Statistical Test for Multi-dimensional Uniformity

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1 Florida Iteratioal Uiversity FIU Digital Commos FIU Electroic Theses ad Dissertatios Uiversity Graduate School Statistical Test for Multi-dimesioal Uiformity Tieyog Hu Florida Iteratioal Uiversity, DOI: /etd.FI20504 Follow this ad additioal works at: Recommeded Citatio Hu, Tieyog, "Statistical Test for Multi-dimesioal Uiformity" (20). FIU Electroic Theses ad Dissertatios This work is brought to you for free ad ope access by the Uiversity Graduate School at FIU Digital Commos. It has bee accepted for iclusio i FIU Electroic Theses ad Dissertatios by a authorized admiistrator of FIU Digital Commos. For more iformatio, please cotact dcc@fiu.edu.

2 FLORIDA INTERNATIONAL UNIVERSITY Miami, Florida STATISTICAL TEST FOR MULTI-DIMENSIONAL UNIFORMITY A thesis submitted i partial fulfillmet of the requiremets for the degree of MASTER OF SCIENCE i STATISTICS by Tieyog Hu 20

3 To: Dea Keeth Furto College of Arts ad Scieces This thesis, writte by Tieyog Hu ad etitled Statistical Test for Multi-dimesioal Uiformity, havig bee approved i respect to style ad itellectual cotet, is referred to you for judgmet. We have read this thesis ad recommed that it be approved. Seh Gulati Gauri L. Ghai Zhemi Che, Major Professor Date of Defese: November 0, 20 The thesis of Tieyog Hu is approved. Dea Keeth Furto College of Arts ad Scieces Dea Lakshmi N. Reddi Uiversity Graduate School Florida Iteratioal Uiversity, 20 ii

4 ACKNOWLEDGMENTS I would like to express my sicere thaks to Dr. Zhemi Che for givig me the opportuity to pursue a Master's degree ad for his support, guidace ad patiece throughout this research. I would like to thak all those who gave me help to complete this thesis. I particular, I would like to thak Dr. Seh Gulati ad Dr. Guari L. Ghai for their time to serve o my committee. Fially, I am eterally grateful to my wife, Tig, for her costat love ad stregth throughout the years. iii

5 ABSTRACT OF THE THESIS STATISTICAL TEST FOR MULTI-DIMENSIONAL UNIFORMITY by Tieyog Hu Florida Iteratioal Uiversity, 20 Miami, Florida Professor Zhemi Che, Major Professor Testig uiformity i the uivariate case has bee studied by may researchers. May papers have bee published o this issue, whereas the multi-dimesioal uiformity test seems to have received less attetio i the literature. A ew test statistic for the multi-dimesioal uiformity is proposed i this thesis. The proposed test statistic ca be used to test whether a uderlyig multivariate probability distributio differs from a multi-dimesioal uiform distributio. Some importat properties of the proposed test statistic are discussed. As a special case, the bivariate test statistic is discussed i detail ad the critical values of test statistic are obtaied. By performig Mote Carlo simulatio, the power of the ew test is compared with the Distace to Boudary test, which was a recetly proposed statistical test for multi-dimesioal uiformity by Berredero, Cuevas ad Vazquez-Grade (2006). It has bee show that the test proposed i this thesis is more powerful tha the Distace to Boudary test i some cases. iv

6 TABLE OF CONTENTS CHAPTER PAGE. INTRODUCTION NEW TEST STATISTIC Geeral Case Bivariate Case CRITICAL VALUES AND POWER STUDY Critical Values Power Study CONCLUSION AND DISCUSSION REFERENCES APPENDIX v

7 LIST OF FIGURES FIGURE PAGE Figure. Power Compariso: Alterative Distributio Figure 2. Power Compariso: Alterative Distributio Figure 3. Power Compariso: Alterative Distributio Figure 4. Power Compariso: Alterative Distributio Figure 5. Power Compariso, Alterative Distributio Figure 6. Power Compariso, Alterative Distributio Figure 7. Power Compariso, Alterative Distributio vi

