POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES*

Kobe Uiversity Ecoomic Review 50(2004) 3 POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES* By HISASHI TANIZAKI There are various kids of oparametric tests. I this paper, we cosider testig populatio mea, usig the empirical likelihood ratio test. The empirical likelihood ratio test is useful i a large sample, but it has size distortio i a small sample. For size correctio, various correctios have bee cosidered. Here, we utilize the Bartlett correctio ad the bootstrap method. The purpose of this paper is to compare the t test ad the empirical likelihood ratio tests with respect to the sample power as well as the empirical size through Mote Carlo experimets.. Itroductio I the case of testig populatio mea, we test the ull hypothesis usig the t test, assumig that the populatio is ormal. However, covetioally it is ot kow whether a populatio is ormal. I a small sample, whe a populatio is ot ormal, we caot obtai proper results if we apply the t test. I a large sample, whe both mea ad variace are fiite, the distributio of sample mea is approximated as a ormal distributio from the cetral limit theorem, where a populatio distributio is ot assumed. I this paper, we cosider testig populatio mea without ay assumptio o the uderlyig distributio. There are umerous papers o the distributio-free test. Here, we itroduce the empirical-likelihood-based test ad compare it with the t test. I the past research, Thomas ad Grukemeier (975) made a attempt to costruct cofidece itervals usig the empirical likelihood ratio. Owe (988) proposed various test statistics usig the empirical likelihood i uivariate cases, which are based o Thomas ad Grukemeier (975). Moreover, Owe (990) obtaied the cofidece itervals i multivariate cases ad Owe (99) cosidered the testig procedure i liear regressio models, usig the empirical likelihood. As for the other studies, see Baggerly (998), Che ad Qi (993), DiCiccio, Hall ad Romao (989), Hall (990), Kitamura (997, 200), Lazar ad Myklad (998), Qi (993) ad Qi ad Lawless (994). Owe (200) is oe of the best refereces i this field. I a lot of literature, e.g., Che (996), Che ad Hall (993), DiCiccio, Hall ad Romao (99), DiCiccio ad Romao (989) ad Hall (990), the empirical likelihood is Bartlett correctable. Also, see Jig ad Wood (996), where the empirical likelihood is ot Bartlett correctable i a special case. I this paper, we show that i a small sample the Bartlettcorrected empirical likelihood ratio test is improved to some extet but it still has size

4 HISASHI TANIZAKI distortio. As show i Namba (2004), the bootstrap empirical likelihood ratio test is preferred. I this paper, the t test ad the empirical-likelihood-based test are compared with respect to the sample powers ad the empirical size through Mote Carlo studies. The empirical-likelihood-based test is a large sample test ad accordigly it has size distortio. To improve the size distortio, we also cosider usig the Bartlett correctio ad the bootstrap method. 2. Empirical Likelihood ratio Test I this sectio, we itroduce the testig procedure o populatio mea, usig the empirical likelihood ratio test. For example, see Owe (200) for the empirical likelihood ratio test. Let X, X 2, X be mutually idepedetly distributed radom variables ad x, x 2,, x be the observed values of X, X 2,, X. For ow, x, x 2,, x are assumed to be the distict values i{x, X 2,, X }. We cosider the discrete approximatio of the distributio fuctio of X, which is approximated as Prob(X i =x i )=p i, where 0< p i < ad p i = have to be satisfied. The joit distributio of X, X 2,, X, i.e., the likelihood fuctio (especially, called the oparametric likelihood), is give by: L(p)= Prob(X i =x i )= p i, where p=(p,p 2,,p ). Let pˆ i be the estimate of p i, which is give by pˆ i =/ because x, x 2, x are regarded as the realizatios radomly geerated from the same distributio. Therefore, the empirical likelihood is represeted as: L(pˆ)= pˆ i =, where pˆ=(pˆ,pˆ 2,,pˆ ). The empirical likelihood ratio R(p) is defied as a ratio of the oparametric likelihood ad the empirical likelihood, which is give by: L(p) R(p)= = p i. () L(pˆ) I this paper, we cosider testig the populatio mea. The expectatio of X is give by: E(X)= x i Prob(X i =x i )= x i p i =µ, which is rewritte as: Π Π Π Π Π (x i-µ)p i= 0. (2) The, give (x, x 2,, x ) ad µ, we wat to obtai the p, p 2,, p which maximize ()

POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES 5 with the two costraits, i.e., p ad (2). Substitutig the solutios of p, p 2,, p ito (), the maximum value of the empirical likelihood ratio is obtaied as a fuctio of µ. That is, we solve the Lagragia: G= log(p i )-λ'( (x i -µ)p i )+γ( p i -), with respect to p, p 2,, p, λ ad γ, where λ ad γ are deoted by the Lagragia multipliers. I the secod term of the Lagragia, is multiplied for simplicity of calculatio. Sice the logarithm of the empirical likelihood ratio, log R(p)= log( p i ), is cocave for all 0<p i <,,2,,, we have a global maximum of G. However, we have the problem that all the solutios of p, p 2,, p, λ ad γ are ot obtaied i a closed form. Solvig the Lagragia, p i is give by: p i =, +(x (3) i - µ)'λ which depeds o λ. I order to obtai the solutio of λ, multiplyig (x i-µ) i both sides of (3) ad summig up with respect to i, we have the followig expressio: x i-µ +(x i -µ)'λ =0. (4) Sice (4) is a implicit fuctio of λ, we may solve (4) with respect to λ by the iterative procedure such as the Newto-Raphso optimizatio method or a simple grid search. Thus, give µ, we ca obtai the λ which satisfies (4). As a computatioal remark, the coditio 0< p i< is required. This coditio is equivalet to: +(x i -µ)'λ>, (5) which comes from (3). Moreover, (5) is rewritte as: - - / <λ<- - /, x x (6) max -µ mi -µ where x mi ad x max represet miimum ad maximum values of x, x 2,, x, respectively. Therefore, we have to perform the above iterative procedure with the coditio (6). Substitutig the solutio of λ ito (3), we ca obtai (p, p 2,, p ), give (x, x 2,, x,) ad µ. I other words, substitutig (3) ito (), the logarithm of the empirical likelihood ratio, log R(p)= log( p i ), with the costraits p i = ad (2) is rewritte as follows: - log(+(x i -µ)'λ)=logr ~ (x;µ), (7) where x = (x, x 2,, x ) ad λ has to satisfy (4). R ~ (x;µ) correspods to the maximum value of R(p) give x, x 2,, x, µ.

6 HISASHI TANIZAKI We have assumed that x, x 2,, x are the distict values i {X, X 2,, X }. Therefore, p i is iterpreted as the probability where X takes x i. Eve whe some of x, x 2,, x take the same value, the above discussio still holds without ay modificatio. However, the iterpretatio of p i should be chaged. Suppose that x i takes the same value as x j for i j, i.e., x i=x j=x *, but it is differet from the others. Prob (X=x * ) = Prob (X=x i) + Prob (X=x j)=p i+p j=p * represets the probability where X takes x i=x j=x *, where p i should be equal to p j. I this case, we eed to iterpret p i as a specific weight, rather tha the probability. Now, we cosider the ull hypothesis H 0 : µ = µ 0. Replace the actual data x i i (7) by the correspodig radom variable X i. The, as goes to ifiity, uder the ull hypothesis H 0 we have the followig asymptotic property: -2 logr ~ (X ;µ 0)=2 log(+(x i-µ 0)'λ) χ 2 (k), (8) which is show i Owe (200), where X=(X, X 2,, X ). Let χ α2 (k) be 00 α percet poit from the χ 2 distributio with k degrees of freedom. Remember that µ is a k vector. Thus, we