Non parametric Statistical Inference for the Two-Sample Location Problem by @Haiyan Yang A thesis submitted to the School of Graduate Studies in partial fulfilment of the requirements for degree of Master of Science Department of Mathematics and Statistics Memorial University of Newfoundland June 2006 Submitted St. John's Newfoundland Canada 1+1 Library and Archives Canada Bibliotheque et Archives Canada Published Heritage Branch Direction du Patrimoine de !'edition 395 Wellington Street Ottawa ON K1A ON4 Canada 395, rue Wellington Ottawa ON K1A ON4 Canada NOTICE: The author has granted a non-exclusive license allowing Library and Archives Canada to reproduce, publish, archive, preserve, conserve, communicate to the public by telecommunication or on the Internet, loan, distribute and sell theses worldwide, for commercial or non-commercial purposes, in microform, paper, electronic and/or any other formats. The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. In compliance with the Canadian Privacy Act some supporting forms may have been removed from this thesis. While these forms may be included in the document page count, their removal does not represent any loss of content from the thesis. • •• Canada AVIS: Your file Votre reference ISBN: 978-0-494-30520-1 Our file Notre reference ISBN: 978-0-494-30520-1 L'auteur a accorde une licence non exclusive permettant a Ia Bibliotheque et Archives Canada de reproduire, publier, archiver, sauvegarder, conserver, transmettre au public par telecommunication ou par !'Internet, preter, distribuer et vendre des theses partout dans le monde, a des fins commerciales ou autres, sur support microforme, papier, electronique et/ou autres formats. L'auteur conserve Ia propriete du droit d'auteur et des droits moraux qui protege cette these. Ni Ia these ni des extraits substantiels de celle-ci ne doivent etre imprimes ou autrement reproduits sans son autorisation. Conformement a Ia loi canadienne sur Ia protection de Ia vie privee, quelques formulaires secondaires ant ete enleves de cette these. Bien que ces formulaires aient inclus dans Ia pagination, il n'y aura aucun contenu manquant. Abstract The two-sample problem occurs in many scientific fields, with a major frequency in en-vironmental and agricultural research. Its primary goal is either to estimate the difference between certain parameters of the two populations or to test the hypotheses about the dif-ference. The well known nonparametric procedure is Mann-Whitney-Wilcoxon test. The aim of this thesis is to develop a new statistical inference procedure for the problem by extending the Mann-Whitney-Wilcoxon test. In particular, simulated probabilities of the new proposed V statistic for the uniform, normal and exponential distributions are consid-ered. Efficient computation algorithms are proposed to obtain the distribution functions of the V statistic. Power studies via simulation compare the new proposed procedure with Mann-Whitney-Wilcoxon procedure. Also included is a real data set. Acknowledgement I would like to express my sincere gratitude to Professor Chu-In Charles Lee for grant-ing me the privilege to write this thesis under his supervision and for his support, both financial and otherwise, through out the course of my program. He has been very generous with his ideas and times. He patiently read through my work several times and offered his very valuable suggestions. This work would have been impossible without his guidance and help. Many thanks go to my wife, Dr. Yuan Yuan, for making available to me the high performance computing facilities and applied mathematical knowledge. I would like to thank professors who have shared with me their wealth of statistical knowledge at various points in the course of my program. I would also like to thank the staff of the Mathematics and Statistics for providing me with a friendly atmosphere and necessary facilities to finish my program. My sincere thanks to my families, friends and classmates whose assistance and encour-agement have seen me to the completion of my program. ii Contents 1 Introduction 2 V Statistic 2.1 Introduction 2.2 Transformation 2.3 The null distribution of V . 2.4 Table value . . . . . . . . 3 Mean and Variance 3.1 Expectations and variances ...... . 3.2 Sampling from the uniform distribution 3.3 Sampling from the normal distribution . 3.4 Sampling from the exponential distribution 3.5 Mean and Variance of V among the tied observations 4 Approximations 4.1 Normal approximation ....... . 4.2 Asymptotically distribution-free of V 4.3 Simulated power comparison . . . . . 111 1 5 5 6 10 15 31 31 50 59 64 68 76 76 80 81 5 Application for Dose-Response Study 5.1 Introduction . . . . . . . .... 5.2 The proposed testing procedure . 5.3 An example . . . . . . . .. 6 Summary and Further research Appendix A Simulation of the Probabilities of Individual V Statistic B Power Test for LY Test and Mann-Whitney Test Bibliography IV 84 84 85 87 90 92 92 94 95 List of Tables 2.1 Distributions, Means and Variances ofV for the cases m = 1, n = 2 and m = 1, n = 3 15 2.2 Cumulative Probabilities of Statistic V for Small Values m and n (U: Uniform, N: Normal, E: Exponential, E*: Negative Exponential) ........ . 2.3 Critical Values for V (U: Uniform, N: Normal, E: Exponential, E*: Negative Exponential ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Critical Values for V When m = n (C: Cauchy, D: Double Exponential) 3.1 Components of It . 3.2 Components of h . 3.3 Components of h . 3.4 Components of I4 . 3.5 Components of( It ,!2) 3.6 Components of (I1, h) 3.7 Components of ( h, I 4) 3.8 Components of (I2, h) 3.9 Components of (h I4) 3.10 Components of ( h I 4) 4.1 Normal Approximation Probability (U: Uniform, N: Normal, E: Exponential) v 17 23 26 32 33 35 37 40 42 43 44 46 48 79 4.2 Normal Approximation Probability(C: Cauchy, D: Double Exponential) 4.3 Power Test for Different Mean (a=0.05) 80 83 5.1 Reaction Times in Seconds of Mice to Stimuli to their Tails . . . . . . . . . . . 87 vi List of Figures 2.1 A plot of distribution of V for different m and n (Exact values when m=l, n=2, Simulated values when n=2, n=3) . . . . . . . . . . . . . 2.2 A plot of distribution of V (simulated) for different m and n 2.3 A plot of distribution of V (simulated) form = n 2.4 A plot of distribution of V (simulated) form = n vii 27 28 29 30 Chapter 1 Introduction In recent years, more and more statisticians have come to appreciate the advantages of non-parametric tests. Not only do nonparametric tests have often surprisingly high efficiency relative to their normal-theory equivalents even under assumption of normality, but they are also less sensitive to the inference of "wild" observations than are the normal-theory equiv-alents. Moreover, in situations where measurements are costly and/or difficult to obtain but ranking of the potential sample data is relatively easy, the use of statistical methods based on ranked set sampling (RSS) can lead to substantial improvement over analogous methods associated with simple random sampling (SRS) schemes. The approach using ranked set sampling has attracted considerable attention in the recent literature, with principal initial interest being driven by environmental and agricultural issues, where it is clear that pre-sampling judgement ranking can be quite inexpensive relative to the cost of detailed mea-surement of many quantities of interest. For example, preliminary supporting data can be easily obtained from a contaminated site and analyzed both quickly and inexpensively be-fore final decisions are reached as to where and how to obtain the specific measurement(s) of interest, such as lead or hazardous material content of the site. A similar approach can 1 be taken when performing standard gasoline octane checks at stations, where preliminary screening at the gasoline pumps can lead to an improved set of ranked set gasoline samples to carry back to the laboratory for more detailed (and costly) analysis. Agricultural settings where such an approach can be used effectively include predicting crop yields or lumber content via preliminary non-destructive measurements (satellite observations or concomi-tant variables, for example). Most of the initial research efforts in ranked set sampling have concentrated on para-metric and nonparametric estimation and testing procedures for the one- and two-sample settings. See, for example, Mcintyre (1952) introduced the concept of ranked set sampling in relation to estimating pasture yield that has promise of application in many other sam-pling problems. Takahasi & Wakimoto (1968) developed an extensive body of theory that is based upon the assumption of perfect ranking. Halls & Dell (1966) tried the procedure in sampling forage yields. Dell & Clutter (1972) proposed that ranked set sampling em-ploys judgment ordering to obtain an estimate of a population mean. Stokes ( 1977) showed that improved estimates of the variance of any population can be produced from rank-set sample. Stokes & Sager (1988) provided a characterization of a ranked-set sample that makes the source of additional information intuitively clear and showed that the empirical distribution function of a ranked-set sample has greater precision than that from a random sample. Bohn & Wolfe (1992, 1994) use the empirical distribution function for ranked-set samples to develop nonparametric inference techniques in the two-sample location prob-lem and discussed the asymptotic relative efficiency comparisons between the simple ran-dom sample Mann-Whitney-Wilcoxon procedures and their ranked-set analogues, and ad-dressed the issues of imperfect judgment rankings that based on the ranked-set samples analog of the Mann-Whitney-Wilcoxon statistic. Koti & Babu ( 1996) introduced a sign test for ranked-set sampling. Lately, some researchers have expressed interest in the appropri-2 ate allocation of order statistics within a ranked set sample. Kaur, Patil & Taillie ( 1997) studied the effects of unequal allocation on estimation of the population mean when the underlying distribution is skewed or symmetric. They provided "near" optimality results based on skewness, kurtosis and/or the coefficient of variation. In the parametric set up, Bhoj (1997) constructed an unbiased estimator of the population mean by using a linear combination of the order statistics from the same set in a ranked set sampling environment. 