8 . INTRODUCTION Testig uiformity i the uivariate case has bee studied by may researchers. May papers have bee published o this issue, whereas the multi-dimesioal uiformity test seems to have received less attetio i the literature. The problem of testig whether the patter of poits i the multi-dimesioal space is distributed uiformly has applicatios i biology, astroomy, computer sciece ad some other fields. A commoly used goodess-of-fit test for uiformity is the chi-square test proposed by Pearso (900). Theoretically, the chi-square test ca be applied for ay multivariate distributio test. However, the problem for the chi-square test is how to determie the cell limits. Two other well-kow methods are the Kolmogorov-Smirov test proposed by Kolmogorov (933) ad Smirov (939) ad the Cramer-vo Mises test proposed by Cramer (952) ad vo Mises. Those two methods are derived from the empirical distributio ad are widely used for uivariate case. However, Berredero, Cuevas ad Vazquez-Grade (2006) poited out that the probability distributios of those multivariate statistics do ot have distributio free property. Therefore, those two methods are ot applied for high dimesioal distributios. Justel, Pea ad Zamar (997) proposed a multivariate goodess-of-fit test based o the Kolmogorov-Smirov test. By usig Roseblatt s trasformatio, they reduce the Kolmogorov-Smirov test from multivariate case to uivariate case. The test has distributio free property ad ca be applied to ay dimesioal case. However, the computatio of the test is complicated especially for over two dimesios. I their paper, the authors preseted bivariate statistic test (d=2) ad performed power study. It

9 showed that the test is powerful whe sample size is moderately large. I the paper by Liag, Fag, Hicherell ad Li (200), several statistical tests are proposed for testig uiformity i multivariate case. Those tests are use umber-theoretic ad quasi-mote Carlo methods for measurig the discrepacy of the poit i multidimesioal uit. They proved that the test statistics ca be approximated by the stadard ormal distributio N (0,) or the chi-squared distributio χ 2 (2) uder the ull hypothesis that the uderlyig distributio is uiformly distributed. The Mote Carlo simulatio is performed for three special cases: the symmetric discrepacy, the cetered discrepacy ad the star discrepacy. They take the sample size as =25, 50, 00, 200. Sice the sample size is moderately large, the approximatio is quite reasoable. It is show that the power will icrease as the dimesio icreases. Berredero, Cuevas ad Vazquez-Grade (2006) proposed a test based o the distace to the boudary. They defie the distace from a poit X to the boudary as DX (, S) = mi{ x S: x X}. They also defie R as the maximum distace to the boudary which ca be obtaied i support S. The, they get the relative distace from X to the boudary which is Y = D( X, S)/ R. I the paper, they prove that the relative depth will follow the beta distributio with parameters ad p (p is dimesioal size ) if the data are uiformly distributed o compact support. I the two dimesioal case, the relative depth Y will follow the beta distributio Beta (, 2) ad it also makes sese i uivariate case. I uivariate case, the relative depth Y will follow the beta distributio Beta (,) which is also the uiform distributio. The, the test will be reduced from p dimesioal case to oe dimesioal case. Fially they used Kolmogorov-Smirov 2

10 method to test whether the distributio of relative depth Y, which is from uderlyig distributio sample, follows the correspodig beta distributio. By performig the Mote Carlo simulatio, uder the same situatios (same sample size ad same dimesio), the Distace to Boudary test is more powerful i the case of mixture model tha the test proposed by Liag, Fag, Hicherell ad Li (200). The other advatage of the Distace to Boudary test is easy to compute ad easy to explai. Che ad Ye (2009) developed a alterative test for uiformity i uivariate case. I that paper, the authors proposed a ew test statistic based o the order statistics i support set [0, ]. The test statistic proposed i the paper is + ( + ) 2, 2,, = ( ( i) ( i ) ). i= + ( ) G X X X X X By performig the Mote Carlo simulatio, it has bee show that the test is more powerful whe the alterative distributio is V-shape distributio ad whe the sample size is small compared with the Kolmogorov-Smirov test. By usig the probability itegral trasformatio, the uiformity test ca be used to check whether the uderlyig distributio follows ay specified distributio. The idea is adopted i this research to develop a test for the multi-dimesioal case. The mai purpose of this research is to propose a ew test statistic for testig multi-dimesioal uiformity. It has bee show that the ewly proposed test improves the power of the multi-dimesioal uiformity tests. Sice the Distace to Boudary test is a recetly published paper i multivariate uiformity test, the power of the proposed test is compared with the power of the Distace to Boudary test whe 3

