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1 This article was dowloaded by: [Hadi Alizadeh Noughabi] O: 12 July 2013, At: 11:39 Publisher: Taylor & Fracis Iforma Ltd Registered i Eglad ad Wales Registered Number: Registered office: Mortimer House, Mortimer Street, Lodo W1T 3JH, UK Commuicatios i Statistics - Theory ad Methods Publicatio details, icludig istructios for authors ad subscriptio iformatio: Testig Normality Usig Trasformed Data Hadi Alizadeh Noughabi a & Naser Reza Arghami a a Departmet of Statistics, Ferdowsi Uiversity of Mashhad, Mashhad, Ira Published olie: 12 Jul To cite this article: Hadi Alizadeh Noughabi & Naser Reza Arghami (2013) Testig Normality Usig Trasformed Data, Commuicatios i Statistics - Theory ad Methods, 42:17, To lik to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Fracis makes every effort to esure the accuracy of all the iformatio (the Cotet ) cotaied i the publicatios o our platform. However, Taylor & Fracis, our agets, ad our licesors make o represetatios or warraties whatsoever as to the accuracy, completeess, or suitability for ay purpose of the Cotet. Ay opiios ad views expressed i this publicatio are the opiios ad views of the authors, ad are ot the views of or edorsed by Taylor & Fracis. The accuracy of the Cotet should ot be relied upo ad should be idepedetly verified with primary sources of iformatio. Taylor ad Fracis shall ot be liable for ay losses, actios, claims, proceedigs, demads, costs, expeses, damages, ad other liabilities whatsoever or howsoever caused arisig directly or idirectly i coectio with, i relatio to or arisig out of the use of the Cotet. This article may be used for research, teachig, ad private study purposes. Ay substatial or systematic reproductio, redistributio, resellig, loa, sub-licesig, systematic supply, or distributio i ay form to ayoe is expressly forbidde. Terms & Coditios of access ad use ca be foud at

2 Commuicatios i Statistics Theory ad Methods, 42: , 2013 Copyright Taylor & Fracis Group, LLC ISSN: prit/ x olie DOI: / Testig Normality Usig Trasformed Data HADI ALIZADEH NOUGHABI AND NASER REZA ARGHAMI Departmet of Statistics, Ferdowsi Uiversity of Mashhad, Mashhad, Ira 1. Itroductio I this article, we first preset a characterizatio of the ormal distributio ad the we itroduce a exact goodess of fit test for ormal distributio. The power of the proposed test uder various alteratives is compared with the existig tests, by simulatio. Keywords Characterizatio; Goodess-of-fit tests; Normal distributio. Mathematics Subject Classificatio Primary 62G86; Secodary 62G10. To make a statistical iferece several assumptios about the data must be fulfilled. Most statistical procedures assume a uderlyig distributio i the derivatio of their results. Therefore, we must check the distributio assumptios carefully. Sice ormality assumptio is idispesable i may statistical methods, testig for ormality is ofte required. Therefore, may ormality tests have bee developed by differet authors. The popular ad powerful tests for ormality are Cramér vo Mises (1931), Kolmogorov (1933), Aderso ad Darlig (1954), Kuiper (1962),Shapiro ad Wilk (1965), Vasicek (1976), ad Jarque ad Bera (1987). For studies about these tests see D Agostio ad Stephes (1986), Thode (2002), Alizadeh Noughabi ad Arghami (2011a), ad refereces therei. Etropy is a measure of ucertaity, ad has bee widely employed i may patter aalysis applicatios. The etropy of a distributio fuctio F with a cotiuous desity fuctio f is defied by Shao (1948) as: Hf = fx log fxdx The problem of estimatio of the Shao etropy has bee cosidered by may authors icludig Vasicek (1976), va Es (1992), Correa (1995), Ebrahimi et al. (1994), ad Alizadeh Noughabi (2010). Received March 25, 2011; Accepted August 3, 2011 Address correspodece to Hadi Alizadeh Noughabi, Departmet of Statistics, Ferdowsi Uiversity of Mashhad, Mashhad, Ira; alizadehhadi@ymail.com 2365

