ALBERT VEXLER, SHULING LIU, LE KANG, AND ALAN DAVID HUTSON

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1 Commuicatios i tatistics imulatio ad Computatio, 38: , 2009 Copyright Taylor & Fracis Group, LLC IN: prit/ olie DOI: / Modificatios of the Empirical Likelihood Iterval Estimatio with Improved Coverage Probabilities ALBERT VEXLER, HULING LIU, LE KANG, AND ALAN DAVID HUTON Departmet of Biostatistics, The tate Uiversity of New York at Buffalo, Buffalo, New York, UA 1. Itroductio The empirical likelihood EL) techique has bee well addressed i both the theoretical ad applied literature i the cotext of powerful oparametric statistical methods for testig ad iterval estimatios. A oparametric versio of Wilks theorem Wilks, 1938) ca usually provide a asymptotic evaluatio of the Type I error of EL ratio-type tests. I this article, we examie the performace of this asymptotic result whe the EL is based o fiite samples that are from various distributios. I the cotext of the Type I error cotrol, we show that the classical EL procedure ad the tudet s t-test have asymptotically a similar structure. Thus, we coclude that modificatios of t-type tests ca be adopted to improve the EL ratio test. We propose the applicatio of the Che 1995) t-test modificatio to the EL ratio test. We display that the Che approach leads to a locatio chage of observed data whereas the classical Bartlett method is kow to be a scale correctio of the data distributio. Fially, we modify the EL ratio test via both the Che ad Bartlett correctios. We support our argumet with theoretical proofs as well as a Mote Carlo study. A real data example studies the proposed approach i practice. Keywords Bartlett Correctio; Empirical likelihood; Iterval estimatio; Likelihood ratio; Modified t-test; Noparametric test; igificace level; Type I error. Mathematics ubject Classificatio 62G10; 62G20. Likelihood methods are commo statistical tools that provide very powerful methods of statistical iferece. The parametric likelihood methodology requires the assumptio that the distributio fuctio of the idividual observatios have kow forms e.g., Vexler et al., 2009). To relax the parametric assumptios, Owe 1988) itroduced the empirical likelihood EL) methodology as a oparametric versio of the parametric likelihood techique. I a wide rage of applicatios, Received March 13, 2009; Accepted August 13, 2009 Address correspodece to Albert Vexler, Departmet of Biostatistics, The tate Uiversity of New York, Farber Hall, Room 246, Buffalo, NY 14214, UA; avexler@buffalo.edu 2171

