let {r such that. = 0, for k=Rn. 1 Zk~ - 1 Zk' = ~ (Y.-X)l:x - 1 (~-X)' has a X2 distribution x '-1 ~_'1 - k -l 1n with 2(m-l) degrees of freedom. Similarly, we can show that '~1 (Y,J._Y)~y-l. By Lemma 2.1,. mi. ,. J-. -. . -. with the t must be th. efficiency (. (¥j-I)' has a X2 distribution with 2(n-l) degrees of freedom. By the fact that. (X U ,X21 ), . . . , (X 1m ,X2m ) and (Y u ,Y 21)' •.. , (Y In'Y 2n) are two independent 1 n 1 m _ random samples, ;~l· (~-~)~x- QS.i-~)' and '~l (¥j-Y)l:y - (¥j-¥)' are. ~. -. ~. -. independent. Therefore, P~ ,nl82 has art P distribution with2(m-l) and 2(n-l) degrees of freedom. (2) and (3) are immediate consequences of (1) and the result in [Cramer (1946), p.242J ~. We ar. Theorem 3, i.e., (analo~. 1. dE(Tn ), 2. There e. This completes the proof. 3. Evaluating the ARE(R,F*) for Bivariate Normal Pistributions / -. The purpose of this section is to obtain the asymptotic relative efficiency of the nonparametric test R (or R*) suggested in Liu (1982) with respect to the parametric test p* for the normal-theory model. Let us first define the asymptotic relative efficiency of test T with respect, . .* ' to test T .. Suppose we have two test statistics Tn and T~ for a hypothesis testina. Droblem, H: 8 E W against A: 8 E Sl-w. Let [8 0 ,8 1 ,8 2 , . . . ] be a sequence tit. - 122. There ~ d>O, v.
(11) On the Efficiency of Centrain Nonpararnetric Tests. constants such that 8 0 specifies a value in wand the remaining 8 1 ,8 2' Sl-w and that. lim. n-+. oo. 8n. . . .. are in. Let {<l>n} and {<l>~} be two sequences of tests all. = 80 ,. of the same size ex, which are based on the test statistics Tn and T~, respectively. Also let {ni} and {n;} be two monotonic increasing sequences of positive integers such that . lim Pcp (8 i) =,lim Pcp**(8i ),. 1-+ 00 ni 1-+ 00 ~. ution -1. '!. with the two limits existing. n~ual. to 0 or 1 (the limiting power of<l>nr at 8 i. must be the same as the limiting power of <l>~~ at 8i ). Then the asymptotic relative 1. efficiency of test T with respect to test T* is defined to be. t that. ndent. )' are. :n-l). 946),. :yof I the. n'. ARE(T, T*) = .lim 1 -+ 00. n~' ~,. if this limit exists.. We are going to apply the following theorem in evaluating the ARE(R,F*).. Theorem 3.1: If T and T* are two tests satisfying the four regular conditions, i.e., (analogous ones for T~) 1. dE(T n )/d8 exists and is nonzero for 8. =80 , and is continuous at 8 0 ,. 2. There exists a positive constant c such that lim dE(Tn)/d8 18 = 80 . n-+ oo. V nVar(TJI8.- 80. - c,. 3. There exists a sequence of alternatives [8 n ] such that for some constant d >0, we have. ipect d. 8n =8 0. +..;n. ~ting. :e of. - 123. <1 :-~. ,. ..IJ ).