6zturk (1999a, b) and 6zrurk & Wolfe (1998, 2000a, b) introduced a design concept for determining the appropriate allocation of the order statistics for the sign and signed rank test statistics. They showed that for all symmetric distributions the best design among all possible allocation procedures is the one that quantifies only the middle observation(s). In this thesis, we extend the same idea to settings where the extention of two-sample ranked set sample Mann-Whitney-Wilcoxon test is appropriate. In Chapter 2, we will introduce some basic ideas for V statistic. We will present trans-formation of V that can be seen an extention of Mann-Whitney-Whicoxon test. We list some values of cumulative probabilities when the data distribution is uniform (0,1), nor-mal (0,1), and exponential. If the population distribution is symmetric, then so does the distribution of V. Furthermore, the V statistic is asymptotically normally distributed. In Chapter 3, we will propose a new test statistic. We study the mean and the variance for different population distribution. We formulate the variance of V and covariances of components of V. We obtain the formula of the mean and variance of V in the presence of tied observations. In Chapter 4, we will consider the problem of large sample data in two sample studies. Hollander(1967) showed that / 4 has an asymptotically distribution-free procedure under H 0 , we will prove that V has asymptotically distribution-free procedure which trends as a unit normal random variable under H0 • We will study simulated power comparison for 3 two test. The evaluation of the simulated power comparison is that the new procedure is significantly more powerful than Mann-Whitney-Wilcoxon's in detecting the difference mean between two-samples when those comparisons are of interest. In Chapter 5, We consider the problem of identifying the minimum effective dose in dose-response studies. Assessing monotone dose-response relationship is frequently en-countered in practice in the context of actively proving a significant monotonous depen-dence of the response on increasing dose or treatments, but the monotone dose-response assumption is not always satisfied. We will propose a new test(LY test) incorporated into the step-down closed testing scheme under the partial dose-response assumption in which the responses in the dose treatments are larger than that in the control. Finally, Chapter 6 will provide a summary of the results of this thesis and further areas of research relating to the problem. 4 Chapter 2 V Statistic 2.1 Introduction We will discuss in detail only of the two-sample problem. Let X 1, • · • , Xm and Yi, · · · , Yn be two independent random samples from continuous distributions with distribution func-tions F1 and F2 , respectively. The excellent properties of the Mann-Whitney-Wilcoxon m n statistic L L I(Yk-X;) for testing H0 : F1 = F2 = F against translation alternatives i=l k=l H 1 : F1 (x) = F(x), F2(x) = F(x- tJ.), tJ. =/:. 0 are well known. Where I(z) = 1 if Z > 0 and = 0 otherwise. We propose a new statistic V and it is a signed rank sum of all possible m n V = L L I(Yk-x,)rank(IYk- Xi I) i=l k=l We assume m ::; n for convenience; if m > n the symbols x and y can be interchanged. V takes values 0, 1, · · ·, mn(mn+l)/2. 5 2.2 Transformation The new test of V statistic is an extension of Mann-Whitney-Wilcoxon statistic. Let Dn(i-1)+k = Yk - Xi and let D(1) ~ D(2) ~ · · · ~ D(mn) be the ordered random vari-ables. It is straight forward that for h < r r when D(r) > 0, rank( ID(r) I) = L l(v(h)+DcrJ) h=1 Therefore m n V L L I(Yk-X;)rank(IYk - Xd) i=1 k=1 mn L rank(ID(r)l) = L l(v(r))rank(ID(r)l) r=1 L L[(D(h)+D(r)) = L L[(Dh+Dr) r~h r~h mn 1 mnmn LLI(Yk-X;) + 2LLLLI(Yk-X;+Yi-Xj) i=1 k=1 i=l k=1 j=l 1=1 (i,k);t(j,l) m n m n n LL 1(Yk-X;) + LLL1(yk~y1 -X;) i=1 k=1 i=1 k=1 1=1 k 0, U2 > 0, ... , Un-1 > OJ. In which U = (U1, U2, ... , Un_1) has a multivariate normal distribution with zero means and correlation coefficient PiJ· The well known expressions for the first four Pn-1 are given below, see McFadden(l960), Abrahamson(1964), and Childs(l967): p1 = t p. 1 1 . -1 2 = 4 + 27rsm P12 p. 1 1 ( . -1 . -1 . -1 ) 3 = 8 + 471" szn P12 + sm P23 + sm P13 P4 = 116 + 8~ 2::.:: sin - 1 (PiJ) iX, Y2 0, X- 2 > 0, Y1 - X> 0) = 6P(T1 > O,T2 > O,T3 > 0) where Y1 + Y3 T1 = Y3 - Y2, T2 = X - 2 , T3 = Y1 - X then and Var(T1 ) = 2, Var(T2 ) = ~. Var(T3 ) = 2 Cov(T1,T2 ) = Cov(Y3,-Jf) = -~ Cov(T2 , T3 ) = Cov(X, -X)+ Cov( -~, Y1) = -~ Cov(T1, T3) = 0 Hence, 12 then and Var(T1) = ~' Var(T2) = ~' Var(T3 ) = ~ Var(T{) = ~' Var(T~) = 2, Var(TD = ~ Cov(T1, T2) = -~, Cov(T2, T3) = -i, Cov(T1, T3) = -i Cov(T{,T~) = -1, Cov(T~,TD = -~, Cov(T{,T~) = t P(V = 2) = 6[~ + 4~ (sin- 1 ( -~) + sin-1 ( -~) + sin-1 ( -~))] +6[~ + 4~(sin- 1 (-~) + sin-1(-4) + sin- 1 (~))] = 0.0735 Similarly, P(V = 3) = 0.1326 P(V = 4) = P(V = 2) = 0.0735 P(V = 5) = P(V = 1) = 0.1102 P(V = 6) = P(V = 0) = 0.25 Using those probabilities, the mean ofV is E(V) = 0.25·0+0.1102·1 +0.0735·2+0.1326·3+0.0735·4+0.1102·5+0.25·6 = 3 The variance of V is Var(V) = 0.25(0- 3)2 + 0.1102(1- 3)2 + 0.0735(2- 3)2 + 0.1326(3- 3)2 13 +0.0735(4- 3)2 + 0.1102(5- 3)2 + 0.25(6- 3)2 = 5.5286 If distribution is exponential with unit mean, then rXJ t rY3 r2 1 P(V = 0) = 6 1o 1o 1o 1o exp( -y1- Y2- Y3- x)dy1dy2dy3dx = 4 1oo loo 1Ya ~Yl 1 P(V = 1) = 6 exp( -y1- Y2- Y3- x)dxdy2dy1dy3 = 12 0 Ya 0 Yl tYa 00 Yl Yi +Y2 Yl +Ya P(V = 2) = 6 { { { 2 { 2 exp( -y1 - Y2- Y3- x)dxdy3dy2dY1 11: 1~~ ~1~2 1!:¥: 1 +6 exp( -y1- Y2- Y3- x)dxdy3dy2dY1 = 24 0 0 Yi tY2 Y3 OO y Yi +Y2 Yl +Y2 P(V = 3) = 6 r r I r 2 r 2 exp( -y1- Y2- Y3- x)dxdy3dy2dY1 1o 1o 1Y2 1Ya 1oo 1Y2 ~Y2 ~YI 3 +6 exp( -y1- Y2- Y3- x)dxdy1dy3dy2 = 20 0100 01Y2 1!2..:!:.!!.3.~t:2 l!ZTia~Y ;~ P(V = 4) = 6 2 exp( -y1- Y2- Y3- x)dxdy1dy3dy2 1:1~1~h~ 3 +6 exp( -y1 - Y2- Y3- x)dxdy1dy3dY2 = 40 01oo 01Y2 Y/,3 Y21:::: 3 P(V = 5) = 6 exp( -y1- Y2- Y3- x)dxdy1dy3dy2 = 20 0 0 Ya Ya P(V = 6) = t Using those probabilities, the mean ofV is E(V) - l · 0 + .l. · 1 + .l. · 2 + ~ · 3 + ~ · 4 + ~ · 5 + l · 6 = 19 - 4 12 24 20 40 20 4 6 14 Table 2.1: Distributions, Means and Variances of V for the cases m = 1, n = 2 and m = 1, n = 3 m=1,n=2 m=1,n=3 v u N E u N E 0 113 113 113 114 0.25 114 1 1/6 116 1/9 118 0.1102 1112 2 1/6 116 2/9 1/16 0.0735 1/24 3 113 1/3 113 118 0.1326 3/20 4 1116 0.0735 3/40 5 118 0.1102 3/20 6 1/4 0.25 114 h(V) J/1. jfl. 14/IJ j j l'J/(J Var(V) 19/12 19/12 128/81 45/8 5.5286 497/90 The variance of V is In Table 2.1, we list distributions of V, means of V, and variances of V for the cases (m = 1, n = 2) and (m = 1, m = 3). The V statistic is symmetric if population distribution function F is normal or uniform. When m =/= n, the V statistic is not symmetric if population distribution function F is exponential. 