Comparison of LR, Score, and Wald Tests in a Non-IID Setting

Transkript

1 oural of multivariate aalysis 60, (1997) article o. MV Compariso of LR, Score, ad Wald Tests i a No-IID Settig C. Radhakrisha Rao The Pesylvaia State Uiversity ad Rahul Mukeree Idia Istitute of Maagemet, Calcultta, Idia Cosiderig a large class of tests, we study higher order power i a possibly oiid set-up. Optimum properties for the likelihood ratio ad score tests are exhibited uder the criteria of secod-order local maximiity ad third-order local average power, respectively. The issue of strigecy with regard to third-order average power has bee addressed. We also compare the power properties of various Bartlett-type adustmets for the tests Academic Press 1. INTRODUCTION Higher order compariso of tests uder cotiguous alteratives has received a cosiderable attetio over the last two decades; see Ghosh (1991) ad Mukeree (1993) for reviews. Most of the results i this area, icludig the differetial geometric oes for the curved expoetial family (Amari (1985), Ch. 6; Kumo ad Amari (1983)), relate to the case of idepedetly ad idetically distributed (iid) observatios. I particular, i the iid case ad uder the absece of uisace parameters, optimality results are ow kow for (a) the likelihood ratio (LR) test i terms of secod-order local maximiity ad (b) the score test i terms of third-order local average power (Mukeree, 1994). Uder a possibly o-iid set-up ivolvig a ukow scalar parameter, Taiguchi (1991) cosidered the problem of third-order compariso of tests. He worked with a large class of tests that icludes the LR, Rao's score ad Received April 26, 1995; revised May 23, AMS subect classificatio: 62F05, 62E20. Key words ad phrases: average power, Bartlett-type adustmet, cotiguous alteratives, likelihood ratio test, local maximiity, Rao's score test, strigecy, Wald's test X Copyright 1997 by Academic Press All rights of reproductio i ay form reserved.

2 100 RAO AND MUKERJEE Wald's tests (as described i Rao (1973, pp ), suggested a Bartletttype adustmet for the tests i the class ad the, o the basis of such adusted versios, explored the poit-by-poit maximizatio of third-order power. Cosequetly, his optimal test is depedet o the alterative hypothesis ad his results are ot comparable with the oes metioed i (a) ad (b) above. We attempt to bridge this gap ad show that the results i (a) ad (b) cotiue to hold i Taiguchi's (1991) possibly o-iid settig. I additio, we discuss the derivatio of a most striget test with respect to thirdorder average power. These results have bee represeted i Sectio 3. The ecessary prelimiaries are give i Sectio 2 where a simplified ad compact expressio for the third-order power fuctio has bee derived. This expressio, more iformative tha what has bee kow so far eve i the iid case, eables us to study the third-order average power ot oly locally but also i its etirety ad hece to address the issue of strigecy as idicated above. Fially, we ote the availability of two more Bartlett-type adustmets for each test statistic i Taiguchi's (1991) class, i additio to the oe proposed by Taiguchi himself, ad i Sectio 4 compare such adusted versios uder the criteria of maximiity ad average power. This substatially stregthes some of the earlier results reported i Rao ad Mukeree (1995) who compared various Bartlett-type adustmets for the score statistic. As i Taiguchi (1991), cosideratio of the possibly o-iid settig allows us to illustrate the results with models which are importat i time series aalysis. It may be oted that our class of tests is slightly larger tha that i Taiguchi (1991). This is eeded to cover ot oly the tests cosidered by Taiguchi (1991) but also the various available Bartlett-type adustmets thereof. Earlier, Amari (1985, pp ) cosidered the issue of strigecy with respect to third-order power. However, ulike him, we cosider a possibly o-iid set-up, make o assumptio regardig curved expoetiality ad compare the tests i their origial forms without ay modificatio for local ubiasedess. Because of the last reaso, our fidigs differ from those of Amari (1985); see e.g., Sectio 3 below. We refer to Madasky (1989) ad Mukeree (1993, 1994) for more discussio o the motivatio for compariso of tests i their origial forms. 2. PRELIMINARY COMPUTATIONS Let X () =(X 1,..., X )$, 1, be a collectio of possibly vector-valued radom variables with desity f (x () ; %), where the parameter % belogs to a ope subset of R 1. Cosider the ull hypothesis H 0 : %=% 0 agaist the alterative %{% 0. We make the assumptios stated i Taiguchi (1991). I