reject the ull hypothesis H 0:µ=µ 0,whe -2logR ~ (x;µ 0) =2 log(+(x i-µ 0)'µ)>χ α2 (k). Note as follows. Uder the ull hypothesis H 0: µ=µ 0, we sometimes have the case where the λ which satisfies (4), (5) ad µ=µ 0 does ot exist. The reaso why we have this case is because the ull hypothesis H 0 : µ=µ 0 does ot hold. Therefore, it is plausible to cosider that the empirical likelihood ratio -2 logr ~ (x;µ 0 ) takes a extremely large value. Bartlett-Corrected Empirical Likelihood Ratio Test: As metioed above, the empirical likelihood ratio test statistic -2 logr ~ (X ;µ 0 ) is asymptotically distributed as a χ 2 (k) radom variable. However, i a small sample, it is ot ecessarily chi-squared ad the approximatio might be quite poor. Therefore, we cosider applyig size correctio to (8). I the past, various size correctio methods have bee proposed. I this paper, we adopt the Bartlett correctio, which is discussed i Che (996), Che ad Hall (993), DiCiccio, Hall ad Romao (99), DiCiccio ad Romao (989), Hall (990), Jig ad Wood (996) ad Owe (200). By the Bartlett correctio, asymptotically we have the followig: or equivaletly, -2 logr ~ a (X ;µ 0)=2 log(+(x i -µ 0 )'λ) (+ )χ 2 (k), -(+ a ) - 2 logr ~ a (X ;µ 0)=(+ ) - 2 log(+(x i-µ 0)'λ) χ 2 (k), where λ has to satisfy (4), (5) ad µ=µ 0, ad a is deoted by a= (m 4 / m 22 )- 3 (m 32 / m 23 ). m j deotes the jth momet of X about the mea Ε(X), i.e., m j =Ε((X-Ε(X)) j ). Sice m j is ukow, replacig m j by its estimate mˆ j =(/) (x i -x ) j, we perform the testig procedure as follows: 2

POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES 7 -(+ â ) - 2 logr ~ â (X ;µ 0)=(+ ) - 2 log(+(x i -µ 0 )'λ) χ 2 (k), (9) where λ has to satisfy (4), (5) ad µ=µ 0, ad â idicates the estimate of a, which is show as: â= (mˆ 4/ mˆ 22 )- (mˆ 32 / mˆ 23 ). Thus, we reject the ull hypothesis H 0 : µ=µ 0, whe -(+â/) - 2 2logR ~ 3 (x;µ 0 ) =(+â/) - 2 log(+(x i -µ 0 )'λ)> χ α2 (k). Bootstrap Empirical Likelihood Ratio Test: As aother size correctio method, we cosider derivig a empirical distributio of -2logR ~ (X;µ) by the bootstrap method. Let x *, x 2*,, x * be the bootstrap sample from x, x 2,, x. That is, oe of the origial data x, x 2,, x is radomly chose for x i*. We repeat this procedure for all,2,, with replacemet ad we have the bootstrap sample x *, x 2*,, x *. We solve () subject to the costraits 0<p i <, p i = ad (x i* -x ) p i =0, where x (/) x i. As a result, i (2) - (7), x i ad µ are replaced by x i* ad x, respectively. That is, istead of (7), we obtai the followig test statistic: -2 logr ~ (x * ;x ), (0) where x * =(x *, x 2*,, x * ). Wheever the bootstrap sample {x *, x 2*,, x * } is resampled from {x, x 2,, x } with replacemet, we have a differet value of (0). Resamplig the bootstrap sample M times, we have M empirical log-likelihood ratios, where M =0 4 is take i the Mote Carlo studies i the ext sectio. This procedure is equivalet to derivig the empirical distributio of -2 logr ~ (X;x ), where the sample mea x is give. Sortig the M values by size, we ca obtai the percet poits from the M empirical log-likelihood ratios. The percet poits based o the M values are compared with the empirical log-likelihood ratio based o the origial data. Thus, we ca test the ull hypothesis H 0 : µ=µ 0. See, for example, Hall (987), Namba (2004) ad Owe (200) for the bootstrap empirical likelihood ratio test. Sometimes we have the case where x i* > x or x i* < x for all,2,,. I this case, we cosider that (0) lies o the side of the distributio ad accordigly takes a extremely large value. 3. Mote Carlo Experimets I Sectio 2, we have itroduced the empirical likelihood ratio tests based o (8), (9) ad (0), which are applied to testig the populatio mea. I this sectio, the t test, the covetioal empirical likelihood ratio test (8), the Bartlett-corrected empirical likelihood ratio test (9) ad the bootstrap empirical likelihood ratio test (0) are compared with respect to empirical size ad sample power through Mote Carlo studies. The setup of the Mote Carlo experimets is as follows: (i) For the uderlyig distributio of uivariate radom variable X i, we take the followig