11 differet alterative distributios are used. The thesis is orgaized as follow. I Sectio 2, we propose a ew test statistic ad discuss some importat properties of the proposed test statistic. As a special case, the bivariate statistic test will be discussed i detail. I Sectio 3, the performace of the proposed test is evaluated usig Mote Carlo simulatio. Critical values of the ew test statistic are obtaied ad tabulated. Power study is coducted for evaluatig the test statistic. Several differet alterative distributios are used i the power study. The fial coclusios are give i Sectio 4. Mote Carlo simulatio is performed to fid the critical values of the ew test statistic ad to coduct power study. The SAS/IML ad SAS/Base are used for statistical simulatio. 4

12 2. NEW TEST STATISTIC 2. Geeral Case I this research, a ew test statistic is proposed. The basic idea of the proposed test statistic follows the paper of Che ad Ye (2009). The uivariate uiformity test is discussed ad evaluatio of the test statistic is performed by power compariso i that paper. The result is exteded to the multi-dimesioal case. Suppose X X2 X X 2 X 22 X 2 X =, X2 =,, X = X X X k 2k k is a radom sample from a k-dimesioal populatio distributio with support set [ ] ( ) 0, k. Here [ ] ( ) 0, k is the k-dimesioal uit cube which the set is defied as Let X (), X ( 2),, X i i ( ) i t t 2 :0 ti ( i=,2,, k). t k be the ordered values of X, X2,, X ( i =,2,, k). i i i The purpose of the multi-dimesioal uiformity test is to test ) H The populatio distributio is a uiform distributio o [ 0,]( k, 0 : H The populatio distributio is ot uiform distributio o [ ] ( ) a : The test statistics proposed i this research is k k ( + ) ( j) ( j ) ( ) i= i2= ik = j= 0, k. ( i j i j) ( ) Gk ( X, X2,, X) = X X. k k

13 Here it is assumed that X ( 0) = 0 ad j ( ) X + = ( j, 2, k) j uderlyig distributio is a uiform distributio o [ ] ( ) 0, k, the k E ( X X ( i ) ( )) j j ij j = j= ( + ) =. It ca be see that if the ( ) i k j =, 2, + ( j =, 2, k) Therefore, if the value of G ( X, X, X ) k 2. is too far away from zero, it could be a idicatio that the uderlyig distributio is ot uiform distributio o [ ] ( ) 0, k. This motivates the followig test procedure. Uder H 0, let Gk, The 0 ( k ( X, X2,, X) k, ) α PG > G α = α (0 < α < ). be a umber such that H should be rejected at sigificat level α if ( X, X,, X ) be show that G ( X, X,, X ) k 2 is always betwee 0 ad. Gk 2 > Gk, α. It ca 2.2 Bivariate Case The bivariate test statistic is discussed i detail as a special case of the multi-dimesioal. The critical values of the test statistic are obtaied ad tabulated. Power compariso is performed i Sectio 3. The purpose of the bivariate uiformity test is to test H The populatio distributio is a uiform distributio o [ 0,] [ 0,] 0 :, H : The populatio distributio is ot uiform distributio o[ 0,] [ 0,]. a I order to raise the power of the proposed test statistic, the defiitio of ( ) G2 X, X2,, X (938). is modified by adoptig the idea of the Kedall s τ statistic 6