3 LR, SCORE, AND WALD TESTS 101 particular, it is assumed that for a appropriate sequece [c ], satisfyig c as, the cumulats, up to the fourth order, of Z i (%)=c &1 _ d i d% log f (X () ; %)&E %{ di i d% log f (X () ; %) (i=1,2,3) =& i possess asymptotic expasio of the form cum % [Z i (%), Z (%)]=k (1) i (%)+c &2 k (2) i (%)+o(c &2 ), cum % [Z i (%), Z (%), Z l (%)]=c &1 k il (%)+o(c &2 ), cum % [Z i (%), Z (%), Z l (%), Z m (%)]=c &2 k ilm (%)+o(c &2 ), where k (1) i (%), k (2) i (%), k il (%), k ilm (%) do ot ivolve. Let I=k (1) 11 (%), J=k(1) 12 0), K=k 111 (% 0 ), L=k (1) 13 0), M=k (1) 22 0), M =M&I &1 J 2, Z i =Z i (% 0 ) (i=1,2,3), W 1 =I &12 Z 1, W 2 =Z 2 &JI &1 Z 1, W 3 =Z 3 &LI &1 Z 1. We shall cosider alteratives of the form % =% 0 +c &1 =. Let F be a class of test statistics for H 0 such that every statistic T i F admits a expasio of the form T=W 2 1 +c&1 (a 1 W 2 W 1 2+a 2 W 3 ) 1 +c &2 (b 1 W 2 +b 1 2W 2 W 2 +b 1 2 3W 4 +b 1 4W 3 W 1 2 +b 5 W 3W 1 3+b 6 W 6) 1 +o(c &2 ), (2.1) over a set with P % -probability 1+o(c &2 ) (uiformly o compact subsets of =), where a 1, a 2 ad b 1,..., b 6 are costats free from. Taiguchi (1991) studied a subclass F 0 of F cosistig of those members of F for which b 6 equals zero. Cosideratio of the wider class F eables us to compare the various Bartlett-type adustmets available for the members of F 0. As oted by Taiguchi (1991), the subclass F 0 is a very atural oe ad icludes, i particular, the LR, score, Wald's ad modified Wald's statistics, the correspodig expressios for a 1 ad a 2 beig give respectively by a 1 (LR)=I &1, a 2 (LR)=& 1 3 I&32 K, (2.2a) a 1 (score)=0, a 2 (score)=0, (2.2b) a 1 (Wald)=2I &1, a 2 (Wald)=I &32 J, (2.2c) a 1 (mod Wald)=2I &1, a 2 (mod Wald)=&I &32 (J+K). (2.2d)