8 HISASHI TANIZAKI seve distributios. Stadard Normal Distributio (N): f (x) = exp(- x 2 ), - < x <. π Uiform distributio (U): f (x) =, - x <. χ 2 () Distributio (X): y- x =, where f (y)= y -/2 exp(- 0 <y < π t(3) Distributio (T): y x =, where f (y)= 2 (+y 2 /3) -2 - < y < π Double Expoetial Distributio (D): y x =, where f (y)= exp(- - < y < Logistic Distributio (L): y exp(y) x =, where f (y)=, - < y < π/ exp(y)) 2 Gumbel Distributio (G): y x =, where f (y)=exp(-(y-α)-e -(y-α) ) ad α -0.5772, - < y < π/ The mea ad variace of X i are ormalized to be zero ad oe for,2,,. (ii) The sample size is give by = 20, 50, 00. (iii) The ull hypothesis H 0 : µ=µ 0 ad the alterative oe H : µ=µ are take for µ 0 =.0 ad µ =.0,.,.2,.4. (iv) The sigificace level α is give by α =.0,.05. (v) For each sigificace level, perform 0 4 simulatio rus ad obtai the umber of rejectios of the ull hypothesis H 0 : µ=µ 0. Dividig the umber of the rejectios by 0 4, the sample power is computed. Uder the setup above, we perform Mote Carlo experimets. The obtaied results are expected as follows:

POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES 9 Sice the ormal distributio is assumed i (N), the t test idicates the same empirical size as the sigificace level ad it gives us the most powerful test. I the case of µ=.0, the power correspods to the size, i.e., the sigificace level α. However, the empirical likelihood ratio test show i (8) idicates a large sample test. Therefore, i a small sample, it might be expected that the empirical likelihood ratio test (8) has size distortio. The Bartlett-corrected empirical likelihood ratio test (9) ad the bootstrap empirical likelihood ratio test (0) should be improved with respect to the empirical size i a small sample. The covetioal empirical likelihood ratio test, the Bartlett-corrected empirical likelihood ratio test ad the bootstrap empirical likelihood ratio test should be equivalet to the t test as is large, because the three test statistics have the same distributio, i.e., chi-square distributio, i a large sample. The results are obtaied i Table, where the three kids of empirical likelihood ratio tests are show with the t test. (8) represets the covetioal empirical likelihood ratio test which simply utilizes the chi-square percet poits with oe degree of freedom. (9) idicates the Bartlett-corrected empirical likelihood ratio test, where the chi-square percet poit with oe degree of freedom is regarded as the percet poit for the test statistic. (0) shows the bootstrap empirical likelihood ratio test, where the distributio of the test statistic is costructed by the bootstrap sample. Let δˆ be a value i the table. The stadard error of δˆ is give by δˆ(-δˆ)/0 4, ad accordigly it is at most.5(-.5)/0 4 =.005. Note that 0 4 i the square root idicates the umber of simulatio rus. The results are summarized as follows. Empirical Size: First, cosider the case of µ =.0 i Table, which represets the empirical size. Let δ ad δˆ be the true size ad the empirical size, respectively. Uder the ull hypothesis H 0: δ = α, asymptotically we have(δˆ -α)/ α( α)/0 4 ~N (0,), which comes from the cetral limit theorem. I the case of µ =.0 i Table, ** ad * show that the ull hypothesis H 0 : δ = α is rejected at sigificace levels % ad 5 % by the both-sided test. That is, i the case of µ =.0, the values without ** ad * are preferred, because they show correct sizes. As for (N), the t test idicates the correct sizes for all α =.0,.05 ad = 20, 50, 00. These results are plausible, because the t test gives us the uiformly most powerful test uder ormality assumptio. For (U), similarly the empirical size δˆ i the t test is very close to the sigificace level α. For (T), (D) ad (L), the t test is ot a appropriate test especially for small, because the size is statistically differet from the sigificace level. That is, the t test gives us the correct sizes ad the powers for (N), but it yields the over-estimated sizes or the uderestimated sizes for (U), (T), (D) ad (L). Whe is small, for ay distributio the covetioal empirical likelihood ratio test (8) has size distortio ad it over-estimates empirical sizes. For example, comparig the empirical likelihood ratio test (8) ad the t test i the case of (N), µ =.0 ad =20, the empirical likelihood ratio test (8) has larger empirical sizes tha the t test for both α =.0 ad.05 (we

20 HISASHI TANIZAKI have 0.340 ad 0.0808 for the empirical likelihood ratio test (8), ad 0.097 ad 0.0503 for the t test). As icreases (for example, =00), the empirical likelihood ratio test (8) as well as the t test idicate the correct sizes. I a small sample, we perform Bartlett correctio to improve the size distortio i the empirical likelihood ratio test. See (8) i Table for the Bartlett-corrected empirical likelihood ratio test. We ca observe that the size distortio decreases but we still have the size distortio to some extet. For example, i the case of (N), =20, µ =.0 ad α =.0, (8) is.340 ad (9) is.26, which idicates that (9) is closer to α =.0 tha (8). (8) is.0808 for α =.05 ad (9) is.0735 for α =.05. Therefore, (9) is better tha (8), but it is ot practical. The bootstrap empirical likelihood ratio test (0) is also implemeted to improve the size distortio. Both (8) ad (9) are size-distorted i most of the cases, but (0) shows a correct empirical size i a lot of cases. Thus, the size distortio is improved whe (0) is applied. As a result, we ca coclude that (0) is better tha (8) ad (9) for the empirical size criterio ad that the empirical size i (0) is very close to the sigificace level. Whe is large, the empirical likelihood ratio tests (8) - (0) ad t test show the size which is equal to the sigificace level. That is, i the case of µ =.0 ad =00, there are a lot of values without the superscript *. Thus, all the tests perform better i a large sample. Sample Power: Next, we cosider the case of µ 0 i Table, which idicates the sample power. I the case of µ = 0., 0.2, 0.4 ad the empirical likelihood ratio tests (8) - (0),,, ad idicate compariso with the t test. Let δt be the value i t Test of Table. We put the superscript whe ( δˆ -δ t )/ δ t( δ t)/0 4 is greater tha.9600, ad the superscript whe it is greater tha 2.5758. We put the superscript if ( δˆ -δ t )/ δ t( δ t)/0 4 is less tha -.9600, ad the superscript if it is less tha -2.5758. Note that i a large sample we have the followig: ( δˆ -δ t )/ δ t( δ t)/0 4 ~N(0,) uder the ull hypothesis H 0 : δ = δ t ad the alterative oe H : δ δ t. Therefore, the values with the superscript idicate a more powerful test tha the t test. I additio, the umber of the superscript shows the degree of the sample power. Cotrarily, the values with the superscript represet a less powerful test tha the t test. Takig a example of the case (N), = 20 ad µ =., the sample power i (8) is give by δˆ = 0.687 while that i t Test is δ t = 0.325. We wat to test H 0 : δ = δ t agaist H : δ δ t. I this case, the test statistic is give by (0.687-0.325) / 0.325(-0.325)/0 4 =0.677, which is greater tha 2.5758. Therefore, H 0 : δ = δ t is