2.4 Table value In this section we give extensive tables of, simulated critical values of V for m ::; n = 2( 1) 10 and m::; n = 10(5)25 and lower a= 0.001, 0.005, 0.01, 0.025, 0.05, 0.10. In Table 2.2 we list some values of cumulative probabilities ofP(V = v) for small values m and n, and Table 15 2.3 we list critical values at a= 0.001, 0.005, 0.01, 0.025, 0.05, 0.10 when F is uniform (0,1), standard normal and unit exponential. From section 2.3, we know that when m -:p n, the V statistic is not symmetric if population distribution F is not symmetric, see Tables 2.1, 2.2 and 2.3. Table 2.4 provides critical values for V for selected values of m and n, m = n, and F is cauchy or double exponential. They are the simulation results using 1,000,000 replications. Tables 2.1 to 2.3 and Figure 2.1 to 2.4 show that if distribution F is uniform or normal, statistic V is symmetric. If distribution F is exponential, the statistic V is not symmetric for small sample size m < n. However, statistic V is symmetric for m = n when distribution F is exponential. Table 2.4lists critical values vat a= 0.001, 0.005, 0.01, 0.025, 0.05, 0.10 when F is cauchy and double exponential. The values are observed from simulation with 1 ,000,000 replications. These values of individual statistic values v are very close to the values of uniform, normal, and exponential distributions at each single point a. 16 Table 2.2: Cumulative Probabilities of Statistic V for Small Values m and n (U: Uniform, N: Normal, E: Exponential, E*: Negative Exponential) v u N E E• v u N h J<; v u N h E• m 2,n=9 7 0.024182 0.021~:>~ 0.019077 0.024954 m=2, n=3 0 0.018188 0.018333 0.018471 0.018215 8 0.026137 0.023019 0.020283 0.026985 0 0.099897 0.099496 0.099662 0.100434 I 0.024307 0.024580 0.024494 0.024349 9 0.028421 0.024722 0.021518 0.029399 I 0.133446 0.136475 0.133315 2 0.029380 0.028885 0.028337 0.029958 10 0.031189 0.026650 0.022898 0.032463 3 0.039152 0.036693 0.034337 0.040761 11 0.033359 0.028395 0.024085 0.034726 m=2, n=4 4 0.044964 0.041499 0.038001 0.047298 12 0.035278 0.029993 0.025211 0.036966 0 0.066775 0.066660 0.066837 0.06659 5 0.048805 0.045512 0.041267 0.051049 13 0.037015 0.031550 0.026276 0.038918 I 0.088742 0.090853 0.088971 0.08886 6 0.053607 0.050081 0.044658 0.056086 14 0.039251 0.033239 0.027401 0.041176 2 0.107158 0.108579 0.103010 0.11083 7 0.059735 0.055292 0.048264 0.062936 15 0.041192 0.034979 0.028567 0.043179 8 0.064532 0.059477 0.051257 0.068267 16 0.043079 0.036548 0.029814 0.045183 m=2, n=5 9 0.069871 0.064177 0.054585 0.074309 17 0.044864 0.038011 0.031046 0.047098 0 0.047408 0.047774 0.047675 0.04760 10 0.076674 0.069560 0.059196 0.082417 18 0.047272 0.039759 0.032273 0.049731 1 0.063264 0.064861 0.063545 0.06354 11 0.082066 0.074448 0.063087 0.088156 19 0.049203 0.041468 0.033485 0.051864 2 0.076526 0.077044 0.073611 0.07866 12 0.086856 0.078984 0.066185 0.093853 20 0.051138 0.043098 0.034606 0.053990 3 0.101657 0.099174 0.089446 0.10953 13 0.091335 0.083359 0.069760 0.098769 21 0.053381 0.044805 0.035796 0.056588 4 0.117091 0.113616 0.099131 14 0.096662 0.088173 0.073314 0.104799 22 0.055368 0.046512 0.036916 0.058751 5 0.127248 0.126090 0.110186 15 0.101512 0.093156 0.076969 23 0.057219 0.048137 0.038054 0.060696 16 0.106025 0.097684 0.080194 24 0.059039 0.049708 0.039219 0.062828 m=2, n=6 17 0.110490 0.102198 0.083910 25 0.060980 0.051326 0.040441 0.064940 0 0.035725 0.036061 0.035621 0.035579 18 0.116245 0.107498 0.087750 26 0.063038 0.053030 0.041622 0.067161 I 0.047619 0.048595 0.047523 0.047286 19 0.121101 0.112434 0.091381 27 0.064932 0.054753 0.042774 0.069274 2 0.057760 0.057410 0.054941 0.058412 20 0.125955 0.117203 0.094953 28 0.066748 0.056341 0.043955 0.071316 3 0.077146 0.073435 0.066911 0.080641 21 0.131836 0.122276 0.099135 29 0.068676 0.058013 0.045181 0.073440 4 0.088607 0.083684 0.074292 0.093614 22 0.136985 0.127516 0.102946 30 0.070627 0.059760 0.046308 0.075594 5 0.096107 0.092429 0.080480 0.100886 31 0.072453 0.061378 0.047438 0.077600 6 0.105512 0.102531 0.088423 m=2, n=lO 32 0.074179 0.063020 0.048641 0.079575 7 0.117935 0.114007 0.099542 0 0.015137 0.015284 0.015340 0.015227 33 0.076153 0.064795 0.049906 0.081856 8 0.127695 0.123161 0.108290 1 0.020201 0.020594 0.020436 0.020257 34 0.078072 0.066501 0.051160 0.084009 2 0.024328 0.024201 0.023624 0.024798 35 0.079965 0.068176 0.052417 0.086126 m=2, n=7 3 0.032331 0.030486 0.028573 0.033576 36 0.081959 0.069841 0.053636 0.088542 0 0.027624 0.027889 0.027869 0.027749 4 0.037235 0.034377 0.031671 0.038799 37 0.083885 0.071633 0.054861 0.090876 1 0.036870 0.037620 0.037094 0.036903 5 0.040413 0.037705 0.034265 0.042074 38 0.085702 0.073429 0.056072 0.092917 2 0.044519 0.044368 0.042886 0.045294 6 0.044520 0.041550 0.036983 0.046369 39 0.087487 0.075113 0.057273 0.094909 3 0.059477 0.056448 0.052167 0.062271 7 0.049564 0.045753 0.039966 0.052026 40 0.089423 0.076848 0.058471 0.096968 4 0.068144 0.064098 0.057941 0.072225 8 0.053558 0.049131 0.042370 0.056418 41 0.091289 0.078671 0.059708 0.099008 5 0.074174 0.070703 0.062586 0.078062 9 0.057934 0.052878 0.044881 0.061494 42 0.093169 0.080435 0.060902 0.101095 6 0.081409 0.078188 0.067649 0.085901 10 0.063392 0.057169 0.047854 0.068013 43 0.094942 0.082171 0.062195 7 0.090604 0.086661 0.073878 0.096656 11 0.067809 0.061108 0.051347 0.072741 44 0.096826 0.083893 0.063437 8 0.098141 0.093535 0.081034 0.104805 12 0.071946 0.064643 0.054392 0.077330 45 0.098768 0.085695 0.064730 9 0.106543 0.100928 0.087647 13 0.075721 0.068134 0.056871 0.081411 46 0.100572 0.087509 0.065984 10 0.116815 0.109519 0.093392 14 0.080016 0.072039 0.059830 0.086427 47 0.102436 0.089211 0.067256 11 0.125077 0.117616 0.099881 15 0.084080 0.075891 0.062593 0.090861 48 0.104338 0.090957 0.068581 12 0.132829 0.125213 0.105656 16 0.087840 0.079419 0.065426 0.095019 49 0.106211 0.092812 0.069840 17 0.091627 0.082973 0.067980 0.099183 50 0.108076 0.094627 0.071114 m=2, n=S 18 0.096509 0.087143 0.070890 0.104921 51 0.109902 0.096390 0.072458 0 0.022276 0.022154 0.022154 0.022238 19 0.100570 0.091022 0.073835 52 0.111865 0.098264 0.073803 1 0.029764 0.029767 0.029715 0.029703 20 0.104557 0.094816 0.076727 53 0.113736 0.100107 0.075169 2 0.035975 0.035066 0.034389 0.036486 21 0.109254 0.098887 0.079555 54 0.115504 0.102026 0.076570 3 0.047846 0.044481 0.041696 0.049900 22 0.113522 0.102921 0.082560 55 0.117378 0.103839 0.077996 4 0.054986 0.050384 0.046313 0.057710 23 0.117308 0.106908 0.085757 56 0.119195 0.105673 0.079393 5 0.059756 0.055531 0.050173 0.062459 24 0.121166 0.110680 0.088827 57 0.121021 0.107493 0.080836 6 0.065507 0.061310 0.054153 0.068574 25 0.125199 0.114624 0.091659 58 0.122808 0.109339 0.082258 7 0.072927 0.067833 0.058570 0.077232 26 0.129425 0.118776 0.094772 59 0.124609 0.111163 0.084455 8 0.078731 0.073191 0.062733 0.083699 27 0.133341 0.122784 0.097806 60 0.126511 0.113092 0.086143 9 0.085389 0.079017 0.068235 0.091322 28 0.137218 0.126877 0.100843 61 0.128286 0.115083 0.087732 10 0.093837 0.085778 0.073608 0.101558 62 0.130071 0.116932 0.089570 11 0.100312 0.091946 0.077840 m=2,n=15 63 0.131915 0.118843 0.091227 12 0.106388 0.097665 0.082533 0 0.007298 0.007440 0.007169 0.007458 64 0.133749 0.120701 0.092836 13 0.111932 0.103155 0.086722 1 0.009836 0.009990 0.009507 0.009941 65 0.135579 0.122558 0.094490 14 0.118445 0.109404 0.091413 2 0.011901 0.011712 0.011053 0.012056 66 0.137392 0.124474 0.096206 15 0.124364 0.115842 0.096179 3 0.015961 0.014546 0.013416 0.016237 67 0.139250 0.126396 0.097741 16 0.129968 0.121674 0.101077 4 0.018318 0.016321 0.014906 0.018698 68 0.141185 0.128370 0.099393 5 0.019821 0.017825 0.016198 0.020334 69 0.143022 0.130293 0.101115 6 0.021766 0.019537 0.017565 0.022309 17 v u N E E• v U N E E• v U N m-3, n=3 22 0.098455 0.098729 0.103143 0.107733 0.112244 0.089238 0.093579 0.097918 0.103240 15 0.033870 0.032928 16 0.035736 0.034797 17 0.037780 0.036677 18 0.039874 0.038667 0 0.049923 0.049893 0.049673 0.050365 23 0.102884 1 0.066515 0.067789 0.066545 0.067136 24 0.107376 2 0.077456 0.080749 0.077486 0.077989 25 0.112043 3 0.097819 0.104668 0.098790 0.099330 19 0.041854 0.040560 4 0.113442 0.121864 0.116818 0.117412 m=3, n=7 20 0.043973 0.042432 0 0.008339 0.008389 0.008283 0.008343 21 0.046360 0.044671 m=3, n=4 0 0.028545 0.028358 0.028720 0.028564 1 0.038123 0.038452 0.038167 0.038140 2 0.044334 0.045518 0.044257 0.044352 3 0.055233 0.058384 0.054425 0.056138 4 0.063479 0.067203 0.062108 0.065659 5 0.072575 0.075823 0.070965 0.076967 6 0.084757 0.086801 0.080633 0.091479 7 0.094323 0.096444 0.090245 0.100784 8 0.102040 0.104444 0.098831 1 O.Ql 1204 O.Ql 1330 O.Ql 1018 O.Ql 1111 22 0.048414 0.046767 2 0.012969 0.013261 0.012778 0.012909 23 0.050519 0.048773 3 0.016171 0.016810 0.015752 0.016310 24 0.052697 0.050835 4 0.018597 0.019175 0.017641 0.018889 25 0.054835 0.052914 5 0.021173 0.021309 0.019413 0.021896 26 0.057024 0.055035 6 0.024383 0.024009 0.021430 0.025710 27 0.059189 0.057228 7 0.026935 0.026507 0.023409 0.028454 28 0.061327 0.059411 8 0.029159 0.028680 0.025241 0.030663 29 0.063599 0.061658 9 0.031857 0.031139 0.027461 0.033553 30 0.065798 0.063896 10 0.034546 0.033935 0.029557 0.036455 31 0.068003 0.066125 11 0.036747 0.036253 0.031503 0.038879 32 0.070267 0.068450 9 0.112406 0.115092 0.109729 m=3, n=5 0 0.018076 0.017747 1 0.024200 0.023891 2 0.028202 0.028124 3 0.035072 0.035736 4 0.039982 0.040921 5 0.045531 0.046038 6 0.052393 0.052309 7 0.058180 0.058105 8 0.063134 0.063138 9 0.068669 0.069175 10 0.074656 0.075902 11 0.079788 0.081208 12 0.085358 0.086884 13 0.091474 0.092987 14 0.098266 0.099606 15 0.104873 0.106102 12 0.039247 0.038657 0.033413 0.041702 33 0.072677 0.070813 13 0.041944 0.041149 0.035491 0.044860 34 0.074977 0.073150 0.017893 0.017756 14 0.044752 0.043889 0.037560 0.048102 35 0.077380 0.075468 0.023802 0.023739 15 0.047325 0.046475 0.039608 0.051077 36 0.079768 0.078012 0.027578 0.027525 16 0.049915 0.049048 0.041756 0.054062 37 0.082334 0.080367 0.033847 0.034728 17 0.052657 0.051799 0.044034 0.057267 38 0.084797 0.082732 0.037893 0.040630 18 0.055615 0.054763 0.046468 0.060874 39 0.087353 0.085243 0.042156 0.047551 19 0.058341 0.057554 0.048711 0.064038 40 0.089873 0.087812 0.047646 0.056060 20 0.061376 0.060446 0.051057 0.067389 41 0.092417 0.090458 0.053092 0.061873 21 0.064883 0.063827 0.053526 0.071595 42 0.094976 0.093072 0.057433 0.066725 22 0.0679% 0.066814 0.056117 0.074885 43 0.097525 0.095641 0.062502 0.073068 23 0.070938 0.069652 0.058638 0.078128 44 0.100155 0.098261 0.068289 0.079228 24 0.074067 0.072660 0.061302 0.081485 45 0.102946 0.101096 0.073231 0.084658 25 