14 Suppose X X X2 X =, 2,, X X = = 2 X X 22 X form a radom sample from a 2 bivariate populatio distributio with support set [ 0,] [ 0,]. X X X X X X i i2 i i2 i i2 ad X X X ad X ad j j2 j j2 X X are said to be cocordat if j j2 are said to be discordat if ( ) X X = 0 X X 0. j2 j i2 i No compariso is made if X =0 2 X. X X X X X X j2 j i2 i X X j2 j i2 i > 0. < 0. are said to be half cocordat ad half discordat if i i Let c be the total umber of cocordat pairs, ad let d be the total umber of discordat pairs. Suppose also that X (), X ( 2 ),, X( ) are the ordered values of X, X2,, X, ad X, X,, X are the ordered values of X 2, X22,, X2. () 2 ( 22 ) ( ) 2 Defie 2 ( + ) ( c d ) () ( ) ( )( ) i= j= G2( X, X2,, X ) = ( X X ) ( X( ) X 2 ( ) 2). i i j j c + d + ( + ) Here it is assumed that X( ) = X 0 ( 02 ) = 0 ad X( ) = X ( 2 ) =. + + It ca be see that if the uderlyig distributio is a uiform distributio o [ 0,] [ 0,], the ( () i ( i ) ) ( j) ( j ) ( ) E X X X X = ( i=, 2,, + ; j =, 2,, + ). + ( ) 2 7

15 Therefore, if the value of G ( X X X ) 2 2,,, is too far away from zero, it could be a idicatio that the uderlyig distributio is ot uiform distributio o [ 0,] [ 0,]. This motivates the followig test procedure. Uder H 0, let The 0 ( 2( X, X2,, X ) ) G α P G > G α = α (0 < α < ). H should be rejected at sigificat level α if G ( X X X ) be a umber such that 2, 2,, > G α. The purpose of icludig the term + c c d + + d is to raise the power of the test whe the two variables of alterative bivariate distributio are correlated. It ca be show that 0 G ( X, X,, X ) for ay X, X, X [ 0,] [ 0,] The lower ad upper bouds of the iequality G ( X X X ) ,,, caot be improved. It meas that oe may costruct bivariate data sets such that the values of ( ) G2 X, X2,, X will reach 0 ad, respectively. 8

16 3. CRITCAL VALUES AND POWER STUDY 3. Critical Values I this research, we focus o the bivariate case of the G ( X X X ),,, test. Critical 2 values of the i bivariate case are obtaied by applyig the Mote Carlo method i SAS/IML. 2 Firstly, a pseudo-radom sample X X X X =, 2,, X X = = 2 X X 22 X is draw 2 from the uiform distributio o the support set [ 0,] [ 0,]. Sice the margial distributios of the bivariate uiform distributio are idepedet ad uiform o [ ] 0,, X, X2,, X ad X2, X22,, X2 ca be geerated by the uivariate uiform distributio. Secodly, the coefficiet of the, 2 ( + ) ( c d + ) ( + 2)( + + ) c d, is calculated by SAS. The we sort the values of X, X2,, X to obtai X, X,, X ad sort the values of X 2, X22,, X2 to obtai () ( 2 ) ( ) X, X,, X. Fially, G value ca be calculated by usig the formula: () 2 ( 22 ) ( ) 2 2 ( + ) ( c d ) () ( ) ( )( ) i= j= G2( X, X2,, X ) = ( X X ) ( X( ) X 2 ( ) 2). i i j j c + d + ( + ) I this research, 00,000 replicatios are used. The the 00,000 calculated G values are sorted i ascedig order. Theoretically, the value of the 95,000 th is treated as the 95 th percetile. For more accuracy, the followig formula is used to calculate the critical values at 95% cofidece level: ( G( ) + G 0.95* rep ( 0.95* rep) ) + G α = Here α = 0.05 is the sigificat level ad rep is the umber of replicatios

17 The critical values of the bivariate are displayed i the Table. The first colum i the table is the sample size (from 5 to 50). Colums 2 to 5 are the critical values for sigificat levelsα = 0.0,0.05,0.025,0.0. I the table, the values are the G test critical value correspodig to the sample size ad sigificat level. The critical values i this research are moderately small, so we keep five decimal places for all the values. 0