4 102 RAO AND MUKERJEE Correspodig to ay statistic T as i (2.1) we shall cosider a critical regio T>z 2 +c &2 d T, where z 2 is the upper :-poit of a cetral chi-square variate with 1 degree of freedom (d.f.) ad d T is a costat, free from, to be so determied that the test has size :+o(c &2 ). For *0 ad positive itegral &, let h &, * ( } ) ad G &, * ( } ) deote respectively the probability desity fuctio ad the cumulative distributio fuctio of a possibly o-cetral chi-square variate with & d.f. ad o-cetrality parameter *. The, aalogously to Theorem 1 i Taiguchi (1991) but with suitable modificatio so as to take care of the extra term b 6 W 6 1 i (2.1), it ca be show that 3 P % (T>z 2 +c &2 d T )=1&G 1, $ (z 2 )&c &1 : where $=I= 2, &c &2 { =0 6 h 1, $(z 2 ) d T + : =0 B (T) G 1+2, $ (z 2 ) A (T) G 1+2, $ (z 2 ) =+o(c&2 ), (2.3) B (T) 0 =& 1 6 (3J+K) =3, B (T) 1 = 1 2 J=3 & 1 2 I &1 (K+3I 32 a 2 ) = B (T) 2 =& 1 2 I32 a 2 = I&1 (K+3I 32 a 2 ) =, (2.4) B (T) 3 = 1 6 (K+3I32 a 2 ) = 3, A (T) 0 0, A (T) 1 1, A (T) 2 2 & 15 2 b 6, A (T) b 6(1&3$), A (T) b 6(3$&$ 2 ), (2.5) A (T) b 6 (15$ 2 &$ 3 ), A (T) b 6 $ 3, the C (T) (0 6) beig as i Taiguchi (1991, p. 228). The quatities C (T) (0 6) are somewhat ivolved ad we require a simplified expressio for 6 =0 A(T) G 1+2, $ (z 2 ) i order to derive our results. To that effect, for positive itegral &, we write h &, $ =h &, $ (z 2 ), G &, $ =G &, $ (z 2 ), ad ote that G &, $ &G &+2, $ =2h &+2, $, $h &+4, $ +&h &+2, $ =z 2 h &, $. The by (2.5) ad the expressios i Taiguchi (1991, p. 228), the coefficiet of, say, b 3 i 6 =0 A(T) G 1+2, $ (z 2 ) equals

5 LR, SCORE, AND WALD TESTS 103 &[ 3 2 (G 3, $&G 5, $ )+3$(G 5, $ &G 7, $ )+ 1 2 $2 (G 7, $ &G 9, $ )] =&[3h 5, $ +6$h 7, $ +$ 2 h 9, $ ] =&[$($h 9, $ +5h 7, $ )+($h 7, $ +3h 5, $ )] =&z 2 ($h 5, $ +h 3, $ )=&z 4 h 1, $. Similar calculatios evetually yield 6 : =0 A (T) G 1+2, $ (z 2 )=9 0 (=, z 2 )&9 1T (z 2 ) h 1, $ &R T ($, z 2 ), (2.6) where 9 0 (=, z 2 ) is free from a 1, a 2, b 1,..., b 6 ad hece is the same for all statistics i the family F, 9 1T (z 2 ) is a costat which is free from = but depedet o T, ad R T ($, z 2 )= 1 2 M $z2 (I &1 h 1, $ a 1 & 1 2 h 3, $a 2 1 ) I &32 $z 2 [z 2 Jh 1, $ & 1 3 $(K+3J) h 3, $] a 2 & 1 4 z4 $h 3, $ a 2 2. (2.7) For positive itegral &, let h & =h &,0 (z 2 ). Now, takig ==0 i (2.3) ad ivokig the size coditio up to o(c &2 ), by (2.4), (2.6), ad (2.7), d T h (0, z 2 )&9 1T (z 2 ) h 1 =0. Solvig for d T, from (2.3) ad (2.6), the third order power fuctio of the test based o T is give by B T (=, z 2 )=P % (T>z 2 +c &2 d T ) 3 =1&G 1, $ (z 2 )&c &1 : =0 B (T) G 1+2, $ (z 2 ) +c &2 [R T ($, z 2 )&9(=, z 2 )]+o(c &2 ), (2.8) where 9(=, z 2 ) is the same for all statistics i F. By (2.5) ad (2.7), the third order power fuctio (2.8) depeds o T oly through a 1 ad a RESULTS ON HIGHER ORDER POWER First cosider the secod order power fuctio which is give by the first three terms i (2.8). By (2.4), the coefficiet of = i each B (T) (0 3) is zero if ad oly if a 2 =& 1 3 I&32 K=a 2 (LR) (vide (2.2a)). Hece exactly as i Mukeree (1992, 1994), uder the criterio of secod-order local maximiity, the LR test is superior to the test give by ay other statistic T i

