Joural of the Korea Data & Iformatio Sciece Society 2018, 29(2), 485 499 http://dx.doi.org/10.7465/jkdi.2018.29.2.485 한국데이터정보과학회지 Resamplig modificatios for the Bagai test Youg Mi Kim 1 Hyug-Tae Ha 2 1 Departmet of Statistics, Kyugpook Natioal Uiversity 2 Departmet of Applied Statistics, Gacho Uiversity Received 17 February 2018, revised 9 March 2018, accepted 14 March 2018 Abstract I this paper, we develop resamplig modificatios for the Bagai statistics to test two competig risks model, ad make their umerical comparisos i terms of coverage probabilities ad statistical powers. We foud that the bias-corrected bootstrap method cosistetly improves o the Jackkife, Efro s percetile bootstrap, ad bootstrap-t methods i both cases of the coverage probability ad statistical power. It is iterestig that this results are differet from the ituitive expectatios i view of the covergece rates of the resamplig methods. The practical importace is also discussed for the distributio free Bagai statistic to test stochastic orderig. Keywords: Bagai statistic, umerical compariso, resamplig methods. 1. Itroductio The cocept of stochastic domiace has bee extesively employed i a wide rage of scietific disciplies icludig ecoomics, fiace, busiess ad psychology. See for istace Levy (1992) for a survey ad Kim ad Kwo (2017) for cesorig. Stochastic domiace betwee two competig risks model explais that the probability of oe variable smaller tha a certai value is greater tha the correspodig probability of the other radom variable. I statistical sese, stochastic domiace deotes a order relatioship betwee cumulative distributio fuctios. A radom variable Y is said to stochastically domiate a radom variable Z whe Pr(Y x) Pr(Z x) for all x with strict iequality. Iferrig stochastic domiace from data samples is importat for may applicatios i ecoometrics ad experimetal studies, but little is kow about the performace of existig iferetial methods. Amog may distributio-free iferetial statistics for testig stochastic domiace, we utilize the Bagai test statistic, which Bagai et al. (1989) proposed for testig stochastic orderig i two competig risks model, to make umerical comparisos of several resamplig This research of Hyug-Tae Ha was supported by the Gacho Uiversity research fud of 2016 (GCU- 2016-0208). 1 Assistat professor, Departmet of Statistics, Kyupook Natial Uiversity, Daegu 41566, Korea. E- mail: kymmyself@ku.ac.kr 2 Correspodig author: Associate professor, Departmet of Applied Statistics, Gacho Uiversity, Sugam-ci, Kyuggi-do 461-701. Korea. E-mail: htha@gacho.ac.kr
486 Youg Mi Kim Hyug-Tae Ha methods. Besides of the popularity of the Bagai statistic as a oparametric statistic for testig stochastic domiace, there are may beefits for the use of the Bagai statistic: 1) The Bagai statistic tests stochastic equality ad domiace at the same time, 2) Its exact ull distributio is kow to be symmetric eve though the explicit expressio of the exact distributio is ot kow, ad its asymptotic distributio is ormal distributio, 3) Bagai statistic is a computatioal efficiet oparametric statistic to apply resamplig methods sice asymptotic ormality ca quickly be achieved eve i small sample. Bootstrap methods are ofte combied to create superior modified distributio-free tests i small samples for providig a specified ull hypothesis about stochastic domiace that is as close as possible to the observed data. See for istaces Hall ad va Keilegom (2005) ad Heathcote et al. (2010), which utilized the usual percetile bootstrappig method. As metioed i Diciccio ad Romao (1988) ad Carpeter ad Bithell (2000), those resamplig methods show differet performaces although the resamplig methods are, i geeral, cosidered to overcome the shortcomigs i statical ifereces whe sample size is small. Despite popular utilizatios of the resamplig methods for testig stochastic domiace, their computatioal performaces of various resamplig methods have ot bee much discussed i coectio with distributio-free iferetial statistics. The percetile ad t bootstrap methods seem to be the most utilized resamplig methods for testig stochastic domiace. It may be because their asymptotic covergece rates are theoretically cocrete as O p ( 1/2 ) ad O p ( 1 ) for percetile ad t bootstrap methods, respectively. I the preset paper, we are aimig to discover the details of practical importaces whe researchers i may scietific fields combie resamplig methods with distributio-free iferetial statistics for testig stochastic domiace ad provide umerical comparisos to show that the bias-corrected bootstrap, which was proposed by Efro (1987) to desig for reflectig properties of asymmetry of samplig distributio i fiite sample, outperforms other methods i such tests. It is eve clearer i small sample cases. This is a iterestig result i practice sice it does t coicide the theoretical examiatios of those resamplig methods icludig Diciccio ad Romao (1988) ad Carpeter ad Bithell (2000). The paper is orgaized as follows. Sectio 2 explais the distributio-free test statistic for stochastic domiace proposed by Bagai et al. (1989) ad proposes four resamplig Bagai statistic for testig for testig oe way stochastic domiace. We explai i detail how the comparisos were coducted i Sectio 3, ad the results of our umerical comparisos ad our coclusios are give i Sectio 4. 2. Resamplig-modified Bagai statistic The Bagai statistic ad its characteristics are briefly itroduced i this sectio. Bagai statistic cosiders the competig risks set-up where a uit is subject to failure due to oe of two risks. O deotig Y ad Z the otioal lifetimes of a uit uder these two risks ad Y 1,..., Y ad Z 1,..., Z idividuals i two radom samples of idepedet observatios of same sizes from two cotiuous distributios, F Y ( ) ad F Z ( ), T = Mi(Y, Z) is the time at which the uit fails ad the T i = Mi(Y i, Z i ) would be observed time to failure from the i th sample. O lettig δ = I(Y > Z) the idicator for the cause of failure, we ca observe (T 1, δ 1 ),..., (T, δ ) where δ i = I(Y i > Z i ) a idicator of the i th uit. O the basis of these