18 Table: Critical values of G 2 test statistic α=0.0 α=0.05 α=0.025 α=0.0 G G G G

19 3.2 Power Study The power of a test is the probability of rejectig the ull hypothesis whe the ull hypothesis is false. It meas the power is the probability of makig correct decisio, so the power is desired to be as large as possible. The power of the test statistic metioed i this research is ( 2( X, X2,, X) > α a) P G G H, where α is the sigificat level ad H a is alterative hypothesis. Theoretically, the power of a statistical test depeds o the sample size, the sigificat level ad the sesitivity of the data. I this research we focus o the case that the sample size is less tha or equal to 50. Sigificat level α = 0.0,0.05,0.0 ad several alterative distributios are used i this power study. I this sectio, the performace of the G ( X X X ) 2 2 power study ad the bivariate case of the G ( X X X ),,, test is evaluated by the 2 2,,, test is compared with other existig test statistics. Several alterative distributios are used to evaluate the performace of the G ( X X X ) 2 2,,, test proposed i this research. As metioed before, the recet research results i the multivariate uiformity test foud i the literature are the discrepacy measures tests proposed by Liag, Fag, Hicherell ad Li (200) ad the Distace to Boudary test proposed by Berredero, Cuevas ad Vazquez-Grade (2006). Sice the Distace to Boudary test is better tha the discrepacy measures tests i may cases, the power of the G ( X X X ) 2 2,,, test is compared with the Distace to Boudary test proposed by Berredero, Cuevas ad Vazquez-Grade (2006). 2

20 Let X X X2 X =, 2,, X X = = 2 X X 22 X be the observed sample poits from 2 the uderlyig populatio distributio i the support set [ 0,] [ 0,]. Let X, X,, X be the ordered values of X, X2,, X, ad let () ( 2 ) ( ) X, X,, X be the ordered values of X 2, X22,, X2. The, the value of the () 2 ( 22 ) ( ) 2 ( ) G2 X, X2,, X ca be calculated by the test statistic formula. To test whether or ot the uderlyig populatio distributio is a uiform distributio i the support set [ 0,] [ 0,], the ull ad alterative hypotheses are H The populatio distributio is a uiform distributio o [ 0,] [ 0,] 0 :, H : The populatio distributio is ot uiform distributio o[ 0,] [ 0,]. a If the value of test statistic G ( X X X ) 2 2 G α,,, is greater tha, the ull hypothesis should be reject at the sigificat levelα. Otherwise, the ull hypothesis is G α ot rejected. Here, is the critical value of the ad ca be foud from the Table correspodig to the differet sigificat level. The Distace to Boudary test is proposed by Berredero, Cuevas ad Vazquez-Grade (2006). They defie the distace from a poit X to the boudary as D( X, S) = mi{ x S : x X. They also defie R as the maximum distace to the boudary which ca be obtaied i support S. The, they get the relative distace from X to the boudary which is Y = D( X, S)/ R. The test is based o the real variables Y = D( X, S)/ R, i =,. It is proved i the paper that if X is from the uiform i i distributio, the relative distace Y would follow the beta distributio with parameters α = ad β = p (p is the umber of dimesio). Let G( y ) be the distributio 3

21 fuctio of Y ad G ( y) be the empirical distributio fuctio associated with the sample. The test statistic is sup G( y) H ( y), where H(y) is the distributio of Y uder the ull hypothesis. y The ull hypothesis assumes that the uderlyig distributio is the bivariate uiform distributio ad the alterative hypothesis is that the uderlyig distributio is ot the bivariate uiform distributio. As for the power study, two differet models of the alterative distributios icludig beta distributio ad meta-type uiform distributio are used. Mote Carlo simulatio is used to fid the power of the ad the Distace to Boudary test. Sample sizes =5,0,5,20,25,30,35,40,45,50 are selected to compare the power ad the selected sigificat levelsα = 0.0,0.05,0.0. Based o the pricipal of efficiecy ad accuracy, 00,000 replicatios are performed i simulatio. The procedure for the power calculatio is summarized as follows:. Geerate X X X2 X =, 2,, X X = = 2 X X 22 X samples i the support set 2 [ 0,] [ 0,] follow the alterative distributio. 2. Let X (), X ( 2 ),, X( ) be the ordered values of X, X2,, X, ad let X (), X 2 ( 22 ),, X( ) be the ordered values of X 2 2, X22,, X2. 3. Calculate the value of the statistics. 4. Compare the value of the test statistic with the critical value for specific sigificat level ad determie whether the ull hypothesis is rejected. 4