3 2366 Alizadeh Noughabi ad Arghami Vasicek s sample etropy has bee most widely used i developig etropybased statistical procedures. Vasicek estimator is give by HV m = 1 { } log 2m X i+m X i m where the widow size m is a positive iteger smaller tha /2, X i = X 1 if i< 1, X i = X if i> ad X 1 X 2 X are order statistics based o a radom sample of size. Vasicek (1976) proved the cosistecy of HV m for the populatio etropy Hf. Amog all distributios that possess a desity fuctio f ad have a give variace 2, the etropy Hf is maximized by the ormal distributio. Based o this property, Vasicek (1976) itroduced the followig statistic for testig ormality: where TV m = exphv m ˆ ˆ = [ = 1/ X 2mˆ i+m X i m ] 1 X i X 2 We see that the test statistic is ivariat with respect to locatio ad scale trasformatios. Small values of TV m idicate that the sample is from a o-ormal distributio. Alizadeh Noughabi ad Arghami (2011b,c), itroduced some goodess-of-fit tests for expoetiality usig trasformed data. They showed that these tests have higher power tha their competitors. I this paper a etropy based test for ormality usig trasformed data is itroduced. Sectio 2 itroduces a goodess-of-fit test for ormality. We obtai the power of the test by simulatio. 2. Testig Normality Usig Trasformed Data 2.1. Test Statistic ad Critical Values Let X 1 X be a i.i.d. (idepedet idetically distributed) sample from a populatio. We wat to test the ull hypothesis H 0 X 1 X is a sample from ormal N vs. the alterative hypothesis that the sample is ot from a ormal populatio, where ad >0 are ukow. I order to obtai a test statistic, we use the followig theorem. Theorem 2.1. Let X 1 ad X 2 be two idepedet observatios from a cotiuous distributio F. The, X 1 + X 2 is distributed as ormal if ad oly if F is ormal.

4 Testig Normality Usig Trasformed Data 2367 Proof. Let M X t deote the momet geeratig fuctio of radom variable X. We have: M X1 +X 2 t = E exptx 1 + tx 2 = E exptx 1 E exptx 2 { ( = exp t + 1 )} 2 2 t2 2 = exp ( 2t + t 2 2) Coversely, we have exp (t + 12 ) t2 2 = M X1 +X 2 t = E exptx 1 + tx 2 = E exptx 1 2 ( 1 E exptx 1 = exp 2 t + 1 ) 4 t2 2 Therefore, the result follows. Let X 1 X 2 X be a radom sample of size. First, we trasform the sample data to Y ij = X i + X j i<j ij= 1 2 By the above theorem, uder the ull hypothesis, each Y ij has a ormal distributio, ad it seems to be appropriate to use the Vasicek s test for ormality (described i the Itroductio) to test the ormality of the distributio of Y ij s ad thus the ormality of the distributio of X i s. Therefore, summary of the test is as where X 1 X Y ij = X i + X j i<j TA m = exphv m ˆ = 2mˆ [ Y i+m Y i m HV m = 1 { } log 2m Y i+m Y i m is Vasicek etropy estimator (1976), = 1/2 ad ˆ = 1 Y i Y 2 ] 1/ Small values of TA m idicate that the sample is from a o-ormal distributio. It is obvious that the distributio of TA m does ot deped o ad therefore the test is exact. By exact we mea that critical poits of the test do ot deped o ay ukow parameters. Lemma 2.1. Let U be a cotiuous, but otherwise arbitrary, fuctio, the 1 UY i E UY 1 where is the sample size, = 1/2 ad Y i s are the trasformed data.