(12) The Journal QfNational Chengchi University Vol. 58, 1988. lim. n-+- oo. IdE (Tn) I d8) 18 = 8n. for the aItemati N01 conditio]. =1. [dE(Tn) / d8] 18 = 80. lim y'Var(Tn) 18 = 8~ n -+00. r. 4. n ~00. y'Var(Tn) 18 - 80. ['fn-E(Tn)] 1 8 = 8 n P( .J Var(Tn) I 8- 8 n ~ z I 8. = 1,. E. = 8 n) = 4>(z), where <Il(z) is the stan E~. normal d. f., then the ARE ofT with respect to T* is. Proof: See {Fraser (1957), p, 273t.. Suppose (Xu ,X 21 ), . . . , (X 1m ,X2m ) and (Y11 ,Y21)' . . . , (YIn'Y 2n) are. where S(u and the a. two iIldependentrandom samples from bivariate normal populations withp.d.f.'s. Let Ul of Xl and:. and. respectively, where p is the known common correlation coefficient,. ~. = VLl,1'2). is the known (or unknown) common mean, a and b are unknown scale parameters. Set 8 = (alb)1/2. Thus, we have two test statistics Rm,n (or R~,n) and F:"n. -124.
(13) On the iEtflCiericy of Centain Nonpararnetric Tests. •for the hypothesis testing problem H': () = I against either one- or two-sided alternatives. Now, it is easy to check that the tests F* and R satisfy the four regular conditions stated in Theorem 3.1, and for every () = (a/b)1/2 € n = (0. 00), we have • _ (n-l) 2 • _ (n-I)2(m+n-3) 4 E(Fm,n) - - - 8 , Var(Fmn) 8 , (n-2) , (m-lXn-2)2(n-3). nE(Rm,n) = mN. [.io [>"NS(u) + (1-~)S(8u)]dS(u)] -m(m+1)/ 2,. Var(~ n) = 2m 2N(1->"N) { ,. II. S(8u} [1-S(8v)] dS(u)dS(v). o<u<v<~. (I->"N). II S(4)[1-S(v)} dS(Ou)dS(Ov)} , >"N o<u<v<". +. re. i. q. where S(u) is the d.f. of U, and U = [(X I -P l )2 + (X 2 - P2)2] 1/2, by Theorem 2.1 and the Chernoff-Savage theorem (see {Chernoff and Savage (1958). n.. .'s In particular,. ;.. [dE~ ,n)/d8] 18=1. 1_. ~. :::: [mn,ious(8u)s(u)du] 18=1' where s(u) is the p.d.f. ofU, = mn,iou[s(u)) 2du, and [Var(~,n)] I 8=1 . tJ. ~.'. == mn(m+n+1)/12.. Let us derive the p.d. f. of U = [(Xl -P I )2 + (X 2 -1l2 )2 ] 1/2 if the joint p. d. f. of Xl and X 2 is. 1. t). f(xl'x 2 ) = -~r:--:;=~ exp [21ray I_p2. [(xl-Ill. -,. ~.'. l~. P - 2p(x l -Ill XX 2-1l2) + (X 2 -1l2)2] ], 2a(1-p2) ~. s.. _oo<xl'x 2 <oo. n - 125.
(14) The Journal of National Chengchi University Vol. 58, 1988. If we set x I -Ill = ucos t and x 2 -1l2 = usin. t~. then the joint I\. rJ, f of U and. Tis 1 u 2(I-psin 2t) g(u,t) = - -__-exp [] , 2rr~ 2a(1-p2). OE;;;u<oo,. o E;;t < 2rr.. Thus the p.d.f. ofU is. s(u) =. 1 2 --a./---J o rr exp 2rr. I_p2. [-. u 2(I-psin 2t). ] dt, 0 <; u < 00.. 2a(I_p2). In or Hence. Lemma 3.. Proof: Se1 where we set r = u 2. Thu. where we set q. = r. a. The. Zl. =. the (19'. - 126.