0.077062 0.075813 0.064188 0.084867 46 0.105691 0.103839 0.078938 0.091008 26 0.080142 0.079005 0.067124 0.088239 47 0.108285 0.106593 0.085233 0.098524 27 0.083297 0.082098 0.070494 0.091772 48 0.111079 0.109362 0.092218 0.105826 28 0.086496 0.085352 0.074435 0.095168 49 0.113813 0.112028 0.100640 29 0.089675 0.088599 0.078174 0.098688 50 0.116587 0.114853 30 0.092953 0.091851 0.081544 0.102141 m=3, n=6 31 0.096048 0.095119 0.084599 m=3, n=9 E E• 0.028029 0.036707 0.029389 0.038814 0.030834 0.041016 0.032386 0.043517 0.033916 0.045748 0.035473 0.048152 0.037174 0.051100 0.038762 0.053495 0.040386 0.055836 0.042088 0.058275 0.043735 0.0606% 0.045499 0.063129 0.047241 0.065639 0.048961 0.068014 0.050914 0.070544 0.052860 0.072978 0.054945 0.075573 0.056971 0.078171 0.059583 0.080911 0.061997 0.083468 0.064347 0.086110 0.067163 0.088837 0.06%46 0.091601 0.071889 0.094304 0.074273 0.097082 0.076600 0.099718 0.078944 0.102549 0.081541 0.083878 0.086215 0.088643 0.090998 0.093417 0.095968 0.098397 0.100933 0 0.011803 0.011929 0.011953 0.011852 32 0.099321 0.098405 0.087861 1 0.015741 0.016066 0.015917 0.015780 33 0.102857 0.101875 0.090956 2 0.018426 0.018971 0.018447 0.018393 34 0.106267 0.105458 0.094224 3 0.023143 0.024033 0.022512 0.023226 35 0.109782 0.108923 0.097336 4 0.026512 0.027404 0.025138 0.026958 36 0.113355 0.112489 0.100773 5 0.029992 0.030576 0.027672 0.031386 0 0.004445 0.004467 0.004460 0.004645 1 0.005991 0.006054 0.005912 0.006131 2 0.007028 0.007091 0.006947 0.007149 3 0.008809 0.008993 0.008539 0.008979 4 0.010101 0.010249 0.009610 0.010299 6 0.034437 0.034465 0.030653 0.036845 7 0.038076 0.038168 0.033941 0.040831 8 0.041290 0.041337 0.036939 0.044041 9 0.045074 0.045072 0.039936 0.048148 10 0.048770 0.049319 0.043091 0.052034 11 0.052075 0.052866 0.046242 0.055659 12 0.055641 0.056443 0.049354 0.059779 13 0.059423 0.060187 0.052202 0.064499 14 0.063351 0.064262 0.055518 0.069208 15 0.067255 0.068311 0.059330 0.073411 16 0.071266 0.072084 0.062876 0.077809 17 0.075238 0.076021 0.066243 0.082347 18 0.079703 0.080574 0.070352 0.087601 19 0.084162 0.084762 0.074417 0.092114 20 0.088692 0.089153 0.078861 0.097046 21 0.093869 0.094260 0.084692 0.103177 0 0.005925 1 0.007888 2 0.009277 3 0.011618 4 0.013354 5 0.015232 6 0.017501 7 0.019386 8 0.020900 9 0.022771 10 0.024656 11 0.026283 12 0.028048 13 0.029931 14 0.031963 5 0.011470 0.011393 0.010549 0.011799 m=3, n=8 6 0.013242 0.012768 0.011593 0.013758 0.006067 0.005923 0.006013 7 0.014566 0.014082 0.012678 0.015253 0.008086 0.007902 0.008050 8 0.015711 0.015218 0.013634 0.016457 0.009532 0.009197 0.009378 9 0.017147 0.016477 0.014608 0.018049 0.012024 0.011299 0.011852 10 0.018637 0.017861 0.015727 0.019532 0.013590 0.012708 0.013688 11 0.019845 0.019101 0.016797 0.020757 0.015142 0.013949 0.015831 12 0.021215 0.020323 0.017795 0.022268 0.017008 0.015408 0.018568 13 0.022635 0.021558 0.018730 0.023993 0.018797 0.016825 0.020550 14 0.024139 0.022961 0.019867 0.025699 0.020343 0.018018 0.022231 15 0.025579 0.024315 0.020888 0.027237 0.022141 0.019436 0.024340 16 0.027027 0.025597 0.021832 0.028805 0.024092 0.021059 0.026400 17 0.028546 0.026960 0.022825 0.030443 0.025787 0.022414 0.028119 18 0.030183 0.028456 0.023941 0.032294 0.027464 0.023758 0.030094 19 0.031629 0.029858 0.024978 0.033961 0.029193 0.025123 0.032383 20 0.033130 0.031272 0.026021 0.035799 0.031062 0.026591 0.034690 21 0.034908 0.032856 0.027180 0.037986 18 v u N 22 0.036406 0.034357 23 0.037931 0.035842 24 0.039530 0.037333 25 0.041125 0.038926 26 0.042817 0.040529 27 0.044406 0.042022 28 0.045980 0.043618 29 0.047657 0.045174 30 0.049302 0.046719 31 0.050940 0.048323 32 0.052551 0.050009 33 0.054314 0.051774 34 0.055933 0.053517 35 0.057628 0.055230 36 0.059453 0.057023 37 0.061164 0.058724 38 0.062878 0.060463 39 0.064751 0.062308 40 0.066619 0.064033 41 0.068383 0.065783 42 0.070249 0.067583 43 0.072055 0.069430 44 0.073895 0.071313 45 0.075906 0.073254 46 0.077857 0.075149 47 0.079772 0.077106 48 0.081684 0.079040 49 0.083666 0.081007 50 0.085663 0.083104 51 0.087676 0.085112 52 0.089611 0.087184 53 0.091602 0.089202 54 0.093651 0.091243 55 0.095688 0.093354 56 0.097706 0.095442 57 0.099884 0.097649 58 0.101921 0.099799 59 0.103902 0.101820 60 0.106091 0.104003 61 0.108263 0.106135 62 0.110527 0.108387 63 0.112717 0.110661 64 0.114862 0.112960 65 0.117044 0.115177 66 0.119205 0.117465 E E* v U N E E* V U N 0.028318 0.039655 13 0.017284 0.016392 0.014129 0.018368 73 0.101471 0.098882 0.029410 0.041402 14 0.018390 0.017385 0.014859 0.019645 74 0.103188 0.100663 0.030632 0.043187 15 0.019535 0.018362 0.015662 0.020875 75 0.104884 0.102411 0.031865 0.044879 16 0.020648 0.019357 0.016395 0.022127 76 0.106602 0.104215 0.033013 0.046655 17 0.021770 0.020394 0.017161 0.023395 77 0.108347 0.106006 0.034184 0.048495 18 0.023070 0.021493 0.018021 0.024838 78 0.110133 0.107883 0.035385 0.050200 19 0.024125 0.022534 0.018833 0.026060 79 0.111931 0.109689 0.036592 0.052091 20 0.025362 0.023629 0.019612 0.027437 80 0.113763 0.111490 0.037876 0.053903 21 0.026734 0.024861 0.020457 0.029130 81 0.115651 0.113309 0.039123 0.055731 22 0.027922 0.025975 0.021265 0.030469 82 0.117447 0.115169 0.040416 0.057538 23 0.029060 0.027107 0.022094 0.031737 83 0.119320 0.117140 0.041819 0.059498 24 0.030316 0.028264 0.022901 0.033068 84 0.121077 0.119062 0.043189 0.061410 25 0.031517 0.029352 0.023736 0.034382 0.014429 0.019114 0.022295 0.027563 m=4, n=4 0.014167 0.019141 0.022476 0.028555 0.044625 0.063304 26 0.032734 0.030474 0.024635 0.035768 0.046040 0.065323 27 0.033993 0.031595 0.025544 0.037176 0 0.047716 0.067377 28 0.035134 0.032831 0.026401 0.038593 I 0.049311 0.069340 29 0.036358 0.033981 0.027288 0.040051 2 0.050867 0.071374 30 0.037562 0.035220 0.028177 0.041449 3 0.052438 0.073364 31 0.038833 0.036413 0.029133 0.042864 0.054075 0.075518 32 0.040068 0.037646 0.030078 0.044244 0.055786 0.077668 33 0.041374 0.038918 0.030958 0.045780 0.057383 0.079741 34 0.042711 0.040119 0.031922 0.047224 0.059221 0.081892 35 0.044001 0.041394 0.032876 0.048709 0.061275 0.084355 36 0.045351 0.042673 0.033811 0.050249 0.063136 0.086460 37 0.046690 0.043953 0.034800 0.051706 0.064848 0.088539 38 0.048006 0.045239 0.035795 0.053227 0.066648 0.090722 39 0.049460 0.046618 0.036853 0.054798 0.068505 0.092785 40 0.050848 0.047983 0.037866 0.056257 0.070254 0.094986 41 0.052164 0.049309 0.039085 0.057778 0.072101 0.097148 42 0.053541 0.050702 0.040192 0.059321 0.073946 0.099308 43 0.054945 0.052054 0.041254 0.060868 0.075723 0.101484 44 0.056324 0.053447 0.042422 0.062463 4 0.031207 0.032622 5 0.035055 0.036765 6 0.039319 0.041696 7 0.044243 0.046559 8 0.048571 0.050727 9 0.053192 0.055636 10 0.059372 0.061500 II 0.064338 0.066213 12 0.068965 0.070891 13 0.073378 0.075581 14 0.078282 0.080896 15 0.083418 0.086185 16 0.088281 0.091222 17 0.093361 0.096238 18 0.099154 0.102373 19 0.105724 0.108416 0.077565 45 0.057947 0.054930 0.043632 0.064349 0.079443 46 0.059377 0.056378 0.044834 0.065989 0.081221 47 0.060873 0.057722 0.045963 0.067585 0.083049 48 0.062306 0.059124 0.047189 0.069115 0.084916 49 0.063739 0.060541 0.048443 0.070750 0.086805 50 0.065170 0.061972 0.049739 0.072449 0.088749 51 0.066627 0.063458 0.051047 0.074138 0.090725 52 0.068030 0.064991 0.052391 0.075792 0.092713 53 0.069525 0.066453 0.053750 0.077458 0.094784 54 0.071004 0.067941 0.055200 0.079118 0.096861 55 0.072494 0.069448 0.056698 0.080734 0.098862 56 0.073946 0.070991 0.058109 0.082455 0.100986 57 0.075466 0.072554 0.059443 0.084238 58 0.077026 0.073996 0.060837 0.086025 m=4, n=5 E E* 0.082744 0.084316 0.086000 0.087636 0.089280 0.090906 0.092582 0.094245 0.095913 0.097665 0.099452 0.101279 0.014361 0.014216 0.018977 0.018977 0.022108 0.022084 0.027367 0.027336 0.031148 0.031072 0.035221 0.035120 0.039933 0.039818 0.044698 0.044723 0.048903 0.048984 0.053952 0.054128 0.060951 0.061066 0.066105 0.066235 0.070353 0.070612 0.074766 0.075081 0.079748 0.080144 0.085070 0.085441 0.089713 0.090036 0.095126 0.095669 0.101767 0.102181 m=3, n=IO 59 0.078562 0.075563 0.062213 0.087741 0 0.007894 I 0.010616 2 0.012359 3 0.015380 4 0.017447 5 0.019483 6 0.021757 7 0.024157 8 0.026402 9 0.028967 10 0.031822 11 0.034399 12 0.036824 13 0.039288 14 0.041940 15 0.044712 16 0.047410 17 0.050048 18 0.053149 19 0.055979 20 0.058861 21 0.062151 22 0.065316 23 0.068474 0.007938 0.007794 0.008019 0.010659 0.010466 0.010788 0.012515 0.012195 0.012534 0.015709 0.015073 0.015453 0.017958 0.016897 0.017529 0.020163 0.018703 0.019773 0.022786 0.020992 0.022301 0.025280 0.023377 0.024965 0.027644 0.025410 0.027348 0.030233 0.027775 0.030151 0.033158 0.030289 0.033723 0.035644 0.032655 0.036489 0.038199 0.035057 0.038895 0.040658 0.037649 0.041361 0.043398 0.040640 0.044259 0.046250 0.044010 0.047096 0.049029 0.046812 0.049620 0.051666 0.049391 0.052493 0.054852 0.052263 0.056040 0.057758 0.055105 0.059430 0 0.003466 0.003579 0.003483 0.003470 60 0.080108 0.077103 0.063610 0.089507 I 0.004677 0.004747 0.004664 0.004632 61 0.081683 0.078719 0.065036 0.091232 2 0.005483 0.005520 0.005433 0.005441 62 0.083208 0.080255 0.066400 0.092877 3 0.006874 0.006880 0.006648 0.006826 63 0.084817 0.081888 0.067859 0.094602 4 0.007857 0.007807 0.007445 0.007894 64 0.086440 0.083530 0.069352 0.096401 5 0.008954 0.008667 0.008143 0.009091 65 0.088084 0.085277 0.070749 0.098183 6 0.010235 0.009678 0.008915 0.010603 66 0.089735 0.086892 0.072223 0.100038 7 0.011299 0.010643 0.009722 0.011720 67 0.091457 0.088580 0.073683 8 0.012173 0.011507 0.010376 0.012630 68 0.093127 0.090276 0.075173 9 0.013189 0.012450 0.011109 0.013803 69 0.094793 0.092034 0.076669 10 0.014274 0.013569 0.011881 0.014933 70 0.096444 0.093776 0.078166 11 0.015218 0.014477 0.012606 0.015969 71 0.098161 0.095520 0.079706 12 0.016228 0.015439 0.013391 0.017119 72 0.099800 0.097201 0.081162 19 0.060687 0.064145 0.067263 0.070401 0.057938 0.062488 0.061314 0.065664 0.064468 0.068973 0.067660 0.071914 v u 24 0.071674 25 0.074740 26 0.078185 27 0.081516 28 0.084811 29 0.088236 30 0.091902 31 0.095318 32 0.098897 33 0.102561 N 0.073636 0.076745 0.080162 0.083526 0.086948 0.090429 0.093905 0.097380 0.100890 0.104681 E E• v U 0.070897 0.075072 48 0.095079 0.074010 0.078355 49 0.097791 0.077471 0.082109 50 0.100463 0.080641 0.085660 51 0.103123 0.083885 0.089280 52 0.105958 0.087259 0.092912 N 0.095990 0.098638 0.101370 0.104074 0.106840 0.090643 0.096729 0.094083 0.100168 0.097379 0.100831 m=4, n=7 0 0.003103 0.003108 I 0.004173 0.004170 2 0.004802 0.004849 3 0.005920 0.006115 m=4, n=6 4 0.006684 0.006951 0 0.004881 0.004778 0.004726 0.004758 5 0.007506 0.007738 I 0.006455 0.006323 0.006333 0.006349 6 0.008430 0.008690 2 0.007506 0.007430 0.007388 0.007420 7 0.009318 0.009626 3 0.009267 0.009401 0.009147 0.009111 8 0.010147 0.010495 4 0.010529 0.010655 0.010261 0.010329 9 0.011063 0.011392 5 0.011753 0.011912 0.011370 0.011641 10 0.012202 0.012394 6 0.013145 0.013403 0.012586 0.013130 II 0.013046 0.013228 7 0.014571 0.014900 0.013860 0.014711 12 0.013890 0.014100 8 0.015889 0.016217 0.015036 0.016146 13 0.014751 0.014971 9 0.017367 0.017784 0.016315 0.017803 14 0.015706 0.015949 10 0.019062 0.019493 0.017679 0.019915 15 0.016584 0.016909 II 0.020543 0.020899 0.018920 0.021541 16 0.017496 0.017851 12 0.021896 0.022340 0.020188 0.022937 17 0.018422 0.018786 13 0.023261 0.023778 0.021436 0.024404 18 0.019475 0.019909 14 0.024789 0.025382 0.022778 0.026008 19 0.020503 0.020976 15 0.026268 0.026943 0.024273 0.027671 20 0.021515 0.022061 16 0.027816 0.028501 0.025709 0.029178 21 0.022601 0.023234 17 0.029370 0.030121 0.027188 0.030750 22 0.023674 0.024309 18 0.031086 0.031865 0.028803 0.032717 23 0.024735 0.025438 19 0.032760 0.033562 0,030408 0.034611 24 0.025798 0.026546 20 0.034366 0.035281 0.032119 0.036394 25 0.026959 0.027697 21 0.036284 0.037146 0.034253 0.038339 26 0.028175 0.028867 22 0.038070 0.038984 0.036065 0.040354 27 0.029355 0.030041 23 0.039883 0.040760 0.037740 0.042151 28 0.030480 0.031234 24 0.041679 0.042512 0.039529 0.044056 29 0.031640 0.032391 25 0.043400 0.044294 0.041364 0.045893 30 0.032800 0.033604 26 0.045356 0.046159 0.043167 0.048146 31 0.034037 0.034800 27 0.047338 0.048042 0.045076 0.050134 32 0.035226 0.036063 28 0.049179 0.049956 0.046929 0.052240 33 0.036471 0.037329 29 0.051127 0.051996 0.048843 0.054309 34 0.037765 0.038655 30 0.053231 0.053955 0.050766 0.056388 35 0.039099 0.039962 31 0.055198 0.055959 0.052575 0.058451 36 0.040457 0.041332 32 0.057285 0.057933 0.054460 0.060635 37 0.041882 0.042700 33 0.059418 0.060114 0.056606 0.063110 38 0.043252 0.044102 34 0.061588 0.062316 0.058760 0.065562 39 0.044618 0.045410 35 0.063782 0.064506 0.060692 0.067924 40 0.045899 0.046761 36 0.066070 0.066785 0.062729 0.070810 41 0.047315 0.048199 37 0.068310 0.069031 0.064821 0.073314 42 0.048779 0.049602 38 0,070605 0.071360 0.066889 0.075832 43 0.050230 0.051028 39 0.072837 0.073691 0.068855 0.078346 44 0.051775 0.052455 40 0.075118 0.076062 0.071024 0.080897 45 0.053357 0.053983 41 0.077468 0.078385 0.073199 0.083285 46 0.054869 0.055514 42 0.080044 0.080790 0.075470 0.085960 47 0.056350 0.057062 43 0.082448 0.083222 0.077642 0.088410 48 0.057865 0.058657 44 0.084898 0.085679 0.079988 0.090947 49 0.059451 0.060209 45 0.087400 0.088259 0.082433 0.093509 50 0.061089 0.061798 46 0.089955 0.090773 0.084840 0.096111 51 0.062729 0.063373 47 0.092458 0.093315 0.087396 0.098554 52 0.064323 0.065015 E E• v U N 0.089865 0.101213 53 0.065922 0.066666 0.092454 54 0.067633 0.068437 0.095127 55 0.069278 0.070175 0.097760 56 0.070987 0.071857 0.100386 57 0.072787 0.073627 58 0.074585 0.075430 59 0.076355 0.077170 0.002992 0.003075 60 0.078213 0.078992 0.003960 0.004113 61 0.080032 0.080836 0.004649 0.004734 62 0.081854 0.082653 0.005787 0.005835 63 0.083679 0.084462 0.006480 0.006651 64 0.085541 0.086306 0.007183 0.007434 65 0.087363 0.088190 0.007910 0.008341 66 0.089277 0.090112 0.008655 0.009348 67 0.091230 0.092026 0.009354 0.010160 68 0.093174 0.094009 0.010184 0.011155 69 0.095141 0.095982 O.ot 1001 0.012433 70 0.097159 0.097981 0.011756 0.013503 71 0.099229 0.099978 0.012449 0.014362 72 0.101308 0.102077 0.013223 0.015289 73 0.103430 0.104243 0.014031 0.016345 74 0.105458 0.106346 0.014800 0.017415 75 0.107515 0.108381 0.015597 0.018325 76 0.109592 0.110426 0.016450 0.019287 0.017333 0.020537 m=4, n=8 0.018193 0.021737 0 0.002033 0.002031 0.019052 0.022893 I 0.002687 0.002709 0.019983 0.024091 2 0.003145 0.003163 0.021009 0.025263 3 0.003912 0.003917 0.021955 0.026416 4 0.004406 0.004423 0.022947 0.027567 5 0.004908 0.004962 0.023993 0.028780 6 0.005508 0.005623 0.025030 0.030080 7 0.006091 0.006279 0.026173 0.031389 8 0.006564 0.006791 0.027472 0.032722 9 0.007175 0.007394 0.028733 0.033988 10 0.007933 0.008096 0.029893 0.035351 11 0.008579 0.008666 0.031038 0.036647 12 0.009138 0.009222 0.032210 0.037978 13 0.009715 0.009785 0.033361 0.039540 14 0.010348 0.010418 0.034583 0.041129 15 0.010918 0.011035 0.035710 0.042637 16 0.011499 0.011623 0.036971 0.044369 17 0.012076 0.012218 0.038206 0.045858 18 0.012793 0.012917 0.039459 0.047420 19 0.013475 0.013585 0.040805 0.049049 20 0.014120 0.014281 0.042056 0.050631 21 0.014831 0.015010 0.043316 0.052145 22 0.015574 0.015717 0.044629 0.053767 23 0.016258 0.016420 0.045977 0.055330 24 0.016951 0.017148 0.047280 0.056995 25 0.017658 0.017879 0.048647 0.058609 26 0.018432 0.018643 0.049970 0.060186 27 0.019244 0.019365 0.051302 0.061781 28 0.020049 0.020157 0.052723 0.063450 29 0.020834 0.020945 0.054095 0.065201 30 0.021604 0.021687 0.055534 0.066846 31 0.022395 0.022496 0.057019 0.068485 32 0.023180 0.023232 0.058481 0.070326 33 0.023997 0.024095 20 E E• 0.059966 0.072050 0.061481 0.073796 0.063004 0.075504 0.064557 0.077299 0.066201 0.079138 0.067907 0.081050 0.069685 0.082830 0.071500 0.084656 0.073204 0.086622 0.075001 0.088480 0.076905 0.090338 0.078731 0.092259 0.080488 0.094197 0.082376 0.096320 0.084193 0.098376 0.086044 0.100400 0.088014 0.089912 0.091768 0.093780 0.095777 0.097782 0.099770 0.101706 0.002000 0.001975 0.002689 0.002649 0.003108 0.003088 0.003879 0.003818 0.004307 0.004349 0.004726 0.004891 0.005207 0.005512 0.005698 0.006148 0.006129 0.006761 0.006630 0.007454 0.007148 0.008258 0.007668 0.008952 0.008109 0.009547 0.008581 0.010198 0.009054 0.010875 0.009591 O.ot 1543 0.010037 0.012216 0.010557 0.012905 0.011118 0.013686 O.ot 1669 0.014459 0.012219 0.015175 0.012794 0.015981 0.013374 0.016768 0.013916 0.017477 0.014494 0.018199 0.015079 0.018992 0.015673 0.019859 0.016317 0.020694 0.016936 0.021558 0.017562 0.022436 0.018262 0.023332 0.018950 0.024210 0.019601 0.025106 0.020357 0.026152 v U N E E* v U N 94 0.095349 0.096262 95 0.096913 0.097876 96 0.098458 0.099491 97 0.099958 0.101069 98 0.101599 0.102655 99 0.103231 0.104299 100 0.104838 0.105894 101 0.106467 0.107520 102 0.108100 0.109161 103 0.109718 0.110759 104 0.111338 0.112387 E 0.086210 0.087660 0.089150 0.090611 0.092154 0.093642 0.095212 0.096787 0.098298 0.099868 0.101491 E• v u 47 0.076992 48 0.079331 49 0.081681 50 0.084010 51 0.086363 52 0.088779 53 0.091246 54 0.093815 55 0.096428 56 0.098902 57 0.101579 N 0.078586 0.080827 0.083060 0.085297 0.087636 0.089998 0.092507 0.094936 0.097348 0.099826 0.102204 34 0.024847 0.024899 0.021129 0.027151 35 0.025689 0.025744 0.021987 0.028116 36 0.026608 0.026621 0.022896 0.029234 37 0.027495 0.027515 0.023673 0.030223 38 0.028367 0.028335 0.024458 0.031224 39 0.029218 0.029262 0.025245 0.032214 40 0.030076 0.030196 0.026030 0.033154 41 0.030966 0.031091 0.026808 0.034144 42 0.03!889 0.032027 0.027685 0.035191 43 0.032831 0.032992 0.028504 0.036214 44 0.033749 0.033879 0.029310 0.037242 45 0.034692 0.034854 0.030160 0.038366 46 0.035688 0.035857 0.030985 0.039377 m=5, n=5 m=5, n=6 47 0.036629 0.036732 0.031858 0.040387 0 48 0.037613 0.037728 0.032719 0.041487 I 49 0.038571 0.038712 0.033643 0.042540 2 50 0.039594 0.039730 0.034546 0.043669 3 51 0.040641 0.040725 0.035432 0.044760 4 0.003989 0.003961 0.003963 0.003997 0 0.002180 0.002222 0.005361 0.005309 0.005318 0.005367 1 0.002887 0.002989 0.006286 0.006266 0.006200 0.006264 2 0.003360 0.003495 0.007744 0.007856 0.007676 0.007627 3 0.004165 0.004349 0.008774 0.008912 0.008663 0.008597 4 0.004728 0.004958 52 0.041632 0.041738 0.036366 0.045869 53 0.042647 0.042780 0.037280 0.047030 54 0.043760 0.043862 0.038227 0.048147 55 0.044808 0.044957 0.039136 0.049339 56 0.045877 0.046051 0.040038 0.050515 57 0.046925 0.047220 0.041034 0.051710 58 0.048011 0.048279 0.041930 0.052877 59 0.049104 0.049383 0.042955 0.054056 60 0.050274 0.050502 0.043951 