5 2368 Alizadeh Noughabi ad Arghami Proof. Let Z i = UY i, we have Var ( ) Z i = VarZ i + i CovZ i Z j j Sice the umber of covariace terms that are equal to zero is 1 2 3, we ca write Var ( ) Z i Var Z VarZ 1 i which the last iequality follows from Coachy-Schwartz iequality CovX Y VarXVarY. Therefore, ( ) Var Z i 2 0 ad thus, by the theorem o Gedeko (1968, p. 250), the proof is complete. Theorem 2.2. Let F be a ukow cotiuous distributio ad F 0 be the ormal distributio with uspecified parameters. The uder H 1, the test TA m is cosistet. Proof. We ca write log TA m = 1 { } log 2m Y i+m Y i m where { = 1 log log 2m Y i+m Y i m fy i fy i = 1 log { fy i } + U m V m log } FY i+m FY i m log FY i+m FY i m U m = 1 { ( log FYi+m FY 2m i m )} { } V m = 1 FYi+m FY i m log fy i ( ) Y i+m Y i m It is clear that where F a = # y i a 2m = F Y i+m F Y i m = 1 I yi a, where I is the idicator fuctio.

6 Testig Normality Usig Trasformed Data 2369 By lettig UY i = I yi a i the Lemma, as m ad m/ 0, we have F Y i+m F Y i m = FY i+m FY i m Also, we kow that FY i+m FY i m fy Y i+m Y i i m ad the approximate equality reduce to equality as m ad m/ 0. By the Lemma we have 1 log { fy i } E log fy 1 Therefore, as m ad m/ 0, we have Thus, log TA m E log fy 1 log = fy logfydy log = Hf log ad the test TA m is cosistet. TA m exphf For small to moderate sample sizes 5, 10, 15, 20, 25, 30, ad 50, we used Mote Carlo methods with 10,000 replicates from stadard ormal distributio to obtai critical values of our procedure. These values are reported i Table Competitor Tests We chose the competitor tests from the class of tests of ormality discussed i Alizadeh Noughabi ad Arghami (2011a). The test statistics of competitor tests are Table 1 Critical values of TA m -statistic m

7 2370 Alizadeh Noughabi ad Arghami as follows. 1. The Kolmogorov Smirov, Cramér vo Mises, Kuiper, ad Aderso-Darlig test statistics based o empirical distributio fuctio are, respectively (see D Agostio ad Stephes, 1986): D = sup F x Z i x { { i = max max 1 i Z i CH = { i V = max 1 i Z i [ A 2 = } ( 2i 1 2 Z i } { max Z i i 1 }} 1 i ) 2 { Z i i 1 } + max 1 i 2i 1 { lz i + l1 Z i+1 } where Z i = X i X i = 1 ad is the cumulative distributio fuctio S X (cdf) of stadard ormal distributio. 2. The test statistic proposed by Shapiro ad Wilk (1965) is W = ( /2 a i+1x i+1 X i X i X 2 The coefficiets a i are tabulated i Pearso ad Hartley (1972). 3. The test statistic proposed by Vasicek (1976) based o sample etropy is: TV m = exphv m ˆ ) 2 [ = 1/ X 2mˆ i+m X i m ] where HV m = 1 log { X 2m i+m X i m } is Vasicek (1976) s etropy estimator. I this power study, we have take the widow sizes m = for sample sizes = , respectively, followig the recommedatio of Vasicek (1976). 4. The test statistic of Jarque ad Bera (1987), based o the skewess ad kurtosis, is: { } c 2 k 32 JB = ] where c = skewess ad k = kurtosis Power Study We compute the powers of the tests based o CH, D, V, W, A 2, TV m, JB, ad TA m statistics by meas of Mote Carlo simulatios uder 20 alteratives. These