22 5. Repeat steps to 4 for 00,000 times. 6. Calculate the rejectio rate which is the power for the. The power of the Distace to Boudary test ca be obtaied by performig the same procedure. I this research, three differet models are chose as the choice of the alterative distributio i power study. I the followig subsectios the alterative distributios are described ad the performace of the two tests uder differet alterative distributios is compared Beta Distributio The Beta distributio family is used by authors as a alterative distributio to coduct power study for uivariate uiformity test. By choosig differet shape parameter ad scale parameter, it gives rich ad flexible result i power compariso. The probability desity fuctio of the Beta distributio is displayed as follow: ( ) f x ( ) ( ) ( ) Γα+β = Γ α Γ β β ( ) < < α x x 0 x 0 elsewhere ( α > β > ) 0, 0. To derive a alterative distributio for k-dimesioal case, let Yi be radom variable followig Beta( α, β ) distributio (i =, 2,, k). It is assumed that i i Y,Y, 2,Yk are idepedet. The, the distributio of Y = (Y,Y 2,,Y k) ca be treated as a k-dimesioal alterative distributio. For coveiece, such a distributio is called the k-dimesioal distributio derived from Beta( α, β ), Beta( α2, β 2),, 5

23 Beta( α, β ) distributios. k k Alterative Distributio The bivariate distributio derived from Beta (5, 2) ad Beta (5, 2). The variace of the margial distributio is Alterative Distributio 2 The bivariate distributio derived from Beta (5, ) ad Beta (5, ). The margial distributio is a left-skewed Beta distributio ad the variace of margial distributio is Alterative Distributio 3 The bivariate distributio derived from Beta (0.5, 0.5) ad Beta (0.5, 0.5). The margial distributio is a symmetric Beta distributio ad the variace of margial distributio is Figures -3 show the power comparisos for the ad the Distace to Boudary test i the coditio that the alterative distributios are derived from Beta( α, β ) ad Beta( α2, β 2). Figure shows that the is more powerful tha the Distace to Boudary test if the margial distributio is ot symmetric ad the variace is large. Figure 2 shows that the is more powerful whe sample size is less tha 25 ad with the sample size icreasig the power of two tests close to each other ad almost close to. I this case, the margial distributio is left skewed ad the variace is ot large. Figure 3 shows that the power of the Distace to Boudary test is sigificatly large tha the whe the alterative distributio 3 is used which is margial distributio is symmetric ad the variace is small. 6

24 The symmetric situatio is preseted i the paper of Berredero, Cuevas ad Vazquez-Grade (2006). It has show the power of Distace to Boudary is much high i this case. After chagig the symmetric coditio, the result seems differet Meta-type Uiform Distributio Meta-type uiform distributio is metioed i the papers of Liag, Fag, Hicherell ad Li (200) ad Berredero, Cuevas ad Vazquez-Grade (2006). They itroduce this distributio for the power compariso. The basic idea for creatig meta-type multivariate distributio is as follows. Let the radom vector X = (, ) have a distributio fuctio ( ) X X 2 F( X ) is the margial distributio fuctio of X ad 2( 2) distributio fuctio of 2 F X.It is defied that F X is the margial X. The radom vector Y as Y = ( ( ), 2( 2) ) F X F X ca be obtaied. The ew radom vector Y has those properties: F( X) ad 2( 2) F X are uiform distributed i the support set [0, ] ad the joit distributio is differet from the bivariate uiform distributio sice F( X ) ad 2( 2) F X are ot idepedet. This kid of multivariate distributio is easily geerated by statistical software ad is useful to check the multivariate uiform distributio. Specifically, two kids of the meta-type uiform distributios are cosidered i power study. Alterative Distributio 4 MNU (Meta Normal Uiform): obtaied from bivariate ormal distributio with μ = [ 0,0 ] 0.5 ad Σ = 0.5. For the cosistece of compariso, the same parameters 7