(15) on the Efficiency ofCentain Nonpatametric Tests ,nd. 1 f2fT (2fT rOO q(2- . So'O-p') 0 10 loq oxp [_ pon 2, i --psin 2,,) 2(1-p2) ] dqdt 2dt 1. =. 4(1 2)2 r2fT .r2fT· -p dt dt Jo Jb 2 1 8fT2(1_p2) (2-psin 2t 1 -psin 2t2)2. _ (l_p2) r2fT r2fT . , 'Jo Jo. 2fT 2. dt dt. 1. (2-psin 2t 1-psin 2t 2). 2. 2. 1. In order to simplify this integral, we need the following lemma.. Lemma 3.1: For a and b real, a> I b I , we have r 2fT Jo. dO. _. (a+bsin 0)2. . 2fTa (a 2_b 2)3/2.. Proof: Setting z = exp(i8), we see that dz == iexp(i8)d8 = izd8. u2. 2fT. Thus,. dO. 10. _. (a+bsin. oi. - Ie. dz/iz. , '. [a+b(z-l/z) / (2i)} 2. since sin 8 = [exp(i8)-exp(-i8)] / (2i) = (z-l/z) / (2i), and where C is the unit circle,. r. :-. a. zdz. 4i. =-Ie - - - - - b2. [z2+(2ai/b)z-l} 2. The function f(z). •. has poles of order 2 at. z [z2+(2ai/b)z-1] 2. zl =. (-a+J a 2 -b 2 )i b ' and z2. the unit circle and z2 outside. (1975), p. 254}) is. =. (-a-./ a 2 -b 2 ) i . . .. b wIth zl bemg mSIde. The residue of f(z) at zl (see {Silverman. ",. - 127-. <. i. I;. \.
(16) The .Journal of National Chel.lgchi Univ"rsity Vol. 5~, .1988. d2~1 2 •.. d·. z lim - , - - [(z-zl) f(z)] = lim - . HZ,-Zt)2 . ] Z ~ Z1 dz2-1 Z ~ Z dz '2 2 1 (Z-Zl) (Z-Z2). Usb. - ---=---..,..,- There By the Residue Theorem (see {Silverman (1975), p. 253 } ) , zdz. fcf(z)dz. =. fc----- [z2+(2ai/b)z-l] 2. = 271'i·----. =--- Theore Hence f~7I'. dO (a+bsinO)2. =-. 4i. b2. zdz. fc-----. be tW( p.d.f.'s. [z2+(2ai/b)z-l] 2. =_._--- =---- and g(Yl. This completes the proof.. -128.
(17) On the Efficiency of Centain NonparametricTests. Using L,emma 3.1, we ,obtain. '. ., (l_p2) . 2 2 . 1 .r; u[s(u)]. ,2du = fo 1r fo 1 r .. = (l_p2) 2. 21r. Therefore,. [dE(~. dt 2dt 1. (2-psin 2t 1 -psin 2t2)2. 21r2 21r fo. 21r(2-psin 2t 1) [(2-psin2t 1 )2_p 2]3/2. dt . 1. ,n) / dO] 10 = 1. = mnf;u [s(u)] 2du _ (l_p2) - mn •. 2'IT. 2. 2t 1) 1.2'IT _ _2'IT(2-psin _ _ _-=-__ c. [(2-psint)2 _p2] 3/2. dt 1. _ mn(1-p2) 2'IT (2-psin t) fo dt. 'IT [(2-psin 2t )2_p2] 3/2. .,4i p". ~i'.J. ,. We now summarize the results in the following theorem. Theorem 3.2: Let (Xu ,X 21 ), . . . , (X 1m ,X2m ) and (Y l1'Y21)' . . . , (Y In'Y2n) be two independent random samples from bivariate normal populations with p.d.f.'s. i~,j1 .· 'j. I. 1 {(x 1-#-Ili-2p(x 1 -#-Il)(x 2 -#-I2)+(x 2-#-I2)2] f(x 1,x 2) =--~....",==exp {},. 2'ITa. I_p2. 2a(l_p2). - 00< xl' x 2 <00,. 1. and g(y l,y2) = -2-'lT~-~1---p2. exp {_. [(y1-#-1 1)2 -2p(y 1-#-Il)(Y2-#-I2)+(Y2-#-12)2] }. 2b(1-p2) -00<Yl'Y2<00,. - 129. ,.