8 Testig Normality Usig Trasformed Data 2371 alteratives were used by Esteba et al. (2001) ad Alizadeh Noughabi ad Arghami (2011a) i their study of power comparisos of several tests for ormality. The alteratives ca be divided ito four groups, depedig o the support ad shape of their desities. From the poit of view of applied statistics, atural alteratives to ormal distributio are i Groups I ad II. For the sake of completeess, we also cosider Groups III ad IV. This fact gives additioal isight to uderstad the behaviour of the tests. Group I: Support, symmetric. Studet t with 1 degree of freedom (i.e., the stadard Cauchy), Studet t with 3 degrees of freedom, Stadard logistic, Stadard double expoetial. Group II: Support, asymmetric. Gumbel with parameters = 0 (locatio) ad = 1 (scale), Gumbel with parameters = 0 (locatio) ad = 2 (scale), Gumbel with parameters = 0 (locatio) ad = 1/2 (scale). Group III: Support 0. Expoetial with mea 1, Gamma with parameters = 1 (scale) ad = 2 (shape), Gamma with parameters = 1 (scale) ad = 1/2 (shape), Logormal with parameters = 0 (scale) ad = 1 (shape), Logormal with parameters = 0 (scale) ad = 2 (shape), Logormal with parameters = 0 (scale) ad = 1/2 (shape), Weibull with parameters = 1 (scale) ad = 1/2 (shape), Weibull with parameters = 1 (scale) ad = 2 (shape). Group IV: Support (0,1). Uiform, Beta (2,2), Beta (0.5,0.5), Beta (3,1.5), Beta (2,1). Uder each alterative we geerated 10,000 samples of size 10, 20, 30. We evaluated for each sample the statistics (CH, D, V, W, A 2, TV m, JB, TA m ad the power of the correspodig test was estimated by the frequecy of the evet the statistic is i the critical regio. Although the required critical values are give i the correspodig articles, we also obtaied them by simulatio, before power simulatios. The power estimates are give i Tables 2 5. For these alteratives, the maximum power was typically attaied by choosig m = 7 for = 10 = 45, m = 20 for = 20 = 190 ad m = 30 for = 30 = 435. With icreasig, a optimal choice of m also icreases, while the ratio m/ teds to zero. From Tables 2 5, we ca see that the tests compared cosiderably differ i power. By average powers, we ca select the tests which are, o average, most powerful agaist the alteratives from the give groups.

9 2372 Alizadeh Noughabi ad Arghami Table 2 Power comparisos of 0.05 tests based o CH, D, V, W, A 2, TV m, JB, ad TA m statistics for sample sizes = uder alteratives from Group I Alteratives CH D V W A 2 TV m JB TA m 10 t t t t t t Logistic Logistic Logistic Double expoetial Double expoetial Double expoetial Average Average Average Table 3 Power comparisos of 0.05 tests based o CH, D, V, W, A 2, TV m, JB, ad TA m statistics for sample sizes = uder alteratives from Group II Alteratives CH D V W A 2 TV m JB TA m 10 Gumbel (0,1) Gumbel (0,1) Gumbel (0,1) Gumbel (0,2) Gumbel (0,2) Gumbel (0,2) Gumbel (0,1/2) Gumbel (0,1/2) Gumbel (0,1/2) Average Average Average I Group I, it is see that the tests JB ad A 2 have the most power ad the test TV m has the least power. The differece of powers the test TV m ad the other tests are substatial. Also, the differece of power the proposed test TA m ad the test TV m is substatial. I Group II, the test W has the most power ad the test V has the least power. For = 10, the test JB ad TA m have the most power ad the differece of powers betwee W, TA m, ad JB is small.

10 Testig Normality Usig Trasformed Data 2373 Table 4 Power comparisos of 0.05 tests based o CH, D, V, W, A 2, TV m, JB, ad TA m statistics for sample sizes = uder alteratives from Group III Alteratives CH D V W A 2 TV m JB TA m 10 Expoetial Expoetial Expoetial Gamma (2) Gamma (2) Gamma (2) Gamma (1/2) Gamma (1/2) Gamma (1/2) Logormal (0,1) Logormal (0,1) Logormal (0,1) Logormal (0,2) Logormal (0,2) Logormal (0,2) Logormal (0,1/2) Logormal (0,1/2) Logormal (0,1/2) Weibull (1/2) Weibull (1/2) Weibull (1/2) Weibull (2) Weibull (2) Weibull (2) Average Average Average I Group III, the proposed test TA m has the most power ad the test D has the least power. The differece of powers betwee the test TA m W TV m ad the other tests are substatial. I Group IV, the test TV m has the most power ad the test JB has the least power. The differece of powers betwee the proposed test TA m ad the other tests are substatial. We observe that the proposed tests perform very well compared with the other tests. However, o sigle test ca be said to perform best for testig ormality agaist all alteratives. We observe that the tests CH, D, ad V do ot have the most power for ay alteratives. We also metio that it ca be said that the proposed tests improve upo Vasicek (1976) s test, which is based o the applicatio of Vasicek s etropy estimator o the origial data.