rejected at sigificace level α =0.0. I the case of (N), the t test is the most powerful test. For small, both (8) ad (9) have size distortio, ad they over-estimate the sample powers. Accordigly, it is ot meaigful to compare (8), (9) ad the t test with respect to the sample powers. The empirical likelihood ratio test (0) is very close to the t test i all the cases of (N). Thus, (0) performs better tha (8) ad (9). For (U), the empirical sizes i (8) ad (9) are over-estimated, while those i (0) ad the t test are sometimes uder-estimated but correctly estimated i may cases. I additio, the sample powers i (8) ad (9) are larger tha those i the t test. (0) is slightly better tha the t

POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES 2 TABLE. Empirical Size ad Sample Power µ \α (8) Empirical Likelihood Ratio Test (9) (0) t Test Normal Distributio(N).0 0.340 ** 0.0808 ** 0.26 ** 0.0735 ** 0.02 0.052 0.097 0.0503 20. 0.687 0.055 0.576 0.0976 0.32 0.0700 0.325 0.0673.2 0.2648 0.802 0.2480 0.666 0.242 0.226 0.280 0.337.4 0.5805 0.4624 0.5624 0.4399 0.504 0.338 0.5306 0.3925.0 0.47 ** 0.067 ** 0.095 ** 0.0583 ** 0.007 0.050 0.0988 0.0473 50. 0.963 0.97 0.89 0.57 0.754 0.0978 0.79 0.029.2 0.490 0.305 0.408 0.2929 0.384 0.255 0.3968 0.2770.4 0.8783 0.8035 0.8745 0.7959 0.8558 0.7428 0.8706 0.7872.0 0.070* 0.057 ** 0.043 0.0558 ** 0.0976 0.0497 0.0992 0.0503 00. 0.2707 0.788 0.2674 0.750 0.2553 0.594 0.2597 0.670.2 0.6379 0.567 0.6325 0.53 0.659 0.4792 0.6290 0.5054.4 0.9905 0.9778 0.9903 0.9766 0.9892 0.9685 0.9903 0.9765 Uifor Distributio(U).0 0.49 ** 0.065 ** 0.057 0.0600 ** 0.0932* 0.0493 0.0923* 0.0469 20. 0.464 0.0885 0.379 0.085 0.232 0.0662 0.237 0.0677.2 0.2456 0.65 0.2338 0.502 0.25 0.96 0.203 0.284.4 0.5805 0.4553 0.5656 0.4386 0.5360 0.372 0.584 0.3767.0 0.056 0.0550* 0.028 0.0524 0.0974 0.0469 0.0962 0.0470 50. 0.853 0.0 0.802 0.065 0.728 0.0972 0.73 0.0966.2 0.447 0.2952 0.4093 0.2893 0.3927 0.2690 0.3926 0.2708.4 0.8933 0.826 0.890 0.873 0.8840 0.797 0.878 0.7929.0 0.0993 0.0527 0.0977 0.058 0.0933* 0.0482 0.0943 0.0483 00. 0.2744 0.764 0.277 0.736 0.2662 0.62 0.2657 0.644.2 0.657 0.5283 0.6490 0.5252 0.6398 0.5054 0.639 0.57.4 0.9935 0.9828 0.9934 0.9822 0.9930 0.9806 0.9924 0.9792 χ 2 () Distributio(X).0 0.777 ** 0.2 ** 0.653 ** 0.096 ** 0.094 ** 0.069 ** 0.540 ** 0.076 ** 20. 0.923 0.90 0.737 0.063 0.0888 0.0390 0.47 0.0628.2 0.365 0.260 0.296 0.966 0.526 0.0552 0.506 0.0643.4 0.884 0.7444 0.802 0.7208 0.6336 0.396 0.577 0.364.0 0.45 ** 0.086 ** 0.363 ** 0.0822 ** 0.0994 0.0594 ** 0.273 ** 0.0829 ** 50. 0.295 0.36 0.2078 0.268 0.359 0.0393 0.45 0.0623.2 0.525 0.4044 0.505 0.3865 0.402 0.925 0.3672 0.240.4 0.9896 0.9785 0.9884 0.9756 0.962 0.82 0.9659 0.908.0 0.258 ** 0.0744 ** 0.24 ** 0.073 ** 0.0949 0.0474 0.5 ** 0.0679 ** 00. 0.2978 0.2058 0.2892 0.97 0.2388 0.0845 0.298 0.44.2 0.7585 0.669 0.7498 0.6496 0.754 0.4799 0.6664 0.4993.4 0.9998 0.9998 0.9998 0.9998 0.9953 0.922 0.9998 0.9992

22 HISASHI TANIZAKI TABLE. Empirical Size ad Sample Power < Cotiued > µ \α (8) Empirical Likelihood Ratio Test (9) (0) t Test t (3) Distributio(T).0 0.590 ** 0.05 ** 0.443 ** 0.0894 ** 0.0998 0.0474 0.0952 0.0445* 20. 0.240 0.44 0.926 0.274 0.420 0.0774 0.49 0.0784.2 0.3434 0.2584 0.3225 0.2367 0.2577 0.623 0.2792 0.803.4 0.6732 0.5849 0.655 0.5600 0.5770 0.4384 0.6355 0.578.0 0.26 ** 0.0736 ** 0.63 ** 0.0668 ** 0.096 ** 0.0446* 0.0922 ** 0.0435 ** 50. 0.2399 0.69 0.2279 0.507 0.946 0.29 0.2020 0.80.2 0.497 0.3878 0.4773 0.3686 0.436 0.3040 0.4679 0.3458.4 0.8838 0.824 0.8737 0.87 0.837 0.7354 0.8947 0.8332.0 0.262 ** 0.0720 ** 0.87 ** 0.0678 ** 0.0967 0.0477 0.0976 0.0475 00. 