(18) The Journal of National Chengchi University Vol. 58', 1988. respectively, where p is the known common correlation coefficient,. ~. = (J.tl ,Ill). is the common mean (either known or unknown), a and b are unknown scale parameters. Set () = (a/b)1/2. For the hypothesis testing problem H': () against either one- or two-sided alternatives, we have '". '". 3(I_p2)2 ARE(R,F"') ;:; ARE(R ,F ) :: 2 [ 7r. i;7r·· . -. [(2-psin ti-p2] 3/2. = ARE(R *,F*) = 3/4.. Particularly, if p = 0 then ARE(R,F*) . '" Proof: Smce e(F mn , ) ::. (2 psint). ([dE(F~,n) I dO]. ,n. dt] 2 .. have tl tions a respect. w~. '". [Var(F m,n)] I 0 =1. nonpara underlYl Fir a disc wi. 4(m-l)(n-3) m+n-3. fi. ([dE(~,n) IdOl I, 0,,:, ,1)2 e(~. ApplYl. 4. Eva. [2(n-l) / (n-2)] 2. and. byTh. 10;:; 1)2. ::------------------------ (n-l)2(m+n-3) I [(m-l)(n-2)2(n-3)] =. =1. ) = ------ [Var(R)] &'IIl,n I 8 = 1 mn(1-p2). (2-psin t). 27r. [fo 7r. dt]. 2. . then EO variables Con Sup. [(2-psin t)2_p2] 3/2. =---------------------------- mn(m+n+l) /12. f( g(. then ARE(R,F '" ). lim. m,n~oo. e(~ ,n). '". e(Fm,n). - 130.
(19) On the Efficiency of Centain Nonparametric Tests. =. 3(1_p2)2 11'2. [J.. 211'. (2-psin t). d ]2 t. [(2-psiI! t)2 _p2] 3/2. 0. =. Applying the technique used in Liu (1981), we can prove that Rm ,n and R~ ,n have the same limiting distribution under both H' and A' if the underlying popula tions are bivariate normal. Hence the tests Rand R * have the same ARE with respect to the test F*, which concludes the proof. .. '\. 4. Evaluating the ARE(R,F**) for Bivariate Uniform Distributions We are now interested in evaluating the asymptotic relative efficiency of the nonparametric test R (or R *) with respect to an appropriate parametric test if the underlying populations have bivariate uniform distributions. First we note that if a random vector (Xl ,X2 ) is uniformly distributed over a disc with p. d. f. f(x l ,X 2 ) = 1/(1I'c 2 ), 0";; (xCJ.LI)2. + (x 2-J.L2)2 ..;; c 2 ,. then E(X I ) = IJ I , E(X 2 ) = 1J2' Var(X I ) = Var(X 2 ) = c2 /4, p = 0, and the random variables Xl and X2 are dependent. Consider the bivariate uniform two-sample scale-model as follows: Suppose (Xu ,X21 ),· .. , (X lffi ,X 2m ) and (Y U ,Y 21 ), .. ·, (Y ln ,Y 2n ) are two independent random samples from bivariate uniform populations with p.d. f. 's. and. f(X I ,X 2) = 1/(1I'C I 2 ),. 0";; (xI-J.LI) 2 + (x 2-J.L2 )2 ..;; c i 2'. g(YI,y2) = 1/(1I'c/),. 0";; (YCJ.L 1)2 + (Y2-J.L2)2..;;c/,. where!!; = (1J 1'1J 2) is the known common mean, c1 and c2 are unknown scale parameters. Set (} =c I /c 2 ' Then the problem is to test the hypothesis H': (} = I against either one- or two-sided alternatives. Since E«X I -IJ 1)2 + (X 2 -1J2)2) = c I 2 /2, and E«Y I -1J 1 )2 + (Y 2 -1J2)2) = c/ /2, we would like to compare the - 131 .