0.055302 61 0.051358 0.051622 0.044938 0.056487 62 0.052474 0.052760 0.045931 0.057723 63 0.053615 0.053884 0.046939 0.058962 64 0.054792 0.055065 0.047943 0.060215 65 0.055930 0.056284 0.049009 0.061471 66 0.057116 0.057491 0.050087 0.062855 67 0.058307 0.058689 0.051215 0.064076 68 0.059574 0.059908 0.052287 0.065442 69 0.060791 0.061177 0.053430 0.066890 70 0.062024 0.062433 0.054640 0.068284 71 0.063280 0.063694 0.055808 0.069610 72 0.064572 0.064997 0.057100 0.071006 73 0.065831 0.066249 0.058265 0.072442 74 0.067107 0.067547 0.059479 0.073831 75 0.068481 0.068945 0.060699 0.075283 76 0.069776 0.070306 0.061887 0.076725 77 0.071073 0.071673 0.063132 0.078117 78 0.072444 0.073031 0.064494 0.079670 79 0.073771 0.074405 0.065806 0.081103 80 0.075158 0.075738 0.067113 0.082482 5 6 0.009733 0.009957 0.010823 0.011181 7 0.011938 0.012372 8 0.012990 0.013474 9 0.014193 0.014697 10 0.015378 0.016107 II 0.016522 0.017304 12 0.017685 0.018562 13 0.018950 0.019852 14 0.020287 0.021224 15 0.021739 0.022596 16 0.023042 0.023929 17 0.024273 0.025217 18 0.025707 0.026653 19 0.027015 0.028043 20 0.028399 0.029443 21 0.029935 0.030994 22 0.031397 0.032464 23 0.032837 0.033876 24 0.034364 0.035400 25 0.035980 0.036980 26 0.037461 0.038619 27 0.039013 0.040208 28 0.040576 0.041804 29 0.042121 0.043441 30 0.043781 0.045196 31 0.045496 0.046958 32 33 81 0.076532 0.077110 0.068391 0.083910 34 82 0.077873 0.078517 0.069699 0.085344 35 83 0.079231 0.079929 0.070954 0.086809 36 84 0.080719 0.081353 0.072296 0.088309 37 85 0.082124 0.082778 0.073600 0.089790 38 86 0.083584 0.084248 0.074958 0.091265 39 87 0.084999 0.085730 0.076399 0.092698 40 88 0.086436 0.087261 0.077760 0.094214 41 89 0.087858 0.088703 0.079146 0.095764 42 90 0.089336 0.090155 0.080551 0.097253 43 91 0.090830 0.091693 0.081964 0.098819 44 92 0.092361 0.093192 0.083369 0.100376 45 93 0.093862 0.094761 0.084802 46 0.047210 0.048951 0.050646 0.052478 0.054297 0.056128 0.057934 0.059933 0.061908 0.063866 0.066032 0.068101 0.070205 0.072431 0.074682 0.048621 0.050369 0.052152 0.054010 0.055865 0.057753 0.059581 0.061602 0.063578 0.065552 0.067635 0.069767 0.071963 0.074149 0.076379 0.009641 0.009492 5 0.005264 0.005546 0.010770 0.010616 6 0.005810 0.006232 0.011919 0.011819 7 0.006443 0.006865 0.013013 0.012817 8 0.006973 0.007450 0.014208 0.013956 9 0.007572 0.008069 0.015481 0.015192 10 0.008220 0.008763 0.016665 0.016316 II 0.008765 0.009355 0.017903 0.017478 12 0.009369 0.009979 0.019168 0.018740 13 0.010020 0.010667 0.020615 0.020174 14 0.010702 0.011318 0.022216 0.021751 15 0.011420 0.012023 0.023547 0.023073 16 0.012057 0.012705 0.024860 0.024369 17 0.012722 0.013352 0.026297 0.025818 18 0.013490 0.014144 0.027735 0.027186 19 0.014224 0.014925 0.029087 0.028597 20 0.014899 0.015725 0.030704 0.030236 21 0.015735 0.016556 0.032224 0.031718 22 0.016549 0.017342 0.033815 0.033242 23 0.017300 0.018109 0.035360 0.034796 24 0.018019 0.018945 0.036930 0.036379 25 0.018784 0.019734 0.038621 0.038030 26 0.019576 0.020567 0.040195 0.039634 27 0.020454 0.021429 0.041769 0.041199 28 0.021277 0.022306 0.043360 0.042853 29 0.022140 0.023142 0.045102 0.044612 30 0.023046 0.024030 0.046780 0.046270 31 0.023887 0.024946 0.048420 0.048008 32 0.024770 0.025826 0.050214 0.049756 33 0.025638 0.026817 0.052094 0.051579 34 0.026540 0.027766 0.053878 0.053363 35 0.027421 0.028658 0.055702 0.055172 36 0.028362 0.029622 0.057701 0.057171 37 0.029287 0.030553 0.059661 0.059061 38 0.030235 0.031573 0.061619 0.060995 39 0.031247 0.032556 0.063532 0.062958 40 0.032206 0.033580 0.065498 0.065002 41 0.033263 0.034646 0.067760 0.067176 42 0.034311 0.035755 0.069942 0.069290 43 0.035386 0.036809 0.072140 0.071416 44 0.036433 0.037962 0.074484 0.073843 45 0.037553 0.039069 0.076724 0.076122 46 0.038625 0.040207 21 E E* 0.078999 0.078343 0.081480 0.080738 0.083889 0.083124 0.086210 0.085542 0.088685 0.088015 0.091255 0.090562 0.093782 0.093014 0.096229 0.095580 0.098927 0.098296 0.101445 0.100736 0.002211 0.002161 0.002931 0.002888 0.003406 0.003375 0.004197 0.004194 0.004715 0.004697 0.005219 0.005215 0.005788 0.005853 0.006396 0.006481 0.006910 0.007072 0.007471 0.007759 0.008115 0.008464 0.008704 0.009077 0.009274 0.009739 0.009884 0.010476 0.010523 0.011230 O.oJ 1142 0.012040 0.011854 0.012758 0.012465 0.013479 0.013224 0.014281 0.013931 0.015026 0.014700 0.015779 0.015576 0.016641 0.016398 0.017440 0.017122 0.018226 0.017933 0.019044 0.018740 0.019874 0.019599 0.020792 0.020454 0.021657 0.021283 0.022508 0.022120 0.023440 0.022962 0.024375 0.023751 0.025297 0.024574 0.026186 0.025497 0.027152 0.026409 0.028134 0.027341 0.029105 0.028296 0.030055 0.029221 0.031095 0.030155 0.032117 0.031097 0.033202 0.032051 0.034237 0.033028 0.035312 0.034000 0.036471 0.035042 0.037630 0.036083 0.038761 0.037098 0.040071 0.038111 0.041290 v u N 47 0.039771 0.041371 48 0.040962 0.042576 49 0.042148 0.043803 50 0.043349 0.045023 51 0.044572 0.046264 52 0.045759 0.047548 53 0.046964 0.048749 54 0.048226 0.050004 55 0.049492 0.051253 56 0.050773 0.052527 57 0.052140 0.053863 58 0.053503 0.055241 59 0.054830 0.056593 60 0.056099 0.057941 61 0.057498 0.059256 62 0.058939 0.060634 63 0.060347 0.062010 64 0.061789 0.063344 65 0.063234 0.064786 66 0.064737 0.066284 67 0.066223 0.067690 68 0.067708 0.069274 69 0.069194 0.070723 70 0.070761 0.072210 71 0.072249 0.073693 72 0.073797 0.075287 73 0.075385 0.076859 74 0.077037 0.078484 75 0.078684 0.080034 76 0.080310 0.081665 77 0.081986 0.083276 78 0.083674 0.084924 79 0.085338 0.086540 80 0.087015 0.088225 81 0.088766 0.089923 82 0.090492 0.091705 83 0.092280 0.093462 84 0.094064 0.095244 85 0.095783 0.097029 86 0.097560 0.098805 87 0.099423 0.100612 88 0.101223 0.102376 89 0.103060 0.104201 E E" v U N E E" 0.039259 0.042522 14 0.006163 0.006519 0.005809 0.006407 0.040415 0.043782 15 0.006600 0.006969 0.006151 0.006896 0.041505 0.044919 16 0.007015 0.007359 0.006486 0.007290 0.042724 0.046189 17 0.007384 0.007751 0.006854 0.007707 0.043894 0.047528 18 0.007786 0.008193 0.007215 0.008185 0.045090 0.048807 19 0.008182 0.008637 0.007555 0.008622 0.046321 0.050102 20 0.008562 0.009051 0.007912 0.009065 0.047589 0.051497 21 0.008964 0.009511 0.008338 0.009560 0.048888 0.052975 22 0.009389 0.009945 0.008748 0.009990 0.050195 0.054283 23 0.009813 0.010371 0.009172 0.010465 0.051609 0.055640 24 0.010213 0.010807 0.009571 0.010957 0.052937 0.056946 25 0.010660 0.011285 0.010015 0.011456 0.054268 0.058357 26 0.011146 0.011770 0.010471 0.011941 0.055616 0.059680 27 0.011606 0.012270 0.010950 0.012442 0.056958 0.061072 28 0.012124 0.012746 O.oJ 1461 0.012955 0.058343 0.062519 29 0.012617 0.013225 0.011950 0.013465 0.059725 0.064012 30 0.013072 0.013693 0.012387 0.014019 0.061127 0.065434 31 0.013581 0.014198 0.012914 0.014551 0.062604 0.066910 32 0.014079 0.014687 0.013406 O.oi5097 0.063991 0.068323 33 0.014642 0.015213 0.013885 0.015662 0.065435 0.069735 34 0.015190 0.015758 0.014406 0.016239 0.067007 0.071206 35 0.015697 0.016306 0.014884 0.016821 0.068539 0.072736 36 0.016215 0.016838 0.015385 0.017443 0.070042 0.074205 37 0.016692 0.017372 0.015871 0.018034 0.071513 0.075741 38 0.017242 0.017945 0.016385 0.018578 0.073076 0.077207 39 0.017784 0.018487 0.016884 0.019180 0.074661 0.078732 40 0.018344 0.019043 0.017443 0.019824 0.076232 0.080247 41 0.018887 0.019585 0.018017 0.020427 0.077899 0.081894 42 0.019481 0.020150 0.018560 0.021061 0.079558 0.083486 43 0.020081 0.020758 0.019089 0.021748 0.081205 0.085199 44 0.020646 0.021337 0.019601 0.022416 0.082964 0.086941 45 0.021255 0.021916 0.020182 0.023076 0.084567 0.088553 46 0.021873 0.022574 0.020757 0.023742 0.086146 0.090230 47 0.022496 0.023197 0.021330 0.024430 0.087788 0.092021 48 0.023119 0.023877 0.021895 0.025141 0.089478 0.093733 49 0.023760 0.024510 0.022531 0.025852 0.091199 0.095446 50 0.024407 0.025174 0.023165 0.026584 0.092830 0.097219 51 0.025064 0.025839 0.023795 0.027339 0.094489 0.098989 52 0.025735 0.026508 0.024442 0.028141 0.096252 0.100821 53 0.026433 0.027175 0.025098 0.028933 0.097862 54 0.027157 0.027883 0.025734 0.029656 0.099596 55 0.027875 0.028559 0.026336 0.030466 0.101321 56 0.028513 0.029284 0.027006 0.031236 57 0.029200 0.030045 0.027656 0.032013 m=5, n=7 58 0.029916 0.030777 0.028360 0.032797 0 0.001303 0.001253 0.001251 0.001273 59 0.030687 0.031505 0.029110 0.033570 I 0.001723 0.001688 0.001682 0.001671 60 0.031433 0.032268 0.029803 0.034333 2 0.002002 0.001966 0.001968 0.001918 61 0.032134 0.033039 0.030532 0.035156 3 0.002488 0.002474 0.002388 0.002370 62 0.032929 0.033810 0.031246 0.035975 4 0.002794 0.002813 0.002656 0.002662 63 0.033726 0.034558 0.032012 0.036822 5 0.003099 0.003141 0.002944 0.003004 64 0.034525 0.035355 0.032744 0.037584 6 0.003441 0.003541 0.003247 0.003382 65 0.035326 0.036147 0.033477 0.038406 7 0.003761 0.003896 0.003609 0.003751 66 0.036115 0.036919 0.034276 0.039210 8 0.004061 0.004225 0.003900 0.004076 67 0.036929 0.037729 0.035065 0.040044 9 0.004405 0.004614 0.004237 0.004445 68 0.037712 0.038536 0.035863 0.040868 10 0.004749 0.005034 0.004568 0.004875 69 0.038533 0.039424 0.036672 0.041801 11 0.005108 0.005409 0.004859 0.005269 70 0.039329 0.040340 0.037508 0.042670 12 0.005429 0.005760 0.005183 0.005612 71 0.040217 0.041175 0.038278 0.043547 13 0.005778 0.006112 0.005505 0.005977 