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: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, Lodo W1T 3JH, UK Commuicatios i Statistics - Theory ad Methods Publicatio details, icludig istructios for authors ad subscriptio iformatio: http://www.tadfolie.com/loi/lsta20 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 2013. To cite this article: Hadi Alizadeh Noughabi & Naser Reza Arghami (2013) Testig Normality Usig Trasformed Data, Commuicatios i Statistics - Theory ad Methods, 42:17, 2365-2375 To lik to this article: http://dx.doi.org/10.1080/03610926.2011.611604 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 http:// www.tadfolie.com/page/terms-ad-coditios
Commuicatios i Statistics Theory ad Methods, 42: 2365 2375, 2013 Copyright Taylor & Fracis Group, LLC ISSN: 0361-0926 prit/1532-415x olie DOI: 10.1080/03610926.2011.611604 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; E-mail: alizadehhadi@ymail.com 2365
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.
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.
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 1 + 1 1 2 3 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.
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 1. 2.2. 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 0.01 0.05 0.10 5 10 2 1.431 1.937 2.159 10 45 7 2.824 3.098 3.197 15 105 15 3.249 3.423 3.496 20 190 20 3.487 3.624 3.684 25 300 25 3.622 3.746 3.793 30 435 30 3.736 3.828 3.870 50 1225 50 3.922 3.981 4.004
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 = 1 12 + { 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 = 2 3 3 for sample sizes = 10 20 30, 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 = + 6 24 ] where c = skewess ad k = kurtosis. 2.3. 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
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.
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 = 10 20 30 uder alteratives from Group I Alteratives CH D V W A 2 TV m JB TA m 10 t 1 0618 0580 0589 0594 0618 0434 0590 0578 20 t 1 0880 0847 0865 0869 0882 0745 0855 0819 30 t 1 0965 0947 0958 0960 0967 0909 0953 0933 10 t 3 0182 0164 0163 0187 0190 0098 0212 0193 20 t 3 0309 0260 0277 0340 0327 0158 0374 0290 30 t 3 0410 0345 0377 0460 0436 0245 0507 0392 10 Logistic 0080 0073 0071 0082 0083 0051 0094 0084 20 Logistic 0106 0087 0090 0123 0113 0052 0140 0095 30 Logistic 0110 0094 0099 0144 0123 0056 0179 0112 10 Double expoetial 0158 0142 0142 0150 0159 0069 0175 0154 20 Double expoetial 0270 0224 0242 0264 0274 0093 0294 0198 30 Double expoetial 0365 0294 0333 0360 0374 0158 0390 0251 10 Average 02595 02398 02413 02533 02625 0163 02678 02523 20 Average 03913 03545 03685 0399 0399 0262 04158 03505 30 Average 04625 04200 04418 0481 0475 0342 05073 0422 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 = 10 20 30 uder alteratives from Group II Alteratives CH D V W A 2 TV m JB TA m 10 Gumbel (0,1) 0137 0121 0117 0153 0147 0106 0160 0158 20 Gumbel (0,1) 0249 0203 0194 0313 0273 0201 0303 0299 30 Gumbel (0,1) 0360 0290 0272 0469 0402 0302 0430 0435 10 Gumbel (0,2) 0136 0121 0117 0150 0144 0104 0159 0160 20 Gumbel (0,2) 0252 0203 0195 0315 0276 0202 0296 0297 30 Gumbel (0,2) 0359 0289 0269 0464 0399 0300 0431 0436 10 Gumbel (0,1/2) 0139 0118 0117 0154 0147 0104 0157 0159 20 Gumbel (0,1/2) 0249 0203 0194 0314 0275 0200 0300 0298 30 Gumbel (0,1/2) 0360 0288 0269 0464 0400 0300 0426 0434 10 Average 01373 0120 0117 01523 0146 01047 01587 0159 20 Average 0250 0203 01943 03140 02747 02010 02997 0298 30 Average 03597 0289 0270 04657 04003 03007 04290 0435 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.