0.38 0.224 0.3074 0.235 0.2746 0.726 0.2950 0.924.2 0.6796 0.5779 0.6688 0.5674 0.6364 0.503 0.6796 0.5764.4 0.975 0.9562 0.9685 0.9553 0.952 0.8880 0.984 0.9676 Double Expoetial Distributio (D).0 0.503 ** 0.0937 ** 0.367 ** 0.083 ** 0.00 0.0455* 0.0924* 0.0408 ** 20. 0.99 0.304 0.769 0.93 0.333 0.0720 0.336 0.0706.2 0.298 0.269 0.2789 0.2022 0.2238 0.295 0.2383 0.469.4 0.597 0.5005 0.5767 0.4762 0.504 0.3483 0.5575 0.4325.0 0.24 ** 0.0737 ** 0.59 ** 0.0695 ** 0.003 0.050 0.0958 0.046 50. 0.2027 0.30 0.99 0.224 0.78 0.0944 0.75 0.08.2 0.4322 0.370 0.472 0.3048 0.383 0.2466 0.46 0.2905.4 0.8653 0.7854 0.8568 0.7757 0.8226 0.6775 0.8685 0.79.0 0.64 ** 0.0666 ** 0.30 ** 0.0635 ** 0.02 0.0502 0.0994 0.048 00. 0.282 0.892 0.276 0.89 0.2530 0.59 0.2705 0.77.2 0.643 0.5239 0.6364 0.559 0.607 0.4649 0.6400 0.585.4 0.9875 0.970 0.9866 0.9692 0.9794 0.932 0.9884 0.9728 Logistic Distributio (L).0 0.348 ** 0.0805 ** 0.235 ** 0.0720 ** 0.0963 0.0465 0.0930* 0.0437 ** 20. 0.755 0.58 0.642 0.060 0.340 0.074 0.292 0.0680.2 0.2735 0.975 0.2590 0.823 0.296 0.265 0.2273 0.387.4 0.5886 0.4787 0.5686 0.454 0.5023 0.3487 0.5424 0.4074.0 0.74 ** 0.0688 ** 0.23 ** 0.0644 ** 0.000 0.0508 0.0956 0.047 50. 0.952 0.207 0.868 0.58 0.690 0.0940 0.728 0.0985.2 0.4239 0.30 0.44 0.300 0.3847 0.2543 0.4042 0.2829.4 0.8740 0.797 0.8675 0.787 0.8432 0.73 0.8709 0.7904.0 0.0 ** 0.0598 ** 0.075* 0.0568 ** 0.0974 0.0484 0.0988 0.0475 00. 0.2764 0.820 0.2723 0.78 0.2554 0.54 0.2660 0.7.2 0.6434 0.5239 0.6385 0.556 0.672 0.4747 0.6393 0.536.4 0.9887 0.974 0.9882 0.9732 0.9847 0.9563 0.9900 0.9758

POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES 23 TABLE. Empirical Size ad Sample Power < Cotiued > µ \α (8) Empirical Likelihood Ratio Test (9) (0) t Test Gumbel Distributio(G).0 0.378 ** 0.0842 ** 0.266 ** 0.0773 ** 0.07 0.0535 0.047 0.0562 ** 20. 0.69 0.0935 0.484 0.0853 0.066 0.040 0.08 0.0464.2 0.264 0.747 0.2453 0.607 0.789 0.0777 0.79 0.0950.4 0.6466 0.5268 0.6253 0.5008 0.5335 0.3094 0.5338 0.3663.0 0.7 ** 0.0672 ** 0.5 ** 0.0640 ** 0.0955 0.0490 0.004 0.0520 50. 0.93 0.46 0.825 0.078 0.52 0.0648 0.480 0.0765.2 0.4449 0.3273 0.4342 0.366 0.3922 0.22 0.38 0.246.4 0.9338 0.8844 0.9297 0.8783 0.995 0.8029 0.908 0.834.0 0.07 ** 0.0588 ** 0.078 ** 0.0565 ** 0.0943 0.0442 ** 0.0990 0.050 00. 0.279 0.847 0.2742 0.776 0.2507 0.297 0.2450 0.446.2 0.680 0.5699 0.6760 0.5638 0.6600 0.5040 0.6434 0.504.4 0.9970 0.9943 0.9966 0.9939 0.9968 0.986 0.9965 0.9909 test but (9) is ot sigificatly differet from the t test. It might be cocluded that (0) is slightly more powerful tha the t test i the case where the uderlyig distributio is give by (U). I the case of (X), (T), (D), (L) ad (G), both (7) ad (8) have size distortio, where all the empirical sizes i (8) ad (9) are over-estimated. Therefore, it is useless for (X), (T), (D), (L) ad (G) to compare (8) ad (9) with the t test. (0) has o size distortio i a lot of cases, while the empirical sizes i the t test are sometimes uder-estimated. I most cases, (0) is less tha the t test with respect to the sample powers. Therefore, the t test is more powerful tha (0), especially i the case of (T), (D) ad (L). I the case of =00, the sample powers take similar values for all µ =0., 0.2, 0.4. Therefore, it is show from Table that all the tests give us the same sample power i a large sample. 4. Summary I this paper, we have compared three empirical likelihood ratio tests with the t test, where we test the populatio mea. I the case of testig the populatio mea, covetioally we assume a ormal distributio. Uder the ormality assumptio, the appropriate test statistic has a t distributio. Thus, we eed the ormality assumptio for the t test. However, the uderlyig distributio is ot kow i geeral. Therefore, the ormality assumptio is ot ecessarily plausible. I this paper, we cosider testig the populatio mea without ay assumptio o the uderlyig distributio. I the case of a large sample, the sample mea is approximated to be ormal whe mea ad