2 2172 Vexler et al. the oparametric EL ratio tests have comparable advatages to those of the parametric likelihood methodologies. EL methods have bee well accepted i the literature as very powerful statistical procedures i the distributio-free settig relative to parametric approaches; e.g., see DiCiccio et al. 1991), Hall ad cala 1990), Owe 1988, 1990, 1991, 2001), Qi ad Lawless 1994), ad Yu et al. 2009). Other results iclude that of Owe 1990) who proved that the asymptotic distributio fuctio of twice the logarithm of a EL ratio-type test has the form of a chi-square distributio fuctio. This asymptotic result is a oparametric versio of the Wilks theorem e.g., Wilks, 1938), which is kow i the cotext of the asymptotic ull distributio of the maximum parametric likelihood ratio test statistic. However, it is well kow that the distace betwee the actual ull distributio of a test statistic ad its asymptotic approximatio ca be sigificat. To acquire the Type I error cotrol related to the EL ratio test, the Bartlett correctio ca be applied; e.g., see Owe 2001). The Bartlett correctio is made by multiplyig the relevat test thresholds by a scale parameter, values of which ca be chose to improve upo Wilks approximatio. I practice, a fiite sample size strogly restricts the Type I error cotrol that is obtaied through Wilks approximatio. I this case, the applicatio of Bartlett s correctio ca be also difficult. This is due to the fact that this correctio is based upo asymptotic priciples, which i tur deped fuctioally o ukow parameters, such as the 3rd ad 4th ukow momets of the uderlyig populatio, ad hece have to be estimated. Whe the sample size is relatively small, the ukow parameters utilized withi Bartlett s correctio ca be poorly estimated such that the efficiecy of the Bartlett s techique deteriorates. It should also be oted that the EL ratio test statistic i certai situatios is ot Bartlett correctable; e.g., see Che ad Cui 2006) ad Lazar ad Myklad 1999). Corcora 1998), as well as Daviso ad tafford 1998), remarked that the Bartlett correctio applied to EL ratios does ot provide sigificatly effective results i certai practical situatios. I this article, we ivestigate the Bartlett-corrected EL ratio test based o samples with fiite sizes. Data with skewed distributios ca also affect the efficiecy of the EL method. I this article, utilizig both theoretical ad empirical argumets, we show that, i the cotext of the Type I error cotrol, whe the data is o ormally distributed ad skewed especially whe the sample size is relatively small), the EL ratio test ca be corrected i a similar maer to that of a tudet s t-test correctio. That is, if some correctio ca improve tudet s t-type tests, the it ca also be adopted to the EL ratio-type test. Che 1995) proposed a modificatio of the t-test that was show to be more accurate ad more powerful tha both the Johso s modified t-test ad the utto s composite test. The Che approach is very efficiet for small sample sizes. Thus, we propose to apply the Che 1995) method to modify the EL ratio test. This correctio is differet from the Bartlett modificatio that rescales the log EL ratio i order to improve the secod-order approximatio via a chi-square distributio. Alteratively, we show that Che correctio trasforms observatios, chagig the locatio of data. To take advatages of both the scale ad locatio correctios, we propose a ew modificatio of the EL ratio test combiig the Che ad Bartlett techiques. To achieve our goals metioed above, sketch theoretical proofs of the ew modificatios are provided. A Mote Carlo study is used to evaluate the cotrol of Type I error related to the proposed procedures. A real data example is provided i order to illustrate the proposed method. ectio 5 cotais our cocludig remarks.

3 Modified Empirical Likelihood Ratio Tests Methods Assume, for simplicity, we observe idepedet idetically distributed i.i.d.) radom variables X 1 X 2 X with E X 1 3 <. We would like to test the hypothesis H 0 E X 1 = vs. H 1 E X 1 1) Furthermore, before we proceed ote that the hypothesis above ca be exteded easily to the more geeral case, amely, H 0 E g X = vs. H 1 E g X where g is a give fuctio. To test the hypothesis above defied at 1) oparametrically, first defie the EL fuctio i the form of L X 1 X = p i where p i = 1. Here, p i s, i = 1, have a sese of Pr X i = x i The values of p i s are ukow ad should be obtaied uder the ull or the alterative hypothesis. Uder the ull hypothesis, the EL methodology e.g., Owe, 2001; Yu et al., 2009) requires oe to fid the value of the p i s that maximize the EL that satisfy the empirical costraits p i = 1 ad p ix i =. Usig Lagrage multipliers, oe ca show that uder the ull hypothesis, the maximum EL fuctio has the followig form: where is a root of L X 1 X = sup 0<p 1 p 2 p <1 p i =1 p i X i = p i = X i 1 + X i = 0 1 2) 1 + X i Uder the alterative hypothesis, just the empirical costrait p i = 1 is i effect. The L X 1 X = sup 0<p 1 p 2 p <1 p i =1 p i = 1 = ) 1 3) Combiig 2) ad 3), we obtai the log EL ratio, test statistic for 1) i the form of l X 1 X = 2 log L X 1 X L X 1 X = 2 log 1 + X i 4)