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 = 10 20 30 uder alteratives from Group III Alteratives CH D V W A 2 TV m JB TA m 10 Expoetial 0390 0301 0360 0442 0416 0424 0371 0456 20 Expoetial 0724 0586 0696 0836 0773 0845 0679 0861 30 Expoetial 0896 0784 0883 0968 0934 0967 0860 0980 10 Gamma (2) 0210 0175 0180 0239 0225 0189 0225 0251 20 Gamma (2) 0425 0326 0353 0532 0467 0457 0430 0538 30 Gamma (2) 0600 0472 0516 0749 0660 0662 0617 0769 10 Gamma (1/2) 0672 0540 0662 0735 0703 0791 0598 0748 20 Gamma (1/2) 0952 0879 0957 0984 0970 0993 0891 0989 30 Gamma (1/2) 09954 09823 09964 09997 09983 09999 09807 09998 10 Logormal (0,1) 0554 0463 0524 0603 0578 0562 0549 0612 20 Logormal (0,1) 0881 0778 0857 0932 0904 0919 0856 0941 30 Logormal (0,1) 0975 0935 0967 0991 0984 0988 0964 0993 10 Logormal (0,2) 0896 0826 0892 0920 0909 0939 0848 0931 20 Logormal (0,2) 09978 0991 09976 09996 09987 09998 0991 09998 30 Logormal (0,2) 1000 09999 1000 1000 1000 1000 09999 1000 10 Logormal (0,1/2) 0220 0182 0187 0245 0233 0177 0241 0253 20 Logormal (0,1/2) 0427 0337 0346 0517 0463 0395 0476 0514 30 Logormal (0,1/2) 0607 0492 0505 0726 0656 0588 0643 0717 10 Weibull (1/2) 0855 0758 0854 0894 0875 0929 0784 0910 20 Weibull (1/2) 09957 09818 09962 09992 09979 09998 0979 09992 30 Weibull (1/2) 09998 09992 09999 1000 1000 1000 09987 1000 10 Weibull (2) 0079 0074 0068 0084 0083 0084 0079 0087 20 Weibull (2) 0120 0103 0095 0156 0132 0129 0133 0157 30 Weibull (2) 0159 0132 0117 0232 0184 0186 0176 0248 10 Average 04845 04149 04659 05203 05028 05119 04619 0531 20 Average 06903 06227 06622 07445 07132 07172 06794 07499 30 Average 07790 07246 07480 08332 08020 07989 07799 08384 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.
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 = 10 20 30 uder alteratives from Group IV Alteratives CH D V W A 2 TV m JB TA m 10 Uiform 0074 0066 0081 0082 0080 0172 0028 0081 20 Uiform 0144 0100 0150 0200 0171 0420 0006 0271 30 Uiform 0230 0145 0230 0381 0299 0653 0010 0526 10 Beta(2,2) 0044 0046 0048 0042 0046 0079 0022 0047 20 Beta(2,2) 0058 0053 0064 0053 0058 0132 0006 0074 30 Beta(2,2) 0072 0060 0080 0080 0080 0187 0003 0128 10 Beta(1/2,1/2) 0229 0162 0237 0299 0268 0515 0096 0274 20 Beta(1/2,1/2) 0509 0318 0490 0727 0618 0911 0058 0795 30 Beta(1/2,1/2) 0738 0507 0707 0944 0862 0992 0230 0976 10 Beta(3,1/2) 0542 0418 0530 0609 0576 0684 0445 0624 20 Beta(3,1/2) 0875 0746 0879 0948 0913 0977 0745 0967 30 Beta(3,1/2) 0979 0934 0981 0997 0991 0999 0922 09993 10 Beta(2,1) 0115 0100 0109 0130 0126 0179 0083 0138 20 Beta(2,1) 0232 0174 0202 0306 0261 0431 0096 0366 30 Beta(2,1) 0359 0268 0315 0515 0428 0651 0136 0630 10 Average 02008 01584 02010 02324 02192 03258 01348 02328 20 Average 03636 02782 03570 04468 04042 05742 01822 04946 30 Average 04756 03828 04626 05834 05320 06964 02602 06519 Refereces Alizadeh Noughabi, H. (2010). A ew estimator of etropy ad its applicatio i testig ormality. J. Statist. Computat. Simul. 80:1151 1162. Alizadeh Noughabi, H., Arghami, N. R. (2011c). Testig expoetiality usig trasformed data. J. Statist. Computat. Simul. 81:511 516. Alizadeh Noughabi, H., Arghami, N. R. (2011a). Mote Carlo compariso of seve ormality tests. J. Statist. Computat. Simul. 81:965 972. Alizadeh Noughabi, H., Arghami, N. R. (2011b). Testig expoetiality based o characterizatios of the expoetial distributio. J. Statist. Computat. Simul. 81:1641 1651. Aderso, T. W., Darlig, D. A. (1954). A test of goodess of fit. J. Ameri. Statist. Assoc. 49:765 769. Correa, J. C. (1995). A ew estimator of etropy. Commu. Statist. Theor. Meth. 24:2439 2449. 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. 20:225 234. 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. 30:761 785. 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. 55:163 172.
Testig Normality Usig Trasformed Data 2375 Kolmogorov, A. N. (1933). Sulla Determiazioe Empirica di ue legge di Distribuzioe. Giorale Dell Itituto Italiao Degli Attuari 4:83 91. Kuiper, N. H. (1962). Test cocerig radom poits o a circle. Proceedigs of the Koiklijke Nederladse Akademie va Weteschappe, Series A. 63:38 47. 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: 379 423; 623 656. Shapiro, S. S., Wilk, M. B. (1965). A aalysis of variace test for ormality. Biometrika 52:591 611. 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. 19:61 72. Vasicek, O. (1976). A test for ormality based o sample etropy. J. Roy. Statist. Soc. Ser. B 38:54 59. vo Mises, R. (1931). Wahrscheilichkeitsrechug ud ihre Awedug i der Statistik ud Theoretische Physik. Leipzig ad Viea: Deuticke.