24 HISASHI TANIZAKI variace exist. This result comes from the cetral limit theorem. Therefore, usig the ormal distributio, we ca test the hypothesis o the populatio mea. The asymptotic distributio of the t distributio is give by the stadard ormal distributio. Thus, whe the sample size is large we ca apply the t test to ay uderlyig distributio. The empirical likelihood ratio test statistic is asymptotically distributed as a chi-square radom variable, which is also a large sample test, where the chi-square percet poits are compared with the empirical likelihood ratio test statistic. The covetioal empirical likelihood ratio test with chi-square percet poits yields the size distortio. Therefore, we have cosidered the empirical likelihood ratio tests based o the Bartlett correctio ad the bootstrap method. I this paper, the ormal distributio, the uiform distributio, the t (3) distributio, the double expoetial distributio, or the logistic distributio is assumed for the uderlyig distributio. Usig the three empirical likelihood ratio tests show i (8) - (0) ad the t test, we have obtaied the empirical sizes ad the sample powers for each distributio through Mote Carlo experimets. Whe the sample size is large, we have obtaied the result that all the tests show the similar empirical sizes ad sample powers. However, whe is small, the covetioal empirical likelihood ratio test (8) idicates a large size distortio, ad accordigly it is ot practically useful. Therefore, we have performed the size correctio by Bartlett correctio, but the empirical likelihood ratio test with Bartlett correctio, show i (9), still has size distortio although it improves to some extet. Next, usig the bootstrap method, we have obtaied the empirical distributio of the the empirical likelihood ratio test statistic ad derived the critical regio. Based o the critical regio, we have performed the empirical likelihood ratio test (0). The obtaied empirical sizes ad sample powers are plausible, usig (0). REFERENCES Baggerly, K.A. (998), "Empirical Likelihood as a Goodess-of-Fit Measure,'' Biometrika, 85, pp. 535-547. Che, S.X. (996), "Empirical Likelihood Cofidece Itervals for Noparametric Desity Estimatio,'' Biometrika, 83, pp. 329-34. Che, S.X. ad Hall, P. (993), "Smoothed Empirical Likelihood Cofidece Itervals for Quatiles,'' The Aals of Statistics, 2, pp. 66-8. Che, J. ad Qi, J. (993), "Empirical Likelihood Estimatio for Fiite Populatios ad the Effective Usage of Auxiliary Iformatio,'' Biometrika, 80, pp. 07-6. DiCiccio, T.J., Hall, P. ad Romao, J.P. (989), "Compariso of Parametric ad Empirical Likelihood Fuctios,'' Biometrika, 76, pp. 465-476. DiCiccio, T.J., Hall, P. ad Romao, J.P. (99), "Empirical Likelihood is Bartlett-Correctable,'' The Aals of Statistics, 9, pp. 053-06. DiCiccio, T.J. ad Romao, J.P. (989), "O Adjustmets Based o the Siged of the Empirical Likelihood Ratio Statistic,'' Biometrika, 76, pp. 447-456. Hall, P. (987), "O the Bootstrap ad Likelihood-Based Cofidece Regios,'' Biometrika, 74, pp. 48-493. Hall, P. (990), "Pseudo-Likelihood Theory for Empirical Likelihood,'' The Aals of Statistics, 8, pp. 2-40. Jig, B.Y. ad Wood, A.T.A. (996), "Expoetial Empirical Likelihood is Not Bartlett Correctable,'' The Aals of Statistics, 24, pp. 365-369. Kitamura, Y. (997), "Empirical Likelihood Methods with Weakly Depedet Processes,'' The Aals of Statistics, 25, pp. 2084-202. Kitamura, Y. (200), "Asymptotic Optimality of Empirical Likelihood for Testig Momet Restrictios,'' Ecoometrica, 69, pp. 66-672.

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