4 2174 Vexler et al. Owe 1988) proved that l X 1 X has a asymptotic chi-square distributio, uder the ull hypothesis. Hece, we ca test H 0 at the approximated sigificace level usig the test l X 1 X = 2 where the test threshold C distributio. Note that oe ca show log 1 + X i >C is the % percetile for a chi-square l X 1 X X i 2 X i 2 as 5) where X i 2 / X i 2 has a asymptotic chi-square distributio uder the ull hypothesis give at 1). The asymptotic propositio 5) ad its proof are similar to results that were demostrated ad utilized by Qi ad Lawless 1994) i a differet cotext. For details, see the Appedix. This asymptotic result motivates us to cosider the t-test statistic T X 1 X = X i X i X 2 X = ) X i / which is asymptotically equivalet to 2 log EL ratio, i.e., by virtue of 5) oe ca show that the 2 log EL ratio is approximated by T 2 that i tur coverges i distributio to a chi-square distributio. This idicates that the EL ratio test ad the tudet s t-test have similar properties i the cotext of the Type I error cotrol. Note that Owe 1990) cocluded that large sample sizes strogly improve the accuracy of the Wilks-type approximatios of the distributio of ull EL ratio test statistic. I the cotext of the Type I error cotrol, we demostrate that the classical EL procedure ad tudet s t-test based o observatios from the same distributios have similar properties. That is, whe data are close to beig ormally distributed, the Type I error cotrol related to the tests based o the data is expected to be accurate eve whe, the sample size, is small. However, whe the sample size is large ad the observatios are from a logormal distributio, either the t-test or the EL ratio test cotrol the Type I error well. I this case, both tests ca be affected strogly by the skewess of data. I ec. 3, we cofirm this coclusio via a Mote Carlo study. I certai situatios, the EL ratio test is Bartlett correctable; e.g., see Owe 2001). The Bartlett correctio is a theoretical adjustmet for the secod-order approximatio of a log-likelihood ratio distributio; e.g., see Hall ad cala 1990). However, the Bartlett correctio caot completely solve the problem of the Type I error cotrol whe the data is skewed. Moreover, whe the Bartlett correctio is applied certai ukow parameters eed to be estimated. The accuracy of this estimatio ca affect the overall efficiecy of the Bartlett correctio. Note that Lazar ad Myklad 1999) proved that whe uisace parameters are preset the EL ratio statistic is ot Bartlett correctable. Whe the Bartlett correctio is i effect, the test statistic is multiplied by a coefficiet. This rescalig of the log likelihood ratio is used to reduce the discrepacy betwee a chi-square distributio ad the

5 Modified Empirical Likelihood Ratio Tests 2175 asymptotic ull distributio of the log likelihood ratio. Thus, we refer to this a scale correctio. Alteratively, sice some modificatios of the t-test, differet from the Bartlett methodology, ca improve the relevat Type I error cotrol, we ca cosider a similar modificatio of the EL ratio test followig the asymptotic equivalece betwee the two tests metioed above. We outlie the approach of Che, showig this method chages the locatio of the sample istead of providig a scalig correctio. I the cotext of the hypothesis H 0 E X = vs. H 1 E X > 6) defie ˆ 1 = X i X 3 / 3, where = X i X 2 / 1 ) 0 5. The the Che modified t-test ca be preseted as where T X 1 X + X 1 X 7) [ X 1 X = 1 { } 2 ] 6 ˆ X [ + 1 { } 2 ] X 9 ˆ 2 X This correctio was obtaied usig a Edgeworth expasio applied to the t-test whe the distributio of data is positively skewed Che, 1995). The test statistic 7) ca be represeted by the regular t-test statistic based o observatios X i = X i + X 1 X i = 1 8) Thus, the Che approach chages the locatio of X i s. Therefore, we ca classify the Che method as a locatio correctio. Assume we chage the locatio of X i s ad defie the EL ratio based o relocated data ad derive a optimal chage of the locatio of the X i s. Towards this ed, the EL ratio test is created utilizig the costraits X i + d p i = ad p i = 1 uder the ull H 0, where d is a shift of the locatio. Let us obtai a form of d to correct the EL ratio test statistic followig the Che approach. By virtue of 5), we have the Type I error { } P H0 l X 1 X >C P X i + d 2 H0 X >C i + d 2 = P H0 { R d > C or R d < C } 9)