Resamplig for Bagai test 487 data types, we are iterested i testig the hypothesis as follows: H 0 : F Y (x) = F Z (x) agaist H A : F Y (x) F Z (x). (2.1) Further deotig R i the rak of T i amog T 1,..., T, Bagai et al. (1989) proposed a test statistic S = 2 (2 1 R i )δ i i=1 3( 1) 2. (2.2) While the explicit fuctioal expressio of its exact ull distributio caot be established, the asymptotic ormality with mea ad variace E(S) = 0 ad V ar(s) = ( 1)(14 13) 6 (2.3) was also discussed. The oparametric resamplig scheme makes o assumptios cocerig the distributio of, or model for, the uderlyig data process. O defiig a rak based test statistic X U(Y, Z) from idepedet paired observatios from two risks Y ad Z, we are iterested i estimatig a parameter, deoted by θ, of the test statistic ad costructig cofidece itervals for its estimator, deoted by ˆθ (X), for statistical ifereces. O lettig F ad F, respectively, the exact distributio ad empirical distributio from the data set, the parameter θ ad its estimator ˆθ are their fuctioals as θ = T (F ) ad ˆθ (X) = T (F ). There are several cofidece iterval estimatio methods for the Bootstrap. Here, we cosider the jackkife ad three bootstrap cofidece iterval estimatio methods such as Efro s percetile bootstrap, bootstrap-t, ad bias-corrected bootstrap. First, the Jackkife itroduced by Queouille (1949) is a popular resamplig method to estimate parameters of iterest o the basis of the rest of the data by deletig oe observatio each time from the origial data. Tukey (1958), who coied the ame of Jackkife, modified the variace estimators of the parameter of iterest, which leads the Jackkife to become a more valuable statistical method. Ad the bootstrap method proposed by Efro (1979) is a computer-itesive method for a large class of statistical iferece issues without cosiderig ay striget structural assumptios o the uderlyig data structure. While Efro s percetile bootstrap does ot eed to estimate variace with coverage error O p ( 1/2 ) of oe-sided cofidece iterval, a ew versio of bootstrappig method proposed by Efro (1982), amely the bootstrap t method, is based o a t distributed pivot usig a variace estimator ad provide a better performace with asymptotic covergece rate O p ( 1 ) whe the variace of the estimator is available. The bias-corrected bootstrap was proposed by Efro (1987) to desig for reflectig properties of asymmetry of samplig distributio i fiite sample. These resamplig methods are examied ad compared i theoretical aspects, see for istaces Diciccio ad Romao (1988) ad Carpeter ad Bithell (2000). 2.1. Jackkife based ˆθ ( i) From cosiderig a estimator (X) = T (F ( i) ) where F ( i) is the empirical distributio fuctio with the 1 observatios obtaied by leavig the i th observatio out ad a
488 Youg Mi Kim Hyug-Tae Ha ˆθ ( i) mea ˆθ( ) = 1 ( i) i=1 of the estimator ˆθ (X) from all the umber of subsamples, the jackkife estimator of the parameter, θ, ad its variace estimator are obtaied as ˆθ JK = 1 i=1 ˆθ ( i) ad ˆσ 2 JK = 1 I additio, Tukey (1958) defied the pseudovalue of costruct a modified Jackkife variace estimator which is ˆσ 2 JKT = 1 ( 1) i=1 i=1 θ ( i) (ˆθ( i) ( θ( i) θ JK ) 2, ) 2 ˆθ( ). = ˆθ + ( 1)(ˆθ ( i) ˆθ ) to where θ JK = ( i) i=1 θ /. Based o these estimators, the procedure of the Jackkife method for oe-sided cofidece iterval estimatio with a level of sigificace α is as follows; Procedure 1. Use a modified data leavig the i th observatio out to compute the Jackkife pesudovalue ( i) θ. 2. Repeat step 1 to obtai pseudovalues. 3. Compute the Jackkife variace estimate, ˆσ 2 JKT. 4. Obtai the Jackkife oe-sided 100(1 α)% cofidece iterval based o ormal approximatio as (, ˆθ ) + z 1 αˆσ JKT where z 1 α is a 100(1 α)% quatile for stadard ormal distributio. 2.2. Efro s percetile bootstrap based The percetile bootstrap method is the most typical bootstrap method to costruct a cofidece iterval for a parameter based o bootstrap replicates. Let X b deote a idepedet ad idetical sample from F, a estimator of F. Suppose that ˆθ is a estimator of b θ ad ˆθ is the b th bootstrap replicate from the b th bootstrap sample, X b for b = 1, 2,..., B, where each bootstrap sample is of size i usual. We order the bootstrap replicates from the smallest to the largest, (ˆθ (1) (B),..., ˆθ ) ad the we compute p (b) (b) = ˆθ ˆθ for b = 1,..., B. The upper limit for the target parameter θ is estimated as ˆθ p (αb) usig the α B quatile of p (αb). Efro (1982) shows that, for the media, the percetile method provides early the same cofidece iterval as the oparametric iterval based o the biomial distributio. The procedure of the Efro s percetile bootstrap method is as follows; Procedure 1. Sample observatios radomly with replacemet from the origial data, X to obtai the b th bootstrap data set, deoted X b.