72 0.041043 0.042034 0.039082 0.044402 22 v U N E E" 73 0.041912 0.042897 0.039862 0.045258 74 0.042785 0.043772 0.040687 0.046138 75 0.043722 0.044676 0.041525 0.047035 76 0.044578 0.045569 0.042397 0.047914 77 0.045468 0.046450 0.043232 0.048880 78 0.046430 0.047351 0.044088 0.049809 79 0.047341 0.048277 0.044940 0.050733 80 0.048287 0.049231 0.045818 0.051707 81 0.049218 0.050157 0.046654 0.052684 82 0.050202 0.051108 0.047550 0.053673 83 0.051129 0.052103 0.048474 0.054659 84 0.052149 0.053071 0.049433 0.055715 85 0.053115 0.054022 0.050417 0.056748 86 0.054116 0.054936 0.051354 0.057729 87 0.055163 0.055871 0.052326 0.058752 88 0.056173 0.056893 0.053278 0.059801 89 0.057199 0.057926 0.054222 0.060799 90 0.058253 0.058936 0.055208 0.061855 91 0.059254 0.060011 0.056217 0.062911 92 0.060277 0.061050 0.057172 0.063998 93 0.061357 0.062124 0.058179 0.065116 94 0.062498 0.063203 0.059166 0.066167 95 0.063598 0.064271 0.060156 0.067269 96 0.064664 0.065374 0.061191 0.068398 97 0.065723 0.066502 0.062221 0.069505 98 0.066876 0.067595 0.063204 0.070627 99 0.068006 0.068652 0.064299 0.071817 100 0.069119 0.069833 0.065340 0.072982 101 0.070294 0.070973 0.066407 0.074123 102 0.071452 0.072164 0.067502 0.075256 103 0.072572 0.073314 0.068586 0.076440 104 0.073702 0.074476 0.069716 0.077592 105 0.074838 0.075643 0.070920 0.078800 106 0.076012 0.076839 0.072054 0.079968 107 0.077185 0.078011 0.073199 0.081171 108 0.078378 0.079223 0.074318 0.082355 109 0.079631 0.080451 0.075424 0.083599 110 0.080859 0.081673 0.076717 0.084885 Ill 0.082124 0.082863 0.077931 0.086157 112 0.083389 0.084108 0.079161 0.087441 113 0.084675 0.085323 0.080397 0.088689 114 0.085926 0.086512 0.081589 0.089986 115 0.087200 0.087803 0.082820 0.091264 116 0.088448 0.089077 0.084062 0.092614 117 0.089759 0.090244 0.085327 0.093955 118 0.091051 0.091534 0.086593 0.095239 119 0.092366 0.092795 0.087862 0.096582 120 0.093689 0.094174 0.089136 0.097942 121 0.095017 0.095459 0.090463 0.099301 122 0.096331 0.096796 0.091712 0.100691 123 0.097644 0.098193 0.092998 124 0.098945 0.099524 0.094284 125 0.100338 0.100906 0.095589 126 0.101725 0.102306 0.096888 127 0.103149 0.103674 0.098228 128 0.104618 0.105060 0.099559 129 0.105974 0.106538 0.100947 Table 2.3: Critical Values for V (U: Uniform, N: Normal, E: Exponential, E*: Negative Exponen-tial) m=2 m=2 m=2 m=2 m=2 m=2 m=2 m=2 m=2 m=2 m=2 m=3 n=3 n=4 n=5 n=6 n=7 n=S n=9 n= 10 n= 15 n=20 n=25 n=3 u .001 N E E* u 2 .005 N 2 E 2 E* 2 u I 3 7 .01 N I 4 10 E I 5 12 E* I 3 7 u 0 I 2 7 15 26 .025 N 0 I 2 9 20 36 E 0 I 2 II 27 50 E* 0 I 2 7 14 25 u 0 I 2 3 5 7 19 37 61 0 .05 N 0 I 2 3 5 8 24 47 79 0 E 0 I 2 4 7 10 33 66 109 0 E* 0 I 2 3 4 6 18 35 57 0 u 0 I 2 5 8 10 14 18 45 83 134 3 .10 N 0 I 3 5 8 12 16 21 52 98 157 2 E 0 I 4 7 11 15 21 27 68 126 202 3 E* I 2 4 7 9 13 17 41 76 123 3 m=3 m=3 m=3 m=3 m=3 m=3 m=3 m=3 m=3 m=3 m=4 m=4 m=4 n=4 n=5 n=6 n=7 n=8 n=9 n= 10 n= 15 n=20 n=25 n=4 n=5 n=6 u 2 7 .001 N 2 8 E 2 9 E* 2 6 u 0 I 10 24 45 0 .005 N 0 I II 29 56 0 E 0 I 13 38 74 0 E* 0 I 9 23 43 0 u 0 2 3 5 22 48 83 0 3 .OJ N 0 2 3 6 26 58 102 0 3 E 0 2 4 7 32 77 138 0 3 E* 0 2 3 5 20 45 79 0 3 u I 3 6 10 14 19 55 107 176 2 7 14 .025 N I 3 6 10 15 21 62 124 207 2 6 13 E I 3 7 12 19 26 79 162 269 2 7 15 E* I 3 5 9 13 18 50 100 165 2 7 13 u 2 5 10 16 22 30 39 100 188 303 8 16 28 .05 N 2 5 10 16 23 31 41 108 206 335 7 16 28 E 2 6 12 19 28 38 50 131 249 404 8 17 29 E* 2 5 9 14 20 27 35 91 173 279 8 16 26 u 7 14 22 32 43 57 72 172 317 505 18 32 49 .10 N 7 14 22 32 44 58 73 178 329 523 17 31 49 E 8 14 24 35 49 65 83 200 370 592 17 32 51 E* 6 13 20 29 40 52 65 !58 291 462 17 30 47 23 m=4 m=4 m=4 m=4 m=4 m=4 m=4 m=5 m=5 m=5 m=5 m=5 m=5 n=7 n=8 n=9 n= 10 n= 15 n=20 n=25 n=5 n=6 n=7 n=8 n=9 n= 10 u 9 29 59 3 7 .001 N 10 33 69 3 6 E 12 40 92 3 7 E* 9 30 58 3 6 u 2 5 8 13 50 109 193 0 4 10 19 29 42 .005 N 2 5 8 13 55 125 222 0 4 9 18 29 42 E 2 5 10 16 70 159 291 0 4 11 20 33 48 E* 2 5 8 12 47 105 180 0 4 10 18 27 39 u 7 13 20 28 87 175 297 5 12 23 37 52 72 .01 N 7 13 20 29 94 196 333 5 12 22 34 52 72 E 8 15 24 34 118 246 422 5 13 24 39 58 80 E* 7 12 19 26 80 166 277 5 12 22 34 49 65 u 23 34 46 61 164 315 514 17 32 50 73 99 129 .025 N 22 34 47 63 173 339 554 16 31 49 71 98 130 E 25 38 54 72 204 399 656 17 32 52 77 107 142 E* 21 31 43 56 151 292 479 17 30 47 68 93 120 u 42 59 79 101 254 474 762 33 55 81 114 150 192 .05 N 42 59 80 103 261 493 794 32 53 80 112 150 192 E 46 65 88 114 293 553 896 32 55 84 119 !59 205 E* 39 55 73 93 235 442 709 32 52 78 108 143 182 u 71 97 126 158 379 695 1106 56 87 124 168 219 277 .10 N 71 96 126 159 382 704 1120 56 86 124 168 219 277 E 75 103 134 170 411 759 1210 55 88 128 175 228 288 E* 67 91 119 150 357 656 1044 55 85 121 164 213 268 m=5 m=5 m=5 m=6 m=6 m=6 m=6 m=6 m=6 m=6 m=6 m=7 m=7 n= 15 n=20 n=25 n=6 n=7 n=8 n=9 n= 10 n = 15 n=20 n=25 n=7 n=8 u 43 105 197 2 7 14 24 108 253 460 8 20 .001 N 44 117 221 2 7 14 23 114 272 503 8 18 E 51 151 281 2 7 15 26 137 322 601 8 20 E* 42 101 191 2 7 13 24 102 237 433 8 19 u 135 279 479 12 25 43 64 88 274 551 936 47 78 .005 N 144 312 527 12 24 41 61 88 278 587 998 45 74 E 170 369 642 13 26 44 67 96 321 670 1148 47 78 E* 126 265 451 13 24 41 60 84 256 522 885 47 74 u 204 406 684 27 47 72 101 137 385 753 1257 79 118 .01 N 215 438 729 25 45 69 99 134 389 782 1302 75 115 E 251 508 860 26 47 73 106 146 437 872 1467 77 120 E* 192 384 639 26 45 68 95 128 357 708 1179 78 114 u 340 646 1059 57 88 127 171 223 580 1101 1799 136 194 .025 N 349 673 1092 54 86 124 169 220 584 1121 1828 133 190 E 386 747 1226 56 88 129 177 234 631 1213 1992 134 196 E* 317 607 995 56 85 121 163 213 550 1046 1710 134 189 u 480 894 1443 89 133 184 243 311 775 1446 2330 195 270 .05 N 485 914 1465 88 131 182 242 310 778 1456 2346 194 270 E 523 982 1587 89 133 188 251 324 823 1541 2494 195 275 E* 454 847 1366 89 130 178 236 302 748 1392 2246 195 269 u 663 1215 1935 134 192 260 338 427 1022 1879 2994 275 371 .10 N 666 1225 1946 133 192 259 338 427 1028 1887 3010 275 372 E 699 1287 2054 135 195 266 346 440 1066 1957 3130 276 377 E* 640 1173 1866 135 191 258 334 422 1007 1839 2930 276 373 24 m=7 m=7 m=7 m=7 m-7 m-8 m-8 m-8 m-8 m-8 m-8 m=9 m=9 n=9 n= 10 n= 15 n=20 n=25 n=8 n=9 n= 10 n= 15 n=20 n=25 n=9 n= 10 u 34 56 216 483 840 37 64 96 361 784 1409 105 157 .001 N 34 50 215 496 895 36 63 95 361 809 1449 103 152 E 37 60 248 576 1048 38 67 103 406 915 1655 107 155 E• 34 54 199 452 814 38 65 97 336 730 1339 107 152 u 113 155 465 941 1562 121 174 239 702 1400 2371 254 340 .005 N 109 150 463 950 1617 117 168 232 698 1430 2429 242 336 E 117 163 517 1065 1805 119 176 245 758 1553 2644 250 340 E• 108 146 431 879 1480 119 170 230 672 1348 2265 249 332 u 166 222 619 1213 1984 177 245 326 904 1775 2934 344 455 .01 N 161 217 619 1230 2044 172 240 322 904 1791 2988 335 449 E 169 231 671 1345 2250 174 249 336 965 1911 3211 341 456 E• 159 210 584 1150 1907 173 242 318 876 1708 2829 341 449 u 262 337 878 1678 2725 275 369 479 1242 2369 3853 499 648 .025 N 258 337 883 1690 2754 271 366 477 1246 2386 3904 494 645 E 267 352 936 1804 2952 275 375 492 1307 2506 4101 502 656 E• 255 328 849 1611 2628 275 366 475 1222 2319 3755 502 646 u 358 456 1134 2122 3421 374 496 630 1570 2938 4727 653 834 .05 N 356 456 1140 2139 3439 372 492 630 1579 2955 4762 652 837 E 365 471 1190 2240 3620 377 502 646 1630 3062 4942 660 849 E• 353 449 1113 2072 3340 377 496 630 1557 2898 4650 659 840 u 483 608 1460 2687 4285 503 655 825 1983 3647 5810 850 1073 .10 N 483 609 1469 2703 4298 503 654 826 1991 3661 5842 852 1079 E 492 624 1508 2785 4448 508 663 840 2034 3757 5992 861 1089 E• 482 608 1447 2655 4226 507 658 829 1977 3619 5764 859 1083 m=9 m=9 m=9 m=10 m=10 m=10 m=10 m= 15 m=15 m=15 m=20 m=20 m=25 n= 15 n=20 n=25 n= 10 n= 15 n=20 n=25 n = 15 n=20 n=25 n=20 n=25 n=25 u 558 1196 2083 233 782 1674 2875 2580 5452 9315 11504 19552 33837 .001 N 552 1188 2134 217 773 1697 2955 2592 5416 9404 11403 19886 34202 E 593 1347 2396 223 834 1853 3263 2585 5610 9815 11503 20225 34394 E• 523 lliO 1986 221 757 1603 2755 2609 5387 9224 11624 19628 34596 u 996 1988 3333 459 1346 2663 4465 3860 7648 12804 15310 25726 42971 .005 N 993 2012 3382 449 1328 2682 4512 3864 7683 12915 15317 25838 43138 E 1050 2164 3679 460 1400 2837 4816 3868 7882 13300 15506 26160 43430 E• 970 1916 3208 459 1308 2584 4324 3865 7665 12736 15512 25818 43620 u 1257 2448 4043 601 1660 3220 5336 4537 8858 14635 17346 28819 47662 .01 N 1247 2468 4084 592 1653 3248 5389 4563 8857 14724 17407 28900 47865 E 1309 2615 4369 604 1725 3380 5677 4574 9079 15108 17580 29261 48215 E• 1226 2385 3931 604 1629 3163 5213 4573 8908 14616 17574 28939 48377 u 1673 3179 5192 838 2172 4135 6723 5636 10734 17508 20561 33617 54948 .025 N 1681 3213 5231 830 2175 4149 6774 5662 10772 17581 20621 33735 55076 E 1735 3352 5473 846 2243 4282 7041 5691 10946 17923 20818 34103 55429 E• 1659 3138 5095 847 2157 4079 6635 5702 10820 17554 20785 33785 55553 u 2084 3886 6263 1068 2662 4983 8022 6676 12482 20145 23467 37956 61440 .05 N 2091 3910 6303 1064 2664 4998 8053 6683 12515 20226 23536 38077 61539 E 2136 4038 6511 1080 2725 5116 8305 6717 12673 20503 23677 38393 61915 E• 2073 3861 6188 1081 2656 4951 7967 6724 12555 20188 23686 38185 61928 u 2589 4756 7587 1358 3271 6022 9618 7924 14586 23322 26959 43211 69167 .10 N 2601 4773 7621 1358 3277 6039 9647 7926 14613 23381 27036 43284 69275 E 2639 4881 7789 1371 3330 6141 9844 7972 14749 23606 27151 43539 69611 E• 2585 4742 7537 1372 3270 6003 9573 7976 14672 23357 27153 43401 69585 25 Table 2.4: Critical Values for V When m = n (C: Cauchy, D: Double Exponential) m=3 m=4 m=5 m=6 m=7 m=8 m=9 m= 10 m= 15 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n= 10 n = 15 .001 c 7 34 98 208 2545 D 8 34 101 211 2567 .005 c 0 11 42 113 237 448 3855 D 0 11 44 113 239 447 3841 .01 c 4 24 72 167 331 590 4580 D 4 25 74 168 332 589 4567 .025 c 2 15 52 130 268 495 843 5733 D 2 16 54 131 270 497 833 5687 .05 c 7 31 86 193 373 659 1083 6767 D 7 31 87 192 371 654 1069 6711 .10 c 2 16 54 133 276 508 862 1378 8025 D 2 17 55 134 274 504 855 1364 7959 26 IKro go·o I>Q'O (•-llld .