6 104 RAO AND MUKERJEE F for which a 2 {& 1 3 I&32 K. O the other had, if a 2 =& 1 3 I&32 K for some statistic T the, by (2.2a), (2.4), the correspodig test will have the same secod order power fuctio as the LR test. Turig to the criterio of average power, let B T($, z 2 )= 1 2 [B T(=, z 2 )+B T (&=, z 2 )], where $=I= 2, ad by (2.4), (2.7), (2.8), observe that B T($, z 2 )=1&G 1, $ (z 2 )+c &2 [R T ($, z 2 )&9 ($, z 2 )]+o(c &2 ), (3.1) 9 ($, z 2 ) beig the same for all statistics i F. Thus, uder this criterio, all statistics i F are equivalet up to the secod order ad a third-order compariso, o the basis of R T ($, z 2 ), is warrated. By (2.7), where R T ($, z 2 )=$V T (z 2 ) h 1 +O($ 2 ), (3.2) V T (z 2 )= 1 2M z 2 (I &1 a 1 & 1 2z 2 a 2 1)+ 1 2z 4 (I &32 Ja 2 & 1 2z 2 a 2 2). (3.3) I view of (3.1) ad (3.2), up to the third order of approximatio ad for small $, the behaviour of the average power fuctio B T($, z 2 ) depeds o T oly through V T (z 2 ), i.e., V T (z 2 ) ca be iterpreted as a measure of third-order local average power associated with T. By (2.2b) ad (3.3), V T (z 2 )=0 for the score statistic. Now, as oted i Taiguchi (1991), the quatity M ca be iterpreted i terms of Efro's (1975) statistical curvature ad is o-egative. If M =0 the, by (3.3), for ay statistic T with a 2 {0, V T (z 2 ) will be egative for large z 2 (i.e., for small :) provided a 2 is free from z 2 ad, therefore, the test associated with T will be iferior to the score test, with regard to third-order local average power, for small test size. If M >0 the, by (3.3), the same coclusio will hold for ay statistic T with (a 1, a 2 ){(0, 0) ad a 1, a 2 free from z 2. I particular, the by (2.2) for small : the score test will be superior to the LR, Wald's ad modified Wald's tests i terms of third-order local average power. Note that ulike Amari (1985) or Taiguchi (1991), we are comparig the tests i their origial forms ad ot via their bias-corrected or Bartletttype adusted versios. Cosequetly, uder the preset approach ad up to the third order of compariso, oe ca discrimiate amog the members of F eve uder models with M =0. This may be cotrasted with the fidigs i Amari (1985) or Taiguchi (1991). I the last two paragraphs we have oted some desirable properties of the LR ad score tests which were earlier kow to hold i the iid case (Mukeree, 1994). The expressio for R T ($, z 2 ), as i (2.7), is i fact more iformative tha the correspodig

7 LR, SCORE, AND WALD TESTS 105 expressio kow so far i the iid settig whe the tests are compared i their origial formssee e.g., Mukeree (1994) who obtaied oly a expressio aalogous to (3.2). This eables us to study the third-order average power fuctio (3.1) from aother stadpoit. Specifically, we brig i the criterio of strigecy with respect to thirdorder average power ad, i view of (3.1), for each T i F cosider the fuctio Q T ($, z 2 )=[sup R T ($, z 2 )]&R T ($, z 2 ), T where, for fixed $ ad z 2, the supremum is over the class F. By (2.7), Q T ($, z 2 )= 1 4 M $z2 [I &1 (h 1, $ h 3, $ )&a 1 ] 2 h 3, $ $z4 [I &32 [J(h 1, $ h 3, $ )& 1 3 $z&2 (K+3J)]&a 2 ] 2 h 3, $. (3.4) A most striget test i F, with respect to third order average power, is oe which miimizes the supremum of Q T ($, z 2 ), with respect to $>0, over F. The aalytical derivatio of such a test is difficult but, as illustrated below, for a give model, usig (3.4) ad well-kow closedform expressio for h 1, $ ad h 3, $, a umerical solutio ca be obtaied. Oe ca as well employ (2.2) ad (3.4) to study the performace of stadard tests uder this criterio. Iterestigly, the first term i (3.4) agrees with the correspodig expressio i Amari (1985, pp ) while the secod term is attributable to cosideratio of tests i their origial forms as doe here. Because of this secod term, the ``uiversal'' ature of the most striget solutio that was observed i Amari's (1985) set-up holds o more i the preset cotext. Example 3.1. Let X () represet a stretch of a Gaussia autoregressive process of order 1 with mea zero ad spectral desity 8 % (+)=_ 2 [2?(1&2% cos ++% 2 )], + #[&?,?], where _ 2 (>0) is kow ad %( % <1) is a ukow parameter. With C = 12, from Propositio 1 i Taiguchi (1986) here I=(1&% 2 0 )&1, K=&3J=6% 0 (1&% 2 0 )&2, M =2(1&% 2 0 )&2. (3.5) (i) By (2.2) ad (3.5), the LR ad Wald tests have idetical secodorder power ad, if % 0 {0, they are superior to the score ad modified Wald tests i terms of secod-order local maximiity. O the other had, if % 0 =0 the all the four tests are equivalet up to the secod order of compariso.