Resamplig for Bagai test 489 2. Calculate the statistic of iterest ˆθ b based o the b th bootstrap sample. 3. Repeat steps 1 ad 2 for the B times to obtai the bootstrap distributio of the estimator ad order them to obtai the bootstrap distributio, (ˆθ (1) (B),..., ˆθ ). 4. Take ˆθ (αb) for the α B quatile of bootstrap replicates to obtai p (αb) 5. ( Costruct oe side bootstrap cofidece iterval for the parameter θ, that is,, ˆθ ) p (αb). 2.3. Bootstrap-t based ˆθ (αb) ˆθ. Efro (1982) proposed bootstrap t method based o a give studetized pivot t = (ˆθ θ)/ˆσ where ˆσ 2 is a variace estimator for ˆθ ad Efro ad Tibshirai (1986) ivestigated that the bootstrap t cofidece itervals are secod-order accurate. Let S b be a estimator of the stadard deviatio for θ based o the b th bootstrap sample. Defie t b = (ˆθ b ˆθ )/S b b for b = 1, 2,..., B. For each of the B bootstrap replicates ˆθ for b = 1, 2,..., B, we ca compute a correspodig t b. For a approximate oe-sided 100(1 α)% cofidece iterval for θ, especially upper limit, we take the iterval [, ˆθ t (αb) S ] where t (αb) is the αb quatile from the ordered values of (t (1),..., t (B) ) ad S is estimated stadard deviatio for ˆθ. It is called the bootstrap t or percetile t oe-sided 100(1 α)% cofidece iterval for the parameter θ. The procedure of the bootstrap-t method is as follows; Procedure 1. Resample observatios radomly with replacemet from the origial data X to obtai the b th bootstrap sample, deoted X b. 2. Compute the b th bootstrap estimator, ˆθ (X b ) ad variace estimator, S b. 3. Calculate t b = (ˆθ b ˆθ )/S b. 4. Repeat steps 1, 2 ad 3 for the B times to obtai the bootstrap distributio of t b ( 5. Costruct bootstrap cofidece itervals for θ as, ˆθ ) t (αb) S for θ. 2.4. Bias-corrected bootstrap based For the bias-corrected percetile method, we cosider a mootoically icreasig fuctio g( ) to cosider φ = g(θ), ˆφ = g(ˆθ ) ad ˆφ = g(ˆθ ) satisfyig ˆφ ˆφ ˆφ φ N( c 1 σ, σ 2 ) for some costat c 1, where θ is the parameter of iterest, ˆθ is a estimator of parameter ad ˆθ is the bootstrap replicate of the parameter. The, based o the bootstrap method cosiderig the bias-correctio, we obtai the oe-sided bootstrap cofidece iterval as (, F 1 ˆθ ) (Φ(2c 1 z α/2 )),.
490 Youg Mi Kim Hyug-Tae Ha where c 1 is estimated by Φ (P 1 (ˆθ ˆθ ) ) ad Φ 1 ( ) is the iverse cumulative Gaussia distributio fuctio. Procedure 1. Obtai observatios radomly with replacemet from the origial data X to produce the b th bootstrap sample, deoted X b. 2. Compute the b th bootstrap estimator, ˆθ (X b ). 3. Repeat steps 1 ad 2 for the B times for the bootstrap distributio of the parameter. 4. Cout the umber of (ˆθ ˆθ ). 1 2 B ˆθ, ˆθ,..., ˆθ that are less tha ˆθ with p = the umber of 5. Obtai c 1 = Φ 1 (p/b) ad calculate Q = B Φ(2c 1 z α/2 ) for the upper edpoit of the bias-corrected cofidece iterval. 6. Estimate the edpoit of the iterval by ˆθ ( Q ), where takes its iteger part, ad ( costruct oe side BC cofidece iterval, ˆθ ) ( Q ). 3. Numerical Comparisos We utilize the distributio-free test statistic that Bagai et al. (1989) proposed for testig stochastic orderig of two competig risk model are examied i terms of coverage probability ad statistical power. The umerical comparisos i various sample sizes = 5, 10, 15, 20, 30, 50, 70 ad 100 are coducted. The coverage probability ad statistical power are ivestigated for the levels of sigificace, α = 1%, 5%, 10% i various examples. For this umerical compariso i case of coverage probability, we cosider ormal (N(µ, σ 2 )), logormal (LogN(µ 1, σ1) 2 where µ 1 ad σ1 2 are mea ad variace o the log scale, i.e., µ 1 ad σ1 2 are mea ad variace of the logarithm of the logormal radom variable), t ad mixture distributios. I this study, we cosider two mixture distributios such as 0.5 N(0, 1)+0.5 N(1.1) ad 0.3 N(0, 1)+0.7 N(1, 1). To compute coverage probabilities of 90%, 95% ad 99% cofidece itervals, we make umerical comparisos i followig cases: 1. C1 : Y ad Z N(0, 1), 2. C2 : Y ad Z LogN(0, 1), 3. C3 : Y ad Z t distributio with degree of freedom 1, 4. C4 : Y ad Z 0.5 N(0, 1) + 0.5 N(1.1) ad 5. C5 : Y ad Z 0.3 N(0, 1) + 0.7 N(1, 1). Note that the examples C1 3 represet stochastic equality ad the other examples C4 ad C5 for stochastic domiace. For statistical power calculatio, 1%, 5% ad 10% levels of sigificace are cosidered. We use the followig examples to express stochastic domiace:
Resamplig for Bagai test 491 1. P1 : Y N(0.5, 10) ad Z N(0, 10), 2. P2 : Y LogN(0, 1) ad Z LogN(0, 1) 0.1, 3. P3 : Y t distributio with degree of freedom 1 ad Z t distributio with degree of freedom 1 0.1, 4. P4 : Y 0.5 N(0, 1) + 0.5 N(1.1) ad Z 0.5 N(0, 1) + 0.5 N(1.1) 0.1 ad 5. P5 : Y 0.3 N(0, 1) + 0.7 N(1, 1) ad Z 0.3 N(0, 1) + 0.7 N(1, 1) 0.1. We geerate 1000 bootstrap replicates for each sample data geerated from the distributios. Ad we simulate the 5000 sample data sets usig Mote Carlo simulatio. I the Tables, JK, PerB, Boot-t ad BC represet the methods of the Jackkife, Efro s percetile, bootstrap t ad bias-corrected bootstrap methods, respectively, ad CP, AL ad SP mea empirical coverage probabilities, average legths ad statistical powers, respectively. 3.1. Coverage probabilities The coverage probability of a cofidece iterval meas the probability of the evets which a give cofidece iterval cotais the true value of iterest or the estimated critical value of the test statistic. The estimated cofidece iterval targets cotaiig the ukow value duratio with a give probability. The probability i coverage probability ca be iterpreted with respect to a set of hypothetical repetitios of the etire data collectio ad aalysis procedure. I these hypothetical repetitios, idepedet data sets followig the same probability distributio as the actual data are cosidered, ad a cofidece iterval is computed from each of these data sets. 3.2. Statistical power The power i statistical hypothesis testig is the probability that the test successfully rejects the ull hypothesis whe the alterative hypothesis is true. See for istace Choi (2017). Ad the statistical power ca i geeral be related to type 2 error, which is the probability of false egative to accept the ull hypothesis whe the alterative hypothesis is true. Therefore, the statistical power ca i geeral be expressed as a fuctio of the possible distributios uder the alterative hypothesis. Power aalysis ca be used to calculate the miimum sample size required so that oe ca be reasoably likely to detect a effect of a give size ad to achieve the give statistical power. 4. Results We summarize i the followig Tables 4.1 4.6 the computatioal comparisos betwee four resamplig modificatios for the Bagai statistics of various competig risk distributios i terms of coverage probabilities ad statistical powers. Table 1, 2 ad 3 preset coverage probabilities of 90%, 95% ad 99% cofideces for 5 differet models from (C1) to (C5), respectively. Table 4, 5 ad 6 preset empirical statistical power of 10%, 5% ad 1% levels of sigificace for (P1) N(0.5, 10 2 ) vs N(0, 10), (P2) LogN(0, 1) vs LogN(0, 1) 0.1, (P3)
492 Youg Mi Kim Hyug-Tae Ha t 1 vs t 1 0.1, (P4) 0.5 N(0, 1) + 0.5 N(1, 1) vs 0.5 N(0, 1) + 0.5 N(1, 1) 0.1, ad (P5) 0.3 N(0, 1) + 0.7 N(1, 1) vs 0.3 N(0, 1) + 0.7 N(1, 1) 0.1, respectively. From these tables we observe the followig poits: The bias-corrected bootstrap method cosistetly outperforms other resamplig methods i various samplig cases of the Bagai statistic. It may be advisable to utilize bias-corrected bootstrap method for the symmetric distributio-free test for stochastic domiace, especially whe the sample size is less tha 50 data poits i coectio with the Bagai statistic. The best performace of the bias-corrected bootstrap method amog other resamplig methods may ot chage from the types of the origial uderlyig distributios or the levels of sigificace. The bias-corrected bootstrap method outperforms i both cases of coverage probability ad statistical power of the Bagai statistic. The out-performace of the bias-corrected bootstrap method becomes greater that ay other resamplig methods whe the sample sizes are very small, say less tha 20. As the sample size grows, the differece betwee resamplig methods is dimiished ad, i our examples, the relative performace differece of four resamplig methods become less tha 1% whe sample sizes are over 70. It should be oted that while, i geeral, the bootstrap-t method has the small order of coverage probabilities, it may ot be advisable to use the bootstrap-t method whe the sample size is large eough. It is, as metioed i the origial paper of Efro (1982), because accurate bootstrap variace estimatio ca be achieved whe sample size is large eough. The performace of the bias-corrected bootstrap method for