(20) The Journal of National Chengchi University Vol. 58, 1988. non parametric test statistic Rm ,n with the test statistic. samples f. variance a. (1). For. i=I, ... , m andj=I, ... ,n, let. X~. (2). = [(Xli-JlI)2+(X2i-Jl2)21/cI2,. Yj = [(YICJlI)2 + (Y2j-Jl2)21/c2 2. Then X'I' ... , X~ and Y'l' ... , the same population and m. ** Fm,n. 2. Y~. which are. ~. Proof:. (. are two independent random samples from. 2. .~ [(Xli-JlI) + (X2CJl2) ] 1=1. 1m. T. =--------~---. .~. [(Y Ij-JlI)2 + (y 2j-Jl2)2] / n. J=1. A. c 2. I =----._------------ c 2. 2. By the fact that 2V' n :. 0011, the Central Limit Theorem, and the result in:. Cramer (1946), p. 254 }, it is easy to see that F~:n, appropriately nonned, is asymptotically normal for every (} = c i /c 2 € n = (0,00). . , The following lemma will give us the approximate values of E(F;n) andl. ** n) to order m- I and n I . Var(F m ,. '. .. Lemma 4.1: Let Xl' . . . , Xm and Y I'. . . . ,. - 132. .:s. Yn be two independent rando..:.
(21) On the Efficiency of Centain Nonparametric Tests. samples from the same population which has finite moments with mean Il and variance a 2 • Then mown.' (1) . X. a2 = 1 +--and. E(~). Y. TIJl2. x .. a2 1 1 (2) Yare-=-) = - ( - +-) , Y 112 m n. which are approximate values to order m- 1 and n- 1 .. x. les from Proof:. Consider -=-=. Y X. Thus,. (~)2 Y. X. X. (Y -1l)2. (Y-Il). (Y-Il)3 + ... ]. = -[1 - - - + -'-- 1l+(Y-Il) Il Il 11 2 -2 . = -X[ 1 11 2. 2(Y -Il) 3{Y _1l)2. ---:----.:.... + - - 11 2. Il. Accurdingly, E( _X ) _ -E{X) -[1 .. Y. )3 4(Y-1l +... ] 11 3. E{Y -Il). Il. 11 3. +. E[(), -1l)2]. E[{Y -1l)3]. +--. 11 2. Il. 11 3. + ... ] a2 m3 1 [ 1 - 0 + - - - - - + ... ] ,where m3 is the nll 2. n 21l 3. third central moment,. a2 = 1 +--, and nll 2. esult in. med, is . ,n) and. X. Yare-=-) Y. X. = E[(-=-)2]. X. Y. _ [E(-=-)] 2 Y. 3(2. 2E(Y-Il). 11 2. Il. = E(-)[ 1 -. andom. - 133. 3E[(Y-1l)2]. +--- 11 2. 4E[(V-Il)3] 11 3.
(22) The Journal of National Chengchi University Vol. 58,1988. (a 2 /m+Jl 2 ). 3a 2. 4m3. Jl2. DIl 2. n 2Jl3. un. - - - - [ 1 - 0 + - - - - + ... ] 2a 2. 2m3. a4. DIl 2. n 2 Jl3. n 2Jl4. H'. AF. - ( 1 + - - - - - + - - + .... ) a2. 1. Prc. 1. =-(-+-), Jl2 m n. when both m and n are so large that terms in m and n of order less than -1 are regarded as negligible. This completes the proof. Next let us derive the p.d.f. of U. = [(Xl -Jl 1 )2 + (X2 -Jl2)2] 1/2. if the joint. p. d. f. of Xl and X2 is. If we set. Xl -. J.l 1. = ucos. t and x 2. - Jl. =usin t, then the joint p. d. f.. of U and. Tis g(u,t) = U/(7TC 2 ),. o~ u ~ c. and 0 ~ t < 27T •. Thus the p.d.f. ofU is. We are ready to evaluate the ARE(R,F**) for bivariate uniform distributions.. By us. Theorem 4.1: Let (X ll ,X 21 ), . . . , (X 1m ,X 2m ) and (Y 11' Y 21)' . . . , (Y In' Y2n ) be two independent random samples from bivariate uniform populations with p.d. f.'s. =E([I. - 134. = Val.