8 106 RAO AND MUKERJEE (ii) Cosider ext the criterio of third-order local average power. Let V LR (z 2 ), V Wald (z 2 ) ad V m Wald (z 2 ) deote the expressios for V T (z 2 ) whe T represets the LR, Wald ad modified Wald statistics respectively. Usig (2.2) ad (3.5) i (3.3), it ca be see that i this example, each of these three quatities is egative wheever z 2 >2. Thus for :=0.10, 0.05 or 0.01, the score is better tha the LR, Wald ad modified Wald tests with regard to third-order local average power. (iii) We ext illustrate the issue of strigecy takig :=0.05 ad % 0 =0. The by (3.5), I=1, K=J=0, M =2 so that by (2.2), a 1 (LR)=1, a 1 (Score)=0, a 1 (Wald)=a 1 (mod Wald)=2, (3.6) ad a 2 equals zero for each of these statistics. Also, by (3.4), Q T ($, z 2 )= 1 2 $z2 [(h 1, $ h 3, $ )&a 1 ] 2 h 3, $ $z4 a 2 2 h 3, $ (3.7) where z 2 = Hece, i order to fid a most striget test with respect to third-order average power, it is eough to restrict to statistics for which a 2 =0. Numerical methods show that for such statistics the optimal choice of a 1 so as to miimize sup Q T ($, z 2 ) over $>0 is a 1 =1.24. Table 3.1 summarizes some more related calculatios. I this table, $* represets the value of $ which maximizes Q T ($, z 2 ) for give T. The etries i colum (4) of Table 3.1 are proportioal to the correspodig etries i Table 6.1 of Amari (1985). I view of the discussio ust precedig this example, this is expected for % 0 =0 sice the we are effectively igorig the secod term i the expressio for Q T ($, z 2 ) ad the fidigs ca be quite differet for % 0 {0. Icidetally, if :=0.05, % 0 =0, the by (2.7), (3.5) ad (3.6), oe ca check that the score test i superior, with regard to third-order average power, to the LR test for 0<$0.86 ad to Wald or modified Wald tests for 0<$3.83. Cosequetly, Table 3.1 (see colum uder $*) is ot i coflict with what has bee stated uder (ii) above. TABLE 3.1 Values of sup Q T ($, z 2 ) over $>0 uder the AR(1) Model (:=0.05, % 0 =0) T a 1 a 2 sup Q T ($, z 2 ) $* LR Score Waldmodified Wald Most striget $>0