small level of sigificaces like 10% to 1% i both cases of coverage probability ad statistical powers may ot be satisfactory for whe the small sample is extremely small, say = 5. It may be because of the difficulty of the techique to correct bias of the symmetricity of the Bagai statistic from small sample data poits. I coclusio, the bias-corrected bootstrap method may be most recommeded amog may other typical resamplig methods i terms of coverage probability ad statistical power whe distributio-free statistics for testig oe way stochastic domiace based o Bagai statistic with the properties of symmetry ad asymptotic ormality are iferred uder small sample situatio. Although the typical percetile ad t bootstrap methods are the most utilized resamplig methods to be combied with distributio free tests due to their theoretical asymptotic covergece rates of O p ( 1/2 ) ad O p ( 1 ), respectively, the biascorrected bootstrap method should also get attetios to practitioers.
Resamplig for Bagai test 493 Table 4.1 Coverage probabilities of 90% cofideces for 5 differet models from (C1) to (C5) JK PerB Boot-t BC CP AL CP AL CP AL CP AL 5 0.7730 9.9 0.8040 13.2 0.7112 9.0 0.9368 20.3 10 0.8468 44.4 0.8792 53.9 0.8484 45.6 0.9152 57.1 15 0.8582 91.9 0.8748 103.0 0.8540 92.7 0.9036 105.7 C1 20 0.8714 147.5 0.8886 161.3 0.8716 149.2 0.9040 163.4 30 0.8812 287.9 0.8912 305.1 0.8818 290.0 0.8994 306.9 50 0.8940 654.0 0.9018 677.9 0.8936 658.1 0.9054 678.9 70 0.8892 1089.6 0.8938 1119.2 0.8894 1095.7 0.8976 1118.9 100 0.8856 1871.1 0.8868 1907.1 0.8844 1879.6 0.8930 1903.3 5 0.7894 10.7 0.8194 14.1 0.7264 9.6 0.9390 20.7 10 0.8486 45.0 0.8852 54.5 0.8512 46.3 0.9182 57.6 15 0.8588 92.2 0.8748 103.2 0.8538 92.9 0.9046 105.9 C2 20 0.8630 149.1 0.8782 162.8 0.8648 150.6 0.8940 165.0 30 0.8824 288.4 0.8906 305.8 0.8828 290.7 0.9012 307.9 50 0.8936 662.0 0.8998 685.4 0.8936 665.8 0.9058 686.5 70 0.8860 1091.8 0.8916 1121.0 0.8862 1097.4 0.8956 1121.5 100 0.8938 1871.9 0.8964 1908.5 0.8936 1880.7 0.8990 1908.5 5 0.7790 10.2 0.8134 13.6 0.7218 9.4 0.9402 20.5 10 0.8378 44.4 0.8738 53.9 0.8362 45.7 0.9108 57.2 15 0.8626 90.9 0.8778 101.9 0.8562 91.6 0.9062 104.6 C3 20 0.8852 154.2 0.9008 168.1 0.8852 155.9 0.9130 170.0 30 0.8862 291.7 0.8952 308.8 0.8860 293.7 0.9082 310.4 50 0.8796 643.3 0.8866 666.8 0.8800 647.4 0.8920 666.2 70 0.8842 1079.9 0.8898 1108.5 0.8848 1085.1 0.8926 1110.1 100 0.8948 1884.2 0.8992 1919.8 0.8962 1892.4 0.9034 1919.2 5 0.7848 10.5 0.8160 13.9 0.7236 9.5 0.9380 20.7 10 0.8394 44.5 0.8788 54.0 0.8396 45.6 0.9166 57.1 15 0.8584 91.9 0.8762 102.9 0.8538 92.6 0.9014 105.5 C4 20 0.8750 149.0 0.8910 162.9 0.8734 150.8 0.9056 164.8 30 0.8838 293.6 0.8914 310.9 0.8826 295.8 0.9036 312.8 50 0.8930 648.1 0.8966 671.7 0.8906 652.0 0.9036 672.7 70 0.8954 1087.5 0.8984 1116.1 0.8948 1093.0 0.9056 1117.3 100 0.8944 1869.8 0.8962 1905.0 0.8936 1876.8 0.8988 1907.4 5 0.7778 10.3 0.8134 13.7 0.7178 9.4 0.9336 20.5 10 0.8468 44.2 0.8832 53.7 0.8474 45.5 0.9194 56.9 15 0.8682 94.1 0.8836 105.0 0.8638 94.8 0.9104 107.5 C5 20 0.8768 151.7 0.8900 165.7 0.8758 153.4 0.9028 167.3 30 0.8794 287.0 0.8888 304.2 0.8780 289.1 0.8986 306.0 50 0.8890 662.1 0.8976 685.3 0.8906 665.7 0.8988 686.6 70 0.8878 1093.5 0.8938 1122.4 0.8886 1099.2 0.8978 1121.9 100 0.8876 1896.7 0.8910 1933.8 0.8868 1905.5 0.8928 1932.2
494 Youg Mi Kim Hyug-Tae Ha Table 4.2 Coverage probabilities of 95% cofideces for for 5 differet models from (C1) to (C5) JK PerB Boot-t BC CP AL CP AL CP AL CP AL 5 0.8406 12.9 0.8406 19.8 0.7466 13.4 0.9374 23.3 10 0.8902 57.6 0.902 68.1 0.8706 57.8 0.9510 71.9 15 0.915 116.6 0.9292 130.4 0.909 117.3 0.9522 132.8 C1 20 0.9206 191.3 0.9298 208.0 0.9138 192.5 0.9522 210.0 30 0.9298 371.5 0.936 392.8 0.9278 373.6 0.9494 394.4 50 0.9314 825.0 0.9364 854.4 0.931 829.3 0.9430 855.0 70 0.9416 1425.7 0.9448 1465.0 0.9412 1434.7 0.9492 1461.7 100 0.9416 2453.2 0.9434 2501.5 0.9414 2466.8 0.9454 2498.0 5 0.8350 12.9 0.8350 19.9 0.7430 13.5 0.9422 23.3 10 0.8990 57.8 0.9102 68.2 0.8788 57.9 0.9538 72.0 15 0.9146 118.9 0.9300 132.9 0.9102 119.8 0.9514 135.1 C2 20 0.9224 191.1 0.9290 207.9 0.9160 192.3 0.9514 209.6 30 0.9278 362.1 0.9328 383.8 0.9234 364.4 0.9462 386.1 50 0.9352 829.3 0.9374 860.2 0.9332 834.6 0.9464 861.1 70 0.9444 1424.3 0.9472 1460.8 0.9428 1431.5 0.9520 1459.1 100 0.9424 2426.4 0.9444 2477.4 0.9434 