Resampling modifications for the Bagai test

Transcription

1 Joural of the Korea Data & Iformatio Sciece Society 2018, 29(2), 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 ). 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 Korea. htha@gacho.ac.kr

2 486 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 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

3 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) 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

4 488 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 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.

5 Resamplig for Bagai test 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) 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 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 )),.

6 490 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) N(1.1) ad 5. C5 : Y ad Z 0.3 N(0, 1) 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:

7 Resamplig for Bagai test 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) N(1.1) ad Z 0.5 N(0, 1) N(1.1) 0.1 ad 5. P5 : Y 0.3 N(0, 1) N(1, 1) ad Z 0.3 N(0, 1) 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 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 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 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)

8 492 Youg Mi Kim Hyug-Tae Ha t 1 vs t 1 0.1, (P4) 0.5 N(0, 1) N(1, 1) vs 0.5 N(0, 1) N(1, 1) 0.1, ad (P5) 0.3 N(0, 1) N(1, 1) vs 0.3 N(0, 1) 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.

9 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 C C C C C

10 494 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 C C C C C

11 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 C C C C C

12 496 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) N(1, 1) vs 0.5 N(0, 1) N(1, 1) 0.1, ad (P5) 0.3 N(0, 1) N(1, 1) vs 0.3 N(0, 1) N(1, 1) 0.1 JK PerB Boot-t BC CP AL CP AL CP AL CP AL P P P P P

13 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 P P P P P

14 498 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 P P P P P

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