9 LR, SCORE, AND WALD TESTS 107 Example 3.2. Let X () represet a stretch of a Gaussia movig average process of order 1 with mea zero ad spectral desity 8* % (+)=[_ 2 (2?)](1&2% cos ++% 2 ), + #[&?,?], where _ 2 (>0) is kow ad %( % <1) is a ukow parameter. With c = 12, by Propositio 1 i Taiguchi (1986) here I=(1&% 2 0 )&1, J=& 2 3 K=4% 0(1&% 2 0 )&2, M =(6&2% 2 0 )(1&%2 0 )&3. (3.8) (i) By (2.2) ad (3.8), the LR ad modified Wald tests have idetical secod-order power ad, if % 0 {0, they are superior to the score ad Wald tests i terms of secod-order local maximiity. O the other had, if % 0 =0 the these four tests are equivalet up to the secod order of compariso. (ii) By (2.2), (3.4), ad (3.8), it ca be see that V LR (z 2 ), V Wald (z 2 ) ad V m Wald (z 2 ) are all egative for :=0.05 or Thus for these values of :, the superiority of the score test, with regard to third-order local average power, follows. (iii) If :=0.05 ad % 0 =0 the by (3.4) ad (3.8), Q T ($, z 2 )= 3 2 $z2 [(h 1, $ h 3, $ )&a 1 ] 2 h 3, $ $z4 a 2 2 h 3, $. Comparig the above with (3.7), as before a statistic with a 1 =1.24, a 2 =0 gives a most striget test with respect to third-order average power. Furthermore, Table 3.1 cotiues to remai valid provided each etry i its last but oe colum is multiplied by COMPARISON OF BARTLETT-TYPE ADJUSTMENTS As idicated earlier, Taiguchi (1991) cosidered a subclass F 0 of F cosistig of those statistics T for which b 6 equals zero (vide (2.1)). For each member of F 0 he suggested a Bartlett-type adustmet. At about the same time, Chadra ad Mukeree (1991) ad Cordeiro ad Ferrari (1991) proposed Bartlett-type adustmets which are applicable to each statistic i F 0. For ay statistic T i F 0 the Bartlett-type adusted versios due to Chadra ad Mukeree (1991), Cordeiro ad Ferrari (1991) ad Taiguchi (1991) will be deoted by T 1, T 2 ad T 3 respectively. While the ull distributio, up to o(c &2 ), of each of T 1, T 2 ad T 3 is cetral chi-square with 1 d.f., their power properties may ot be idetical up to that order. Therefore, give T, a comparative power study will help i choosig oe of the adusted versios depedig o the cotext. Rao ad Mukeree (1995) co-

10 108 RAO AND MUKERJEE sidered this problem whe T represets the score statistic. The preset geeral framework eables us to hadle all members of F 0. Give ay statistic T i F 0, it ca be see that each of T 1, T 2 ad T 3 belogs to F ad that the expressios for a 1 ad a 2 associated with them are give by a 1 (T 1 )=a 1 (T), a 2 (T 1 )=& 1 3 I&32 K, (4.1a) a 1 (T 2 )=a 1 (T), a 2 (T 2 )=a 2 (T), (4.1b) a 1 (T 3 )=a 1 (T), a 2 (T 3 )=& 1 3 I&32 K, (4.1c) where a 1 (T) ad a 2 (T) correspod to T. The relatios (4.1b) ad (4.1c) follow from Cordeiro ad Ferrari's (1991) equatio (14) ad Taiguchi's (1991) equatio (3.5) respectively. The relatio (4.1a) ca be proved essetially alog the lie of Chadra ad Mukeree's (1991) Theorem 2.1 ad we omit the details. By (4.1), T 2 differs from T by O(c &2 ) while this may ot be the case with T 1 or T 3 i geeral. Hece, compared to T 1 or T 3 the adustmet T 2 perturbs T to a smaller extet (cf. Rao ad Mukeree, 1995). From Cordeiro ad Ferrari's (1991) equatio (14), ote that T 2 may belog to F&F 0 ad this ustifies our cosideratio of the wider class F. From (2.4), (2.7) ad (2.8), recall that the third-order power fuctio depeds o ay statistic oly via a 1 ad a 2. Hece by (4.1), the third-order power fuctios correspodig to T 3 ad T are idetical with those associated with T 1 ad T 2 respectively. Thus, it will suffice to compare T 1 ad T 2. I fact, if a 2 (T)=& 1 3 I&32 K the, by (4.1), T, T 1, T 2 ad T 3 will lead to idetical power fuctios up to the third order. Therefore, to avoid trivialities, let a 2 (T){& 1 3 I&32 K. The by (2.2a), (4.1), a 2 (T 1 )= a 2 (LR){a 2 (T 2 ) so that, as i the first paragraph of Sectio 3, T 1 (or equivaletly, T 3 ) is superior to T 2 with regard to secod-order local maximiity. Furthermore, by (3.3) ad (4.1), where V T1 (z 2 )&V T2 (z 2 )= 1 36 z4 I &3 s T (z 2 ), s T (z 2 )=z 2 [9I 3 a 2 2 (T)&K2 ]&6J[K+3I 32 a 2 (T)], (4.2) ad, uder the criterio of third-order local average power, T 1 is superior (iferior) to T 2 if s T (z 2 ) is positive (egative). Oe ca as well employ (2.7), i couctio with (4.1), to compare the etire third order average power fuctios of T 1 ad T 2. Some of the results i Rao ad Mukeree (1995), which relate to the scalar parameter case, follow from the above cosideratios. For further