2440.8 0.9486 2472.1 5 0.8414 13.1 0.8414 20.0 0.7456 13.6 0.9356 23.3 10 0.8980 56.9 0.9088 67.1 0.8758 56.8 0.9580 71.2 15 0.9170 120.0 0.931 134.1 0.9132 120.9 0.9524 136.1 C3 20 0.9292 192.9 0.9356 210.0 0.9214 194.4 0.9528 211.2 30 0.9376 377.3 0.9416 399.2 0.9338 379.7 0.9522 399.3 50 0.9364 834.8 0.9402 864.1 0.9354 839.3 0.9442 865.0 70 0.9406 1395.5 0.9434 1435.2 0.9396 1405.3 0.9476 1431.7 100 0.9446 2436.1 0.9492 2488.1 0.9458 2453.1 0.9514 2479.9 5 0.8462 13.4 0.8462 20.4 0.7580 14.0 0.9408 23.5 10 0.8906 57.2 0.9010 67.4 0.8714 57.1 0.9498 71.3 15 0.9146 118.0 0.9308 131.8 0.9102 118.7 0.9544 134.1 C4 20 0.9200 191.8 0.9272 208.6 0.9142 193.0 0.9492 210.4 30 0.9330 371.9 0.9390 393.7 0.9290 374.5 0.9492 394.5 50 0.9392 835.2 0.9434 866.3 0.9388 841.0 0.9476 865.6 70 0.9406 1397.8 0.9452 1436.7 0.9414 1406.4 0.9470 1434.5 100 0.9448 2451.0 0.9462 2499.5 0.9446 2464.5 0.9490 2496.0 5 0.8466 13.1 0.8466 20.2 0.7582 14.0 0.9392 23.5 10 0.8946 58.0 0.9044 68.5 0.8744 58.1 0.9518 71.9 15 0.9158 119.2 0.9312 133.1 0.9104 119.9 0.9550 135.5 C5 20 0.9192 189.7 0.9282 206.5 0.9106 190.9 0.9502 208.5 30 0.9346 371.3 0.9401 393.0 0.9338 373.6 0.9526 394.8 50 0.9356 827.1 0.9404 857.7 0.9352 832.6 0.9444 857.6 70 0.9410 1422.0 0.9434 1461.0 0.9402 1430.5 0.9492 1460.1 100 0.9478 2463.6 0.9500 2511.9 0.9470 2476.4 0.9532 2508.1
Resamplig for Bagai test 495 Table 4.3 Coverage probabilities of 99% cofideces for 5 differet models from (C1) to (C5) JK PerB Boot-t BC CP AL CP AL CP AL CP AL 5 0.8702 18.2 0.8416 23.1 0.7452 15.6 0.9376 26.6 10 0.9454 81.0 0.9370 92.5 0.9232 78.3 0.9884 95.4 15 0.9678 167.9 0.9648 184.1 0.9552 165.8 0.9890 185.2 C1 20 0.9724 272.9 0.9686 292.9 0.9630 271.2 0.9872 293.2 30 0.9792 527.3 0.9782 553.8 0.9756 526.7 0.9880 551.8 50 0.9856 1180.3 0.9854 1219.0 0.9844 1183.1 0.9904 1212.7 70 0.9890 1978.4 0.9880 2030.2 0.9862 1987.9 0.9918 2017.0 100 0.9852 3426.7 0.9852 3492.0 0.9846 3441.8 0.9880 3470.2 5 0.8808 18.9 0.8434 24.1 0.7552 16.4 0.9430 26.8 10 0.9462 81.2 0.9380 92.2 0.9188 78.2 0.9884 95.8 15 0.9690 167.8 0.9658 183.7 0.9556 165.4 0.9892 185.4 C2 20 0.9716 270.8 0.9696 291.1 0.9614 269.2 0.9884 291.5 30 0.9804 523.5 0.9774 549.7 0.9720 522.5 0.9886 548.3 50 0.9838 1191.5 0.9826 1230.5 0.9804 1194.7 0.9884 1222.8 70 0.9818 1976.4 0.9810 2025.3 0.9802 1983.3 0.9864 2015.1 100 0.9888 3423.7 0.9886 3492.5 0.9868 3442.3 0.9900 3466.8 5 0.8824 19.0 0.8544 24.3 0.7678 16.6 0.9436 26.9 10 0.9458 81.0 0.9390 92.2 0.9210 78.1 0.9860 95.5 15 0.9652 167.0 0.9632 182.9 0.9528 164.6 0.9858 184.5 C3 2 0.9748 274.6 0.9732 294.7 0.9656 273.1 0.9876 294.8 30 0.9760 525.9 0.9752 553.4 0.9690 525.8 0.9866 551.2 50 0.9846 1183.8 0.9826 1222.6 0.9804 1186.6 0.9886 1214.5 70 0.9866 2002.8 0.9866 2053.3 0.9852 2010.8 0.9898 2037.4 100 0.9866 3445.5 0.9848 3513.5 0.9836 3462.9 0.9882 3484.0 5 0.8832 18.9 0.8510 24.0 0.7582 16.3 0.9436 26.8 10 0.9478 80.3 0.9412 91.4 0.9198 77.4 0.9888 95.0 15 0.9678 168.6 0.9658 184.8 0.9534 166.4 0.9868 185.7 C4 20 0.9740 272.1 0.9728 291.9 0.9628 270.2 0.9880 292.2 30 0.9820 531.8 0.9812 558.8 0.9770 531.5 0.9894 556.2 50 0.9822 1176.2 0.9828 1215.5 0.9798 1179.8 0.9878 1206.0 70 0.9866 2005.3 0.9864 2057.0 0.9860 2015.1 0.9890 2038.9 100 0.9906 3474.1 0.9902 3540.8 0.9896 3492.1 0.9910 3510.0 5 0.8746 18.6 0.8408 23.6 0.7356 16.0 0.9346 26.5 10 0.9550 82.0 0.9472 93.5 0.9228 79.4 0.9884 96.5 15 0.9680 168.3 0.9660 184.7 0.9552 166.4 0.9870 185.7 C5 20 0.9698 272.3 0.9670 292.3 0.9578 270.6 0.9848 292.6 30 0.9824 529.5 0.9818 556.9 0.9776 529.3 0.9912 554.2 50 0.9846 1197.1 0.9842 1234.9 0.9806 1199.3 0.9890 1228.4 70 0.9882 2010.5 0.9874 2062.1 0.9860 2019.0 0.9908 2045.9 100 0.9858 3437.8 0.9854 3501.8 0.9848 3452.2 0.9880 3477.1