11 2374 Alizadeh Noughabi ad Arghami Table 5 Power comparisos of 0.05 tests based o CH, D, V, W, A 2, TV m, JB, ad TA m statistics for sample sizes = uder alteratives from Group IV Alteratives CH D V W A 2 TV m JB TA m 10 Uiform Uiform Uiform Beta(2,2) Beta(2,2) Beta(2,2) Beta(1/2,1/2) Beta(1/2,1/2) Beta(1/2,1/2) Beta(3,1/2) Beta(3,1/2) Beta(3,1/2) Beta(2,1) Beta(2,1) Beta(2,1) Average Average Average Refereces Alizadeh Noughabi, H. (2010). A ew estimator of etropy ad its applicatio i testig ormality. J. Statist. Computat. Simul. 80: Alizadeh Noughabi, H., Arghami, N. R. (2011c). Testig expoetiality usig trasformed data. J. Statist. Computat. Simul. 81: Alizadeh Noughabi, H., Arghami, N. R. (2011a). Mote Carlo compariso of seve ormality tests. J. Statist. Computat. Simul. 81: Alizadeh Noughabi, H., Arghami, N. R. (2011b). Testig expoetiality based o characterizatios of the expoetial distributio. J. Statist. Computat. Simul. 81: Aderso, T. W., Darlig, D. A. (1954). A test of goodess of fit. J. Ameri. Statist. Assoc. 49: Correa, J. C. (1995). A ew estimator of etropy. Commu. Statist. Theor. Meth. 24: D Agostio, R. B., Stephes, M. A. (1986). Goodess-of-Fit Techiques. New York: Marcel Dekker, Ic. Ebrahimi, N., Pflughoeft, K., Soofi, E. (1994). Two measures of sample etropy. Statist. Probab. Lett. 20: Esteba, M. D., Castellaos, M. E., Morales, D., Vajda, I. (2001). Mote Carlo compariso of four ormality tests usig differet etropy estimates. Commui. Statist. Simul. Computat. 30: Gedeko, B. V. (1968). The Theory of Probability. New York: Chelsea Publishig Compay. Jarque, C. M., Bera, A. K. (1987). A test ormality of observatios ad regressio residuals. It. Statist. Rev. 55:

12 Testig Normality Usig Trasformed Data 2375 Kolmogorov, A. N. (1933). Sulla Determiazioe Empirica di ue legge di Distribuzioe. Giorale Dell Itituto Italiao Degli Attuari 4: Kuiper, N. H. (1962). Test cocerig radom poits o a circle. Proceedigs of the Koiklijke Nederladse Akademie va Weteschappe, Series A. 63: Pearso, E. S., Hartley H. O. (1972). Biometrika Tables for Statisticias. Cambridge: Cambridge Uiversity Press. Shao, C. E. (1948). A mathematical theory of commuicatios. Bell Syst. Tech. J. 27: ; Shapiro, S. S., Wilk, M. B. (1965). A aalysis of variace test for ormality. Biometrika 52: Thode, H. Jr. (2002). Testig for Normality New York: Marcel Dekker. va Es, B. (1992). Estimatig fuctioals related to a desity by class of statistics based o spacigs. Scad. J. Statist. 19: Vasicek, O. (1976). A test for ormality based o sample etropy. J. Roy. Statist. Soc. Ser. B 38: vo Mises, R. (1931). Wahrscheilichkeitsrechug ud ihre Awedug i der Statistik ud Theoretische Physik. Leipzig ad Viea: Deuticke.

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