(23) On the Efficiency of CentainNonparametric Tests. and g(y l,y2). =. respectively, where If. =. (/J.l ,/J.2). unknown scale parameters. H':. o ~ (yl-~li + (y2-~2)2 ~c22. 1/(1Tc/),. Set. is the known common mean, c l abd c 2 are c 1 fc2.. (J =. For the hypothesis testing problem. 0=1 against either one- or two-sided alternatives, we have ARE(R,F**) =. ARE(R *,F**) = 1.. Proof:. Since. [dE(Rm,n)/d9] 19 = 1. = mn f;' u[s(u)] 2du. = mn f~l u(2u/c ?)2du. = mn, and. [Var(~ ,n)]. I9 = 1. = mn(m+n+l)/12, it implies that. e(~,n). =. ([dE(~ ,n)/d9] 19 = 1)2. Var(~,n) 19 =1. (mn)2 =. mn(m+n+l)/12. = 12mn/(m+n+l).. By using the fact that E([(Xl-~li + (X2-~2)2] / c l 2). = E([(Y C~1)2 + (Y 2-~2)2] / c/) = 1/2, Var([(Xl-~1)2 + (X2-~2)2] / c l 2). = Var([(Yl-~l)2+(Y2-~2)2]. /c/) = 1/12, and Lemma 4.1,weobtain. - 135.
(24) The Journal of National Chengchi University Vol. 58, 1988. App have tiom reSpt .. = 0 2 (1 +. 1/12. .). Refel. n(1/2)2 = 0 2 (1. Ander. 1. + 3n ), and. 1. Chern(. ** [Var(F n)] m. ,. .~. [(X U -1-I 1)2 + (X 2i -1-I2)2] / (c 1 2m). 1=1. O=l=Var ( - - - - - - - - - - - - - ). .~. J=l. [(Y 1J·-1-I1)2 + (Y 2i-1-I2)2] / (c/n) . 1 1 1 =-(-+-) . 3 m n 1 .. 4(1+_)2 3n. ** ** ([dE(Fm,n)/dO] 10=1) 2 e(Fm n) = - - - - - - , ** [Var(F m n)] 10=1. 1 1 1 -(-+-) 3 m n. ,. 1 = 12rnn(1 + _)2 / (m+n). 3n. Therefore ARE(R,F ** ) =. e(Rm n) , lim - - m,n ~ 00 e(F ~* n). ,. =. lim. mn~oo. ,. (. Uu, M J. Silverm. 1/12 1 1 ---(-+-) (1/2)2 m n. Hence. Cramel Fraser, 1 Uu, M.. 12rnn/(m+n+ 1) 1 2 12rnn(1 + - ) / (m+n) 3n. = 1.. - 136.
(25) On the Efficiency of Centain Nonparametric Tests. Applying the technique used in Liu (1981), we can prove that Rm ,n and R~ ,n have the same limiting distribution under both H' and A' if the underlying popula , tions are bivaria te uniform. Ii. Hence the tests Rand R * have the same ARE with. respect to the test p**, which concludes the proof.. References Anderson, T. W., An Introduction to Multivariate Statistical Analysis, John Wiley and Sons, Inc., New York, 1958. Chernoff, H. and Savage, I. R, Asymptotic Normality and Efficiency of Certain Nonparametric Test Statistics, Ann. Math. Stat., 29(1958),972·994. Cramer, H., Mathematical Methods of Statistics, Princeton University Press, Princeton, 1946. Fraser, D. AS., Nonparametric Methods in Statistics, John Wiley and Sons, Inc., New York, 1957. tiu, M. R, On the Asymptotic Normality of Certain Nonparametric Test Statistics, J. National Chengchi University, 43(1981), 65· 78. tiu, M. R, On the Large Sample Properties of Centain Nonparametric Tests for Dispersion, 1. National Cheng chi University, 45(1982), 25-42. Silverman, H., Complex Variables, Houghton Mifflin Company, Boston, 1975.. - 137.
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