496 Youg Mi Kim Hyug-Tae Ha Table 4.4 Empirical statistical power of 10% level of sigificace for (P1) N(0.5, 10 2 ) vs N(0, 10), (P2) LogN(0, 1) vs LogN(0, 1) 0.1, (P3) t 1 vs t 1 0.1, (P4) 0.5 N(0, 1) + 0.5 N(1, 1) vs 0.5 N(0, 1) + 0.5 N(1, 1) 0.1, ad (P5) 0.3 N(0, 1) + 0.7 N(1, 1) vs 0.3 N(0, 1) + 0.7 N(1, 1) 0.1 JK PerB Boot-t BC CP AL CP AL CP AL CP AL 5 0.7974 11.0 0.8270 14.3 0.7280 9.9 0.9304 20.7 10 0.8590 48.1 0.8956 57.6 0.8588 49.5 0.9248 60.7 15 0.8792 101.2 0.8928 112.4 0.8766 102.1 0.9176 114.3 P1 20 0.8958 166.9 0.9092 181.1 0.8948 168.9 0.9218 182.1 30 0.9054 325.6 0.9160 343.3 0.9070 328.2 0.9230 343.5 50 0.9200 765.6 0.9244 790.5 0.9198 770.9 0.9284 789.3 70 0.9316 1324.6 0.9360 1356.3 0.9322 1332.6 0.9356 1351.1 100 0.9346 2326.9 0.9360 2364.1 0.9348 2335.4 0.9368 2359.7 5 0.8414 13.2 0.8772 16.9 0.7788 11.9 0.9342 21.5 10 0.9072 57.6 0.9350 67.5 0.9074 59.2 0.9538 68.5 15 0.9200 120.7 0.9318 132.9 0.9180 122.4 0.9466 131.2 P2 20 0.9400 203.5 0.9504 218.9 0.9402 206.5 0.9582 215.9 30 0.9560 417.0 0.9592 436.6 0.9556 421.4 0.9636 430.7 50 0.9706 996.3 0.9738 1023.8 0.9712 1003.9 0.9756 1012.3 70 0.9764 1778.9 0.9776 1814.5 0.9760 1790.3 0.9776 1797.8 100 0.9850 3286.0 0.9856 3330.5 0.9850 3301.5 0.9864 3303.9 5 0.8010 11.3 0.8324 14.8 0.7440 10.2 0.9420 20.9 10 0.8746 49.7 0.9038 59.2 0.8728 51.0 0.9318 62.0 15 0.8860 101.4 0.9014 112.5 0.8810 102.2 0.9238 114.5 P3 20 0.8960 168.2 0.9138 182.3 0.8970 170.0 0.9272 183.2 30 0.9162 333.0 0.9234 350.8 0.9154 335.5 0.9318 350.6 50 0.9296 774.4 0.9324 798.0 0.9292 778.6 0.9372 796.4 70 0.9370 1349.9 0.9402 1380.4 0.9374 1357.0 0.9422 1375.4 100 0.9468 2406.7 0.9500 2443.8 0.9470 2415.5 0.9502 2436.2 5 0.8196 12.1 0.8462 15.6 0.7538 10.9 0.9364 21.4 10 0.8822 52.1 0.9090 61.7 0.8836 53.5 0.9348 64.0 15 0.8976 108.5 0.9100 119.8 0.8954 109.5 0.9322 120.8 P4 20 0.9152 176.6 0.9280 191.1 0.9156 178.9 0.9378 190.7 30 0.9342 370.4 0.9412 388.4 0.9334 373.3 0.9462 386.6 50 0.9426 848.3 0.9460 873.6 0.9430 853.7 0.9490 869.3 70 0.9498 1491.7 0.9512 1522.4 0.9500 1499.1 0.9518 1515.8 100 0.9588 2710.2 0.9608 2751.0 0.9588 2722.7 0.9622 2740.7 5 0.8622 14.4 0.8832 18.0 0.7840 12.5 0.9242 21.9 10 0.9338 65.8 0.9536 75.5 0.9290 66.7 0.9672 75.8 15 0.9530 140.5 0.9608 152.6 0.9508 142.3 0.9692 150.5 P5 20 0.9732 242.8 0.9802 258.0 0.9730 245.7 0.9832 253.9 30 0.9790 497.6 0.9818 516.8 0.9786 501.9 0.9846 509.5 50 0.9896 1227.5 0.9912 1254.7 0.9898 1235.0 0.9914 1240.0 70 0.9944 2264.3 0.9946 2300.2 0.9946 2276.4 0.9948 2276.3 100 0.9966 4251.6 0.9968 4295.5 0.9966 4267.7 0.9968 4262.4
Resamplig for Bagai test 497 Table 4.5 Empirical statistical power of 5% level of sigificace for 5 differet models from (P1) to (P5) JK PerB Boot-t BC CP AL CP AL CP AL CP AL 5 0.8564 13.9 0.8564 21.0 0.7600 14.3 0.9350 23.5 10 0.9080 61.3 0.9176 71.9 0.8896 61.6 0.9600 74.8 15 0.9340 129.0 0.9458 143.4 0.9302 130.2 0.9650 144.2 P1 20 0.9440 210.9 0.9508 228.6 0.9394 213.0 0.9648 228.4 30 0.9538 411.2 0.9580 434.1 0.9510 414.5 0.9664 432.4 50 0.9548 943.1 0.9588 975.4 0.9546 950.1 0.9632 970.2 70 0.9626 1615.2 0.9640 1656.3 0.9618 1626.1 0.9664 1648.0 100 0.9692 2882.5 0.9696 2936.2 0.9688 2900.6 0.9718 2924.4 5 0.8912 15.8 0.8912 23.3 0.7896 16.2 0.9342 24.2 10 0.9390 68.6 0.9484 80.6 0.9234 69.8 0.9716 79.8 15 0.9582 146.3 0.9664 162.2 0.9564 148.9 0.9774 158.8 P2 20 0.9660 241.9 0.9704 262.0 0.9626 246.2 0.9802 255.5 30 0.9752 491.4 0.9782 517.6 0.9750 498.0 0.9824 507.5 50 0.9854 1171.8 0.9870 1210.1 0.9846 1184.7 0.9878 1190.1 70 0.9892 2059.6 0.9890 2109.6 0.9884 2079.2 0.9894 2078.6 100 0.9940 3820.6 0.9948 3885.4 0.9940 3850.6 0.9950 3839.8 5 0.8590 14.1 0.8590 21.2 0.7678 14.5 0.9362 23.6 10 0.9116 60.8 0.9180 71.2 0.8892 60.7 0.9614 74.2 15 0.9300 126.6 0.9408 140.4 0.9240 127.3 0.9560 141.7 P3 20 0.9434 208.7 0.9496 225.7 0.9382 210.1 0.9654 225.8 30 0.9572 419.1 0.9612 440.5 0.9520 421.0 0.9688 439.1 50 0.9652 972.6 0.9664 1001.9 0.9636 976.6 0.9716 997.0 70 0.9692 1658.3 0.9712 1694.9 0.9670 1665.2 0.9728 1686.0 100 0.9708 2881.7 0.9714 2928.8 0.9710 2892.5 0.9736 2913.2 5 0.8676 14.6 0.8676 21.7 0.7780 14.9 0.9408 23.9 10 0.9236 64.3 0.9286 74.9 0.9004 64.3 0.9672 76.9 15 0.9402 135.7 0.9502 150.0 0.9346 136.9 0.9694 149.9 P4 20 0.9542 220.3 0.9598 237.6 0.9504 222.2 0.9734 236.2 30 0.9620 441.5 0.9662 463.8 0.9596 444.5 0.9744 459.2 50 0.9732 1041.3 0.9748 1072.7 0.9724 1047.0 0.9796 1063.4 70 0.9816 1801.0 0.9828 1838.7 0.9810 1808.9 0.9840 1825.0 100 0.9838 3249.5 0.9832 3300.0 0.9830 3263.7 0.9848 3275.4 5 0.9106 17.6 0.9106 25.2 0.8032 17.2 0.9210 24.4 10 0.9642 78.9 0.9664 90.9 0.9496 79.8 0.9834 88.6 15 0.9736 166.9 0.9782 182.6 0.9706 169.2 0.9854 177.9 P5 20 0.9834 279.1 0.9858 298.7 0.9810 283.1 0.9922 290.8 30 0.9920 580.6 0.9926 606.4 0.9910 586.9 0.9942 592.4 50 0.9978 1413.8 0.9978 1450.5 0.9972 1425.2 0.9984 1424.6 70 0.9982 2563.3 0.9984 2608.9 0.9982 2579.4 0.9988 2572.5 100 0.9994 4799.0 0.9994 4861.1 0.9994 4824.7 0.9992 4809.7