11 LR, SCORE, AND WALD TESTS 109 elucidatio, let T represet the Wald statistic for the rest of this sectio. The by (2.2c) ad (4.2), a 2 (T){& 1 3 I&32 K if ad oly if K+3J{0, ad s T (z 2 )=z 2 (9J 2 &K 2 )&6J(K+3J). (4.3) Example 3.1 (cotiued). By (3.5), here K+3J=0 ad T, T 1, T 2 ad T 3 lead to idetical power fuctios whatever % 0 might be. Example 3.2 (cotiued). By (3.8), here K+3J{0 for % 0 {0. Hece for % 0 {0, T 1 (or T 3 ) is better tha T 2 i terms of secod-order local maximiity. Also, the by (3.8) ad (4.3), s T (z 2 )=9J 2 ( 3 4z 2 &1), which is positive for :=0.10, 0.05 or Thus, for these values of :, T 1 (or T 3 )is superior to T 2 also uder the criterio of third-order local average power. I fact, by (2.2c), (2.7), (3.8) ad (4.1), R T1 ($, z 2 )&R T2 ($, z 2 )=$z 2 % 2 0 (1&%2 0 )&1 [(2$+3z 2 ) h 3, $ &4z 2 h 1, $ ] ad if :=0.10, 0.05 or 0.01 ad % 0 {0 the this is see to be positive for every $>0. Hece, for these values of :, T 1 (or T 3 ) domiates T 2 uiformly i $ (>0) with respect to third order average power. Cosequetly, i this example, if % 0 {0 the there is eough reaso to prefer the adustmet T 1 (or T 3 ) for the Wald statistic to T 2. I the preset paper, we have worked uder the framework of cotiguous alterative. It will be of iterest to compare the tests from cosideratio of Bahadur efficiecy. It is, however, likely that such a study will ivolve the use of etirely differet tools. ACKNOWLEDGMENTS We are thakful to the referees, a editor, ad the chief editor for very costructive suggestios. The work of Rahul Mukeree was supported by a grat from the Ceter for Maagemet ad Developmet Studies, Idia Istitute of Maagemet, Calcutta, ad that of C. R. Rao by the Army Research Office uder Grat DAAH04-93-G REFERENCES [1] Amari, S. (1985). Differetial Geometrical Methods i Statistics. Spriger-Verlag, Berli. [2] Chadra, T. K., ad Mukeree, R. (1991). Bartlett-type modificatio for Rao's efficiet score statistic. J. Multivariate Aal [3] Cordeiro, G. M., ad Ferrari, S. L. P. (1991). A modified score test statistic havig chisquared distributio to order &1. Biometrika [4] Efro, B. (1975). Defiig the curvature of a statistical problem (with applicatios to secod order efficiecy). A. Statist

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