6 2176 Vexler et al. where R d = X i + d X i + d 2 ) X d = X + i + d 2 / X i + d 2 Usig the Taylor expasio with d aroud 0, we obtai that R d = T X 1 X + d X i 2 + o d2 as d 0 Here, the correspodig remaied term has a order of d 2 ad ca be igored. ice R d > C ad R d T X 1 X + d / X i 2, the by the Che methodology we must set up d 1 d 0 5 X i 2) 0 5 = X 1 X as i 7) i order to correct the Type I error. Thus, we coclude that the optimal chage i the data locatio give the rule R d > C is d = 1 X 1 X By virtue of 9), ad i order to approximate the decisio rule l X >C, we should also cosider R d < C. I a similar maer to the aalysis metioed above, for a egatively skewed populatio, we should set up d = 1 X 1 X Here, whe we cosidered R d < C, to preserve the statemet of 6), we utilized observatios X 1 X keepig the sig > of the alterative hypothesis of 6). That explais the form of d = 0 5 X 1 X, comparig with the locatio chage 0 5 X 1 X give the rule R d > C. Thus, we propose the followig modificatio of the EL ratio test: 1. Calculate values of the statistics l 1 = l X 1 + X 1 X X + X ) 1 X ad l 2 = l X 1 + X 1 X X + X ) 1 X 2. Reject the ull hypothesis of 1) if l Z 1 ) 2 or l Z 1 ) 2

7 Modified Empirical Likelihood Ratio Tests 2177 where Z 1 is the percetile of the stadard ormal distributio. Note that, if we ca presuppose the uderlyig distributio fuctio of observatios is positively or egatively skewed, we the ca defie the test statistic l = l X 1 + X 1 X for positively skewed data; l = l for egatively skewed data, X i + X 1 X X + X 1 X ) X + X 1 X ) respectively, ad reject the ull hypothesis if l 2 1 1, where is the % percetile of the chi-square distributio with the degree of freedom of 1. Hece, we showed that the Che approach ca be also adapted to be applied to the EL ratio test. I accordace with the previously metioed remarks, we ca cosider the Bartlett ad Che techiques as two differet test-modificatios. The Bartlett correctio ca reduce the slope caused by the differece betwee the chi-square distributio ad the log EL ratio distributio ad the Che approach corrects the bias caused by the skewess of a ull distributio of observatios. Thus, we coclude to combie these modificatios ad to propose the ext decisio rule. We suggest rejectig the ull hypothesis at 1) if l Z 1 ) 1 + a 0 5 or l 2 ) Z 1 ) 1 + a ) where a is a estimator of 1 E X EX 4 2 E X EX 2 1 E X EX E X EX 2 3 Note that, by virtue of the relatio betwee the testig ad iterval estimatio, we ca obtai the cofidece iterval estimator of i the form of CI 1 = X T C 1 where T presets a test statistic for the hypothesis 1) ad C 1 is the correspodig threshold, assumig that the omial coverage probability is 1. Therefore, the previoulsy cosidered issue of the Type I error cotrol is also relevat to a problem to improve the accuracy of the iterval estimatio. 3. Mote Carlo tudy I this sectio, we provide a Mote Carlo study to compare the followig five tests: 1) the stadard EL ratio test; 2) the proposed Che-corrected EL ratio test; 3) the Bartlett-corrected EL ratio test; 4) the proposed Che Bartlett-corrected EL ratio test; ad 5) the tudet s t-test. Towards this ed, the ull distributio