498 Youg Mi Kim Hyug-Tae Ha Table 4.6 Empirical statistical power of 1% level of sigificace for 5 differet models from (P1) to (P5) JK PerB Boot-t BC CP AL CP AL CP AL CP AL 5 0.8896 19.8 0.8634 25.3 0.7760 17.4 0.9422 26.9 10 0.9574 84.7 0.9502 96.9 0.9306 82.6 0.9924 98.2 15 0.9702 175.5 0.9682 192.6 0.9574 174.1 0.9898 191.3 P1 20 0.9804 288.9 0.9782 310.6 0.9704 288.7 0.9914 306.7 30 0.9856 565.0 0.9850 594.1 0.9814 566.2 0.9918 586.7 50 0.9908 1292.5 0.9908 1335.0 0.9886 1298.8 0.9948 1316.6 70 0.9914 2199.6 0.9910 2256.0 0.9896 2213.9 0.9934 2230.4 100 0.9956 3893.3 0.9960 3967.6 0.9954 3915.2 0.9970 3929.4 5 0.9122 21.3 0.8880 27.7 0.7884 19.0 0.9354 26.9 10 0.9716 93.6 0.9650 108.5 0.9530 93.6 0.9956 103.2 15 0.9854 194.2 0.9832 215.3 0.9764 196.5 0.9948 205.3 P2 20 0.9918 324.6 0.9908 351.3 0.9864 329.1 0.9966 336.1 30 0.9940 647.2 0.9936 683.7 0.9920 655.9 0.9968 658.6 50 0.9976 1509.8 0.9976 1567.3 0.9970 1530.5 0.9988 1519.0 70 0.9982 2654.4 0.9986 2729.1 0.9980 2686.3 0.9984 2658.5 100 0.9990 4823.6 0.9988 4925.6 0.9986 4875.0 0.9992 4818.9 5 0.8794 19.2 0.8494 24.6 0.7496 16.7 0.9338 26.6 10 0.9582 85.5 0.9520 97.8 0.9314 83.6 0.9912 98.7 15 0.9732 178.9 0.9710 196.1 0.9628 177.6 0.9926 194.3 P3 2 0.9828 291.7 0.9820 313.3 0.9740 291.6 0.9926 309.3 30 0.9856 570.3 0.9854 599.4 0.9810 572.0 0.9924 592.0 50 0.9932 1300.4 0.9928 1342.7 0.9922 1307.1 0.9952 1323.7 70 0.9920 2215.5 0.9916 2271.6 0.9908 2228.8 0.9940 2241.9 100 0.9962 3949.3 0.9960 4021.9 0.9958 3971.2 0.9960 3974.7 5 0.9006 20.3 0.8732 26.0 0.7760 17.7 0.9376 26.9 10 0.9588 87.8 0.9524 100.6 0.9352 86.3 0.9908 100.1 15 0.9774 185.2 0.9754 203.8 0.9686 185.2 0.9922 199.5 P4 20 0.9852 300.6 0.9842 323.5 0.9784 301.4 0.9940 317.4 30 0.9902 598.0 0.9902 628.9 0.9878 601.4 0.9948 617.2 50 0.9926 1370.5 0.9924 1416.5 0.9924 1380.5 0.9952 1392.9 70 0.9958 2394.3 0.9958 2456.0 0.9954 2414.5 0.9968 2415.0 100 0.9970 4240.8 0.9962 4322.1 0.9958 4272.0 0.9970 4260.8 5 0.9312 23.0 0.9138 29.9 0.8128 20.5 0.9234 26.8 10 0.9834 101.2 0.9794 116.9 0.9664 101.7 0.9928 108.4 15 0.9934 216.6 0.9918 239.1 0.9886 220.3 0.9980 225.0 P5 20 0.9940 361.3 0.9938 390.0 0.9924 367.8 0.9994 369.6 30 0.9976 732.1 0.9972 771.8 0.9956 744.2 0.9988 738.5 50 0.9992 1762.4 0.9996 1824.0 0.9994 1787.9 0.9998 1762.6 70 0.9998 3154.9 1.0000 3233.3 0.9998 3191.5 1.0000 3146.8 100 0.9994 5786.9 0.9992 5901.5 0.9992 5849.0 0.9994 770.8
Resamplig for Bagai test 499 Refereces Bagai, I., Deshpade, J. V. ad Kochar, S. C. (1989). Distributio free tests for stochastic orderig i the competig risks model. Biometrika, 76, 775-781. Carpeter, J. ad Bithell, J. (2000). Bootstrap cofidece itervals: Whe, which, what? A practical guide for medical statisticias. Statistics i Medicie, 19, 1141-1164. Choi Y.-H. (2017). Power compariso for 3 3 split plot factorial desig. Joural of the Korea Data & Iformatio Sciece Society, 28, 143-152. Diciccio, T. J. ad Romao, J. P. (1988). A review of bootstrap cofidece itervals. Joural of the Royal Statistical Society B, 50, 338-354. Efro, B. (1979). Bootstrap methods: Aother look at the jackkife. The Aals of Statistics, 7, 1-26. Efro, B. (1981). No-parametric stadard errors ad cofidece itervals. Caadia Joural of Statistics, 9, 139-172. Efro, B. (1982). The jackkife, the bootstrap ad other resamplig plas. I Regioal Coferece Series i Applied Mathematics, 38, SIAM, Philadelphia. Efro, B. (1985). Bootstrap cofidece itervals for a class of parametric problems. Biometrika, 72, 45-58. Efro, B. (1987). Better bootstrap cofidece itervals. Joural of the America Statistical Associatio, 82, 171-185. Efro, B. ad Tibshirai, R. (1986). Bootstrap methods for stadard errors, cofidece itervals, ad other measures of statistical accuracy. Statistical Sciece, 1, 54-77. Hall, P. ad va Keilegom, I. (2005). Testig for mootoe icreasig hazard rate. The Aals of Statistics, 33, 1109-1137. Heathcote, A, Brow, S., Wagemakers, E. J. ad Eidels, A. (2010). Distributio-free tests of stochastic domiace for small samples. Joural of Mathematical Psychology, 54, 454-463. Kim, Y.-J. ad Kwo D.-Y. (2017). Nopararmetric estimatio for iterval cesored competig risk data. Joural of the Korea Data & Iformatio Sciece Society, 28, 947-955. Levy, H. (1992). Stochastic domiace ad expected utility. Maagemet Sciece, 38, 555-593. Shao, J. ad Tu, D. (1955). The jackkife ad bootstrap, Spriger.