25 are chose as i the paper of Berredero, Cuevas ad Vazquez-Grade (2006). Alterative Distributio 5 MTU (Meta T distributio Uiform): obtaied from bivariate Studet s-t distributio with = [ ] μ 0,0 0.5, Σ = 0.5 ad 5 degree of freedom. The power compariso results uder meta-type uiform distributio are show i the Figure 4 to 5. The is more powerful tha Distace to Boudary test i each case. With the sample size icreasig, the power of icrease ad the power of distace to boudary test still do t chage too much. Based o the paper of Liag, Fag, Hicherell ad Li (200), the is also powerful compare with the discrepacy measures tests Dirichlet Distributio Dirichlet distributio is a family of cotiuous multivariate probability distributios with parameter vector α. It is the multivariate geeralizatio of the beta distributio. I this research, the two dimesioal Dirichlet distributio is used i power compariso. The probability desity fuctio of the two dimesioal Dirichlet distributio is displayed as follow: ( α) ( α2) ( α3) ( ) ( ) ( ) Γ ( α + α + α ) Γ Γ Γ f( x, x ;,, ) x x x x α α 2 α 3 2 α α2 α3 = x > 0, x > 0, x x > 0 ad α, α2, α 3 > 0 for

26 Alterative Distributio 6 Choose α = ( ) Alterative Distributio 7 Choose α = ( ) 2, 2, 2. This is the symmetrical Dirichlet distributio. 3, 4, 9. This is the osymmetrical Dirichlet distributio. Figures 6-7 show the power comparisos for the ad the Distace to Boudary test i the coditio that the alterative distributio is Drichlet distributio. The is more powerful tha Distace to Boudary test i each case. Oe is symmetrical Dirichlet distributio ad the other is osymmetrical Dirichlet distributio Summary of the Power Compariso. I i.i.d case which is the margial distributio are idepedet ad idetical distributed, if the margial of the Beta distributio is ot symmetric ad variace is high, the performs better tha the Distace to Boudary test. 2. I i.i.d case, if the margial of the Beta distributio is left-skewed ad variace is ot high, the is more powerful tha the Distace to Boudary test uder the sample size of 25. Whe sample size goes over 25, two tests perform i same way. 3. I i.i.d case, if the margial of the Beta distributio is symmetric ad the variace is low, the Distace to Boudary test performs much better tha the G test. 4. For the meta-type uiform distributio, the has more power tha the 9

27 Distace to Boudary test both i MNU ad MTU cases. 5. For the Dirichlet distributio, the has more power tha the Distace to Boudary test especially i osymmetrical case. Figure. Power Compariso: Alterative Distributio The bivariate distributio derived from Beta (5, 2) ad Beta (5, 2) 20

28 Figure 2. Power Compariso: Alterative Distributio 2 The bivariate distributio derived from Beta (5, ) ad Beta (5, ) Figure 3. Power Compariso: Alterative Distributio 3 The bivariate distributio derived from Beta (0.5, 0.5) ad Beta (0.5, 0.5) 2

29 Figure 4. Power Compariso: Alterative Distributio 4 Meta-type Uiform Distributio: MNU Figure 5. Power Compariso: Alterative Distributio 5 Meta-type Uiform Distributio: MTU 22

30 Figure 6. Power Compariso: Alterative Distributio 6 Dirichlet distributio: Dir (2, 2, 2) Figure 7. Power Compariso: Alterative Distributio 7 Dirichlet distributio: Dir (3, 4, 9) 23