8 2178 Vexler et al. fuctios were chose to be i Normal ad Logormal forms, sice the asymptotic result metioed i ec. 2 showed that we ca expect a good approximatio to the test-statistic distributio whe data are from a ormally distributed populatio, whereas the approximatio ca be expected to be worse i the case with logormally distributed data. We also geerated data followig a chi-square distributio to ivestigate the proposed method. Radom variables were draw from the ull distributios with differet parameters. At each baselie distributio ad each set of parameters, we repeated 10,000 times data geeratios. The Mote Carlo Type I errors are preseted i Tables 1 3. As is show i Table 1, the t-test demostrates the actual Type I errors that are closer to the expected 0.05 tha those of the cosidered tests, uder the ormal distributio assumptio. The stadard EL ratio method has well cotrolled Type I errors whe the sample sizes are greater tha 25. The Bartlett ad Che correctios are assumed to reduce a bias caused by the differece betwee the chi-square distributio ad log EL ratio asymptotic distributio. I this case, this estimatio is ot ecessary; however, the proposed Che Bartlett correctio is readily available. Table 2 displays situatios whe data follow a logormal distributio. The otatio LN a b correspods to a logormal distributio where a ad b are the mea ad stadard deviatio of the logarithm. I accordace with the aalysis metioed i ec. 2, we ca expect correctios of the test statistics are eeded i this case. Here, the Mote Carlo study cofirms this coclusio. The Che Bartlettcorrected EL ratio test is the best of all the five test statistics. Especially whe the data are strogly skewed, e.g., i the cases with LN 1 2, LN 2 2, = 25, the Mote Carlo Type I errors for the EL ratio test ad the t-test are greater tha 0.3 we expected the sigificace level at 0.05). Whe the combiatio of Che Table 1 The Mote Carlo Type I Errors of the tests based o data from the ull ormal distributios: X Normal 0 2 ad H 0 EX= 0 The expected Type I error is 0.05) ample size ELR Che Bartlett Che & Bartlett T-test

9 Modified Empirical Likelihood Ratio Tests 2179 Table 2 The Mote Carlo Type I errors correspoded to ull logormal distributios: X Logormal 2 exp ad H 0 EX= 0 The expected Type I error is 0.05) ample size ELR Che Bartlett Che & Bartlett T-test 25 0, 1.0) , 1.5) , 2.0) , 1.0) , 1.5) , 2.0) , 2.0) , 1.0) , 1.5) , 2.0) , 1.0) , 1.5) , 2.0) , 2.0) , 1.0) , 1.5) , 2.0) , 1.0) , 1.5) , 2.0) , 2.0) , 1.0) , 1.5) , 2.0) , 1.0) , 1.5) , 2.0) , 2.0) Table 3 The Mote Carlo Type I errors correspoded to ull chi-square distributios: X ad H 0 EX= 0 The expected Type I error is 0.05) ample size ELR Che Bartlett Che & Bartlett T-test

10 2180 Vexler et al. ad Bartlett correctios were applied, the Type I error was about two times better approximated tha that of the two other tests. Therefore, we ca coclude that the proposed method ca sigificatly improve the Type I error cotrol ad the accuracy to cover probabilities of the iterval estimatio. Table 3 is related to the cases where data follow a chi-square distributio. From Table 3, the proposed Che Bartlett-corrected EL ratio test is show to have the best ability to cotrol the Type I error especially whe the sample sizes are relatively small. With the icreasig of the sample size, all the tests ted to perform well. 4. A Glucose Data Example I this sectio, we use the proposed method to costruct the cofidece iterval estimators based o data from a study that evaluates biomarkers related to atherosclerotic coroary heart disease. We cosider the biomarker glucose that ca quatify differet phases of oxidative stress ad atioxidat status process of a idividual e.g., chisterma et al., 2001). A populatio-based sample of radomly selected residets of Erie ad Niagara couties, years of age, was the focus of this ivestigatio chisterma et al., 2001). The samplig frame for adults betwee the ages of 35 ad 65 was from the New York tate Departmet of Motor Vehicle drivers licese rolls, while the elderly sample age 65 79) was radomly chose from the Health Care Fiacig Admiistratio database. A cohort of 917 me ad wome were selected for the aalyses to preset a populatio without myocardial ifarctio. Participats provided a 12-h fastig food specime for biochemical aalysis at baselie, ad a umber of parameters were examied from fresh blood samples. We illustrate the proposed method based o the biomarker glucose that has values with a right-skewed distributio. The strategy was that a sample with size N was radomly selected from the glucose data to estimate the cofidece itervals at the level 95% of the mea, say a b ; the rest of the data was used to calculate the average of the glucose levels, say c. The value of 917-N was chose to be large, ad hece we believe the computed average c is very close to the theoretical mea of the glucose levels. We repeated this strategy 10,000 times calculatig the frequecies of the evet c a b. This bootstrap-type test shows that the Che Bartlett ad Che corrected EL ratio tests provides the coverage probabilities that are closer to the target 0.95 tha those of the EL ratio test ad the Bartlett corrected EL ratio test ad the t-test. Table 4 presets these results for the differet sample sizes of Table 4 The coverage probabilities related to the glucose biomaker The expected coverage probability is 0.95) ample size ELR Che Bartlett Che & Bartlett T-test