(j!t!CfOqOJd ~ S! 0 .; "' oi0 oi 0 "' 0 0 0 7 c oi I E .e ~ g :5 o~·o ao·o 90'0 ~ S! 0 .; "' oi0 oi "1 0 "' 0 0 0 1 "i E .e 'ii ~ 0 z oro s1ro oo·o I>Q'O ~o·o (••llld .(jmq-eqOJd ~ c .. E .e 'ii !l z oc·o s~·o w·o s ~ ·o (•·llld .(jmq-eqOJd ~ S! 0 .; "' oi0 oi 1 oi I E .e 'ii i ! .. E "1 .e - 'ii "' 0 0 0 l w Figure 2.1: A plot of distribution of V for different m and n (Exact values when m= 1, n=2, Simu-lated values when n=2, n=3) 27 ·'. ·: ··.:.: =r. .. ~ .. ~ ~ .. .. ·~ . . .. '. ,. . .. ·. ·. ·:.··· . . ·. •• 1 · .: . · .. ·:· ·' I ·.: :. ·:: .. ·: ' .. .. ~ ·: .. ~ '' . ~ .. ·. .. ' i: ... ~ . I ... . .. ··':. "' .. "' ~ ' "' ! ·: ! ! ... 1 : 1 -': .. I ,I E .. E .. E .e .. .e , . .e ' .,. § .. § § ]! e "ii ·' .... c ~ ·. ~ · .. · ~ :·~ ::> · .. :;::.; z ·. ·. ! ·. ;. ' ·. , .. ·::· .. ... . · . .. :. ·: ~- :, . .. ii! ·. ·. ii! ii! .. . .. , ., ·! ~ ·:· I ··. ·:·:.\. I, I I ·. , ,. .. .. '· · . : I • .. .····: . ··: 0 soo·o 900"0 1'00"0 ~oo·o soo·o 900"0 1'00"0 ~oo·o 010"0 soo·o 900"0 1'00"0 ~oo·o (A•I\)d . z w ·. 0 ~ ~ N : ·. : s~o·o o~o·o s~o·o ow·o soo·o 0£()"0 s~o·o ~o·o 510"0 o~o·o soo·o 0£0"0 s~o·o o~·o s~o·o o~o·o (A•i\)d .\l!)!ql!q0Jd (A-Nd .llmqvqOJd (A•A)d Al!l!qvqOJd Figure 2.2: A plot of distribution of V (simulated) for different m and n 28 9(ro ~·o aro z:o·o lo·o (A•Ald ~mq&qwd !i : : : ·. llO'O 800'0 tOO'O (•·rJd -'l!lt Fi), i=1,2, ... ,k. When ties occur in the Ni observations, the values of J.L(Vi) are used and a modification of variance of Vi is obtained by reducing the expression Ni-tni l: t(t- 1)(13t- 20) [1 5288N (ni) 1 5288 . (Ni-t) 24Ni(Ni- 1) + . ,_1 3 + . n1 3 + 6.5778 (Ni-1) (ni)l l:t(t -1)(t- 2)(t- 3) 3 3 Ni(Ni- 1)(Ni- 2)(Ni- 3) 86 Table 5.1: Reaction Times in Seconds of Mice to Stimuli to their Tails Reaction time GroupO 2.4 3.0 3.0 2.2 2.2 2.2 2.2 2.8 2.0 3.0 Group 1 2.8 2.2 3.8 9.4 8.4 3.0 3.2 4.4 3.2 7.4 Group 2 9.8 3.2 5.8 7.8 2.6 2.2 6.2 9.4 7.8 3.4 Group 3 7.0 9.8 9.4 8.8 8.8 3.4 9.0 8.4 2.4 7.8 As noted in Tamhane at al. (1996), the family of null hypotheses H = HOi, where Ho; : (F0 = F1 = · · · = F;_1 = F;) fori= 1, 2, ... , k, is closed under intersection in the sense that H0; E Hand Hoi E H imply H0; nH0i E H. Hence, a level-a closed procedure that includes separate level-a tests of individual H 0; applied in a step-down manner can be employed in finding the MED. To estimate the MED, we first let k1 = k and find V{~t)' where V(~t) is the maximum of v;_•, V2*, ... , Vk~. Since the results in Terpstra (1952) imply that the statistics Vt, v;•, ... , Vk~ are stochastically independent and we have showed that they are standard normal approach under the null hypothesis Hok1 • Therefore, we observe that P{Vk'1 ~ z(a)IHokJ = [P{v;_• ~ z(a)IHokJ]k1 ~ (1 - a)k1 , where z(a) is the upper ath percentile of the standard normal distribution. let a(k1) = 1 - (1 - a) 1/kt. Define d(k1) to be antirank of V{~t) that V{~t) = Vd(k 1 ). Then, if \1(~1 ) 2: z(a(ki)), reject H0i, j = d(k1 ), •.. , k1. and go to the second step with k2 = d(k1 ) - 1; otherwise, stop testing and accept all hypothesis. 5.3 An example Consider the data set in Table 5.1 analyzed in Shirley (1977) which contains three dose levels and a zero-dose control, there are 10 observations in each group. (1) LY test: the V statistics, their corresponding means and standard deviation are 87 obtained in the following:V1 = 4829, V2 = 16274, V3 = 39633, J.L(Vt) = 2525, J.L(V2 ) = 10050, J.L(V3) = 22575, O"(Vt) = 881.32, O"(V2) = 3058, O"(V3 ) = 6490, Vt = 2.614, \12* = 2.035, V3* = 2.628. Note that the largest statistic among the three Vt's is V3*, so d(3)=3. Since at the level a =0.05, V3* = 2.628 > z(1 - (.95) 113) = 2.1212, we go to the second step with k2 = 2. We observe that d(l)=l and Vt = 2.614 > z(1 - (.95)) = 1.6448. Therefore, we estimate that, at the 5% significant level, the MED is the first dose level. (2) Chen test: the statistic is: i-1 n; ni Ti = L L L f(Yiu-Yjv}• i = 1, 2, ... , k, j=O u=1 v=O let i = 1, 2, ... , k, where J.L(Ti) = niNi_tf2 and 0"2 (7i) = niNi-1(Ni + 1)/12, with Ni the mean and variance of Ti under Hoi· If there are ties among the Ni observations, a modification ofTt is obtained by replacing the Ni + 1 in 0"2 (Ti) with Ni + 1- L t(t2 - 1)/[Ni(Ni- 1)], their corresponding means and variances are obtained in the following: T1 = 84, T2 145, T3 = 235, J.L(Tt) = 50, J.L(T2 ) = 100, J.L(T3 ) = 150, 0"2 (T1 ) = 170.79, 0"2 (T2 ) 510.69, 0"2 (T3 ) = 1018.94, T{ = 2.60, r; = 1.99, T3 = 2.66. We estimate that, at the 5% significant level, the MED is the first dose level. (3) Williams test: the statistic is: i i L - I: - 2 1 1 1/2 ti =[max ( n·~·/ n·)- ~o]/[O"· (- + -)] 1:'SJL:'Si . 3 3 . 3 z ni n0 ' J=U J=U i=1,2, ... ,k, h 2 N;(N;+l} . 1 2 k Wh . . h ki k w ere O"i = 12 , z = , , ... , ,. en tles occur m t e ran ngs, average ran s are used and corrections for ties in the rankings are made by reducing the expression Ni ( Ni + 1)/12 by L t(t2 - 1)/12(Ni- 1). 88 The mean of the ranks for the three groups are: R3o = 8.25 R31 = 20.70 R32 = 23.65 R33 = 29.40 R2o = 7.8 R21 = 18.3 R22 = 20.4 R10 = 6.7 R11 = 14.3 The variance a} for the three treatment groups are: a? = 34.16, a~ = 76.60, a~ = 135.86. The statistics ti for the three treatment groups are t 1 = 2.91, t2 = 3.22, t3 = 4.06. The MED is the first dose level. 89 Chapter 6 Summary and Further research The application of this new test we have described does not require any assumption that the distribution(s) from which the samples were drawn is normal or any other specific shape. The test has the additional advantage of being directly applicable to non-numerical ordi-nal data. Observations which are ordered but not measured on any scale arise frequently, especially in research in the behavioral sciences. Exact inference has important place in statistical inference of discrete data, in partic-ular for sparse contingency table problems for which large-sample chi-squared statistics are often unreliable. However, approximate results are sometimes more useful than exact results, because of the inherent conservativeness of exact methods. Mann and Whitney (1947) proved that the limit distribution of U is normal if m and n go to infinity in any arbitrary manner, and they suggest that this approximation may be used to obtain critical values of U when both m and n is greater than 20. Statistic V is normal approximation, and is asymptotically normal for m and n larger than 10. A well-known distribution-free test is the rank sum W-test of Wilcoxon (1945) which introduced a ranking method for determining the significance of the difference between 90 two treatments. Unfortunately not all statistics are distribution-free. In this thesis, we show that V statistic is asymptotically distribution-free. We are investigated the relative power of the new test with respect to the U test. We applied the new procedure to independent two-sample problems, and found that the test is efficient. The power comparison show that the new procedure is consistently more powerful than Mann-Whitney-Wilcoxon's in detecting mean difference between two-samples. The new procedure is efficient and powerful nonparametric test, it may be a reasonable method and test fork-sample problems. 91 Appendix A Simulation of the Probabilities of Individual V Statistic # simulation of the probabilities of individual V statistic k<-O;rn<-value;n<-value;N<-rn*n*(rn*n+1)/2+1 dy<-rnatrix(O,rn,n);udy<-rnatrix(O,rn,n);edy<-rnatrix(O,rn,n) c215<-rep(O,N);uc215<-rep(O,N); ec215<-rep(O,N) while(k<1000000) k<-k+1; tr<-O;utr<-O;etr<-0 y0<-rnorrn(rn,0,1);uy0<-runif(rn,0,1);ey0<-rexp(rn,1) y1<-rnorrn(n,0,1);uy1<-runif(n,0,1);ey1<-rexp(n,1) for (i in 1 :rn) { for (j in 1:n) {dy[i,k]<-y1[j]-y0[i] udy[i,k]<-uy1[j]-uy0[i] edy[i,k]<-ey1[j]-ey0[i] 92 dr<-matrix(rank(abs(dy)),m,n) udr<-matrix(rank(abs(udy)),m,n) edr<-matrix(rank(abs(edy)),m,n) for(i in 1:m) { for( j in 1:n) {if(dy[i,k]>O) tr<-tr+dr[i,k] if(udy[i,k]>O) utr<-utr+udr[i,k] if(edy[i,k]>O) etr<-etr+edr[i,k] c215[tr+1]<-c215[tr+1]+1 uc215[utr+1]<-uc215[utr+1]+1 ec215[etr+1]<-ec215[etr+1]+1 sc<-O;suc<-O;sec<-0 for(i in 1:N) { suc<-suc+uc215[i] sc<-sc+c215[i] sec<-sec+ec215[i] cat(i-1," ", suc/1000000, sc/1000000, " " ' '' " ' sec/1000000, "\n",file="filename",append=TRUE) 93 AppendixB Power Test for LY Test and Mann-Whitney Test # Power test for LY test and Mann-Whitney test rnus<-0 sd<-sqrt(variance) nsirn<-10000; k<-0 n<-# of variables csl<-O;cs0<-0 ccsl<-O;ccs0<-0 while(k

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