31 4. CONCLUSION AND DISCUSSION I this research, a ew multi-dimesioal uiformity test is proposed. The basic idea is from the uivariate uiformity test proposed by Che ad Ye (2009). The method is exteded to the multi-dimesioal case ad the bivariate case is discussed i detail. The ew test ca be used to test whether a uderlyig multivariate probability distributio differs from a uiform distributio. The critical values of bivariate uiformity test are obtaied ad the power study is performed by comparig with the recetly published multivariate uiformity test. O the basis of the cetral limit theorem, if the sample size is sufficietly large with fiite mea ad variace, the distributio of the sample poits will be approximately ormally distributed. I this research, we focus o the small sample study which is the case that the sample size is less tha or equal to 50. The Distace to Boudary test is a recetly published multivariate uiformity test by Berredero, Cuevas ad Vazquez-Grade (2006). The result of the power compariso shows that the test proposed i this research is more powerful tha the Distace to Boudary test i some cases. Especially, whe the margial distributio of alterative distributio is ot symmetric ad all the margial distributios are idepedet ad idetical distributed. The meta-type uiform distributio is itroduced i this research. We geerate the meta-type uiform distributio from ormal distributio ad studet s t distributio. The power study shows that ew test is more powerful tha Distace to the Boudary test i this case. The two dimesioal Dirichlet Distributio is used as a alterative distributio. The has more power tha the 24

32 Distace to Boudary test both i symmetrical case ad osymmetrical case. Theoretically, we ca geerate the multivariate distributio uder the coditio that all the margial distributios are idepedet ad idetical distributed. However, whe radom samples are draw from a multivariate populatio distributio, the margial distributios are usually depedet. Because of the iclusio of the term ( c d + ) ( + + ) c d i the bivariate case, the power of the modified test statistic will be raised sigificatly whe the margial distributios are ot idepedet. 25

33 REFERENCES K. Pearso (900) O the Criterio that a Give System of Deviatios from the Probable i the Case of a Correlated System of Variables is such that it ca be reasoably Supposed to Have Arise from Radom Samplig, Philosophical Magazie, 5, A. N. Komogorov (933) Sulla Determiazioe Empirica di Ua Legge di Distribuzioe, Giorale dell Istituto degli Attuari, 4, N. V. Smirov (939) Estimate of Deviatio Betwee Empirical Distributios (Russia), Bulleti Moscow Uiversity, 2, 3-6. H. Cramer (928) O the compositio of elemetary errors. Skadiavisk Aktuarietidskrift,, 3-74, A. Justel, D. Pea ad R. Zamar (997) A multivariate Kolmogorov-Smirov test of goodess-of-fit, Statistics & Probability Letters, 35, J. J. Liag, K. T. Fag, F. J. Hickerell ad R. Li (200) Testig multivariate uiformity ad its applicatio. Mathematics of Computatio, 70, J. R. Berredero, A. Cuevas ad F. Vazquez-Grade (2006) Testig multivariate uiformity: The distace-to-boudary method. The Caadia Joural of Statistics, 34, Zhemi Che ad Chumiao Ye (2009) A alterative test for uiformity. Iteratioal Joural of Reliability, Quality ad Safety Egieerig, 4, Kedall s, M. (938) A New Measure of Rak Correlatio, Biomtrika, 30(-2),

34 APPENDIX Tables 2-7: Power of ad Distace to Boudary test The bivariate distributio derived from Beta (5, 2) ad Beta (5, 2) sample Size (α=0.0) (α=0.0) (α=0.05) (α=0.05) (α=0.0) (α=0.0) The bivariate distributio derived from Beta (5, ) ad Beta (5, ) sample Size (α=0.0) (α=0.0) (α=0.05) (α=0.05) (α=0.0) (α=0.0)

35 The bivariate distributio derived from Beta (0.5, 0.5) ad Beta (0.5, 0.5) sample Size (α=0.0) (α=0.0) (α=0.05) (α=0.05) (α=0.0) (α=0.0) Meta-type uiform distributio: MNU sample Size (α=0.0) (α=0.0) (α=0.05) (α=0.05) (α=0.0) (α=0.0)

36 Meta-type uiform distributio: MTU sample Size (α=0.0) (α=0.0) (α=0.05) (α=0.05) (α=0.0) (α=0.0) Dirichlet distributio: Dir (2, 2, 2) sample Size (α=0.0) (α=0.0) (α=0.05) (α=0.05) (α=0.0) (α=0.0)

37 Dirichlet distributio: Dir (3, 4, 9) sample Size (α=0.0) (α=0.0) (α=0.05) (α=0.05) (α=0.0) (α=0.0)

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