11 Modified Empirical Likelihood Ratio Tests 2181 Table 5 The cofidece iterval estimators at the level 95% of the mea of the glucose biomarker measured from 917 patiets without myocardial ifarctio ELR , ) Che , ) Bartlett , ) Che&Bartlett , ) T-test , ) N = 15 25, 35, 50, ad 100. It ca be oticed that whe the sample size is above 50, the actual coverage probabilities is very close to the omial oe. Thus, we propose to use the Che Bartlett ad Che corrected EL ratio statistics to estimate the cofidece iterval for the mea of glucose measuremets. The ivestigated test statistics provided the cofidece itervals at the level 95% of the mea of the glucose biomarker measured from 917 patiets without myocardial ifarctio the healthy populatio). The results are show i Table 5. The ELR, Bartlett corrected ELR, ad T-test show approximately similar results, the use of which ca be misleadig, takig ito accout the outputs of Table 4 ad that glucose measuremets have a right skewed distributio. 5. Coclusios The EL methodology has bee well addressed i the literature as a oparametric versio of its powerful parametric likelihood couterparts. A importat property is that the asymptotic distributio of 2 log EL ratio test statistics follow a chi-square distributio. However, this approximatio ca be biased especially whe the sample size is small. I this article, we illustrated the skewess of the data distributio ca also affect the efficiecy of the EL ratio tests. With respect to improvig the Type I error cotrol of the tests ad coverage probability of the iterval estimatio the EL techique ad the Bartlett correctio of the EL ratio tests are well accepted i the literature. We proved that the EL ratio test ca be corrected i a similar maer to a classic t-test s correctio. Thus, the Che 1995) approach, which was proposed to correct the skewess of the t-test distributio, was adapted to be applied to the EL ratio test. The Bartlett correctio was show to be based o scalig modified data, whereas the Che method was preseted via chagig the locatio. We oted that the Bartlett techique requires certai additioal coditios, e.g., o uisace parameters related to the statemet of problem, Cramér s coditios, which are uecessary whe the Che correctio is assumed to be applied. I this article, we also proposed to combie both the Bartlett ad Che techiques as the ew modificatio of the EL ratio test i order to improve the accuracy of the iterval estimatios. Our argumets are cofirmed via both the theoretical cosideratio ad the Mote Carlo study. The Bartlett Che-corrected EL ratio test was show to perform much better tha the stadard EL ratio test ad also the classic tudet s t-test whe data were from skewed distributio. To demostrate that the proposed

12 2182 Vexler et al. method ca be utilized i practice, we applied the proposed method to costruct the cofidece itervals for the glucose data that is from a coroary heart disease study. The proposed methodology ca also be applied to differet EL-type procedures. Appedix A.1. ketch Proof of the Asymptotic Chi-quare Distributio of l X 1 X We have l X 1 X = 2 log L X 1 X = 2 log 1 + X i where is a root of the equatio g = X i = 0 A1) 1 + X i Applyig the Taylor expasio, we ca expad g with aroud 0. Hece, by A1), we coclude that X i X i 2 as A2) ice l X 1 X = 2 log 1 + X i, ow cosider l X 1 X with Taylor series where is close to 0 ad after pluggig A2) to the statistic 2 log 1 + X i we obtai l X 1 X X i 2 X A3) i 2 Note that the test statistic T X 1 X 2 has a asymptotic chi-square distributio as follows T X 1 X = X i p N 0 1 X i X 2 ad hece l X 1 X T 2 p X 1 X 2 1. This completes the brief proof. Ackowledgmet The authors are grateful to the Editor ad referee for their helpful commets that clearly improved this article. Refereces Che, L. 1995). Testig the mea of skewed distributios. Joural of the America tatistical Associatio 90:

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