Statistic Distribution Models for Some Nonparametric Goodness-of-Fit Tests in Testing Composite Hypotheses

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1 Communications in Statistics - Theory and Methods ISSN: (Print) X (Online) Journal homepage: Statistic Distribution Models for Some Nonparametric Goodness-of-Fit Tests in Testing Composite Hypotheses B. Yu Lemeshko, S. B. Lemeshko & S. N. Postovalov To cite this article: B. Yu Lemeshko, S. B. Lemeshko & S. N. Postovalov (2) Statistic Distribution Models for Some Nonparametric Goodness-of-Fit Tests in Testing Composite Hypotheses, Communications in Statistics - Theory and Methods, 39:3, 46-47, DOI:.8/ To link to this article: Published online: 6 Jan 2. Submit your article to this journal Article views: View related articles Citing articles: 8 View citing articles Full Terms & Conditions of access and use can be found at Download by: [Novosibirsk State Technical University] Date: 4 November 26, At: 2:53

2 Communications in Statistics Theory and Methods, 39: 46 47, 2 Copyright Taylor & Francis Group, LLC ISSN: print/532-45x online DOI:.8/ Goodness-of-Fit Tests Statistic Distribution Models for Some Nonparametric Goodness-of-Fit Tests in Testing Composite Hypotheses B. YU. LEMESHKO, S. B. LEMESHKO, AND S. N. POSTOVALOV Department of Applied Mathematics, Novosibirsk State Technical University, Novosibirsk, Russia The results (tables of percentage points and statistic distribution models) for the Kolmogorov, Cramer Von Mises Smirnov, and Anderson Darling tests, when unknown parameters are estimated by their MLEs, are presented in this article. Keywords Anderson Darling test; Composite hypotheses testing; Cramer Von Mises Smirnov test; Goodness-of-fit test; Kolmogorov test. Mathematics Subject Classification 62G.. Introduction In composite hypothesis testing of the form H Fx Fx, when the estimate ˆ of the scalar or vector distribution parameter Fx is calculated by the same sample, nonparametric goodness-of-fit Kolmogorov, 2 Cramer Von Mises Smirnov, and 2 Anderson Darling tests lose the free distribution property. The value D n = sup F n x Fx x< where F n x is the empirical distribution function and n is the sample size, is used in Kolmogorov test as a distance between the empirical and theoretical laws. In testing hypotheses, a statistic with Bolshev (Bolshev and Smirnov, 983) correction Received June 24, 28; Accepted June 24, 29 Address correspondence to B. Yu. Lemeshko, Department of Applied Mathematics, Novosibirsk State Technical University, K.Marx pr. 2, Novosibirsk 6392, Russia; Lemeshko@fpm.ami.nstu.ru 46

3 Statistic Distribution Models 46 of the form S K = 6nD n + 6 () n where D n = maxd + n D n, D + n { } { i = max i n n Fx i D n = max Fx i i } i n n n is the sample size, x x 2 x n are sample values in increasing order is usually used. The limit probability distribution function (pdf) of test () in testing simple hypotheses follows the Kolmogorov s pdf KS = k= k e 2k2 s 2. In 2 Cramer Mises Smirnov test, one uses a statistic of the form S = n 2 n = { n 2n + Fx i 2i } 2 (2) 2n and in test of 2 Anderson Darling type, the statistic of the form S = n 2 { n 2i i= 2n i= ( ln Fx i+ 2i ) } ln Fx 2n i (3) In testing a simple hypothesis, statistic (2) follows the pdf (Bolshev and Smirnov, 983) of the form as = j + /2 } 4j + 4j + 2 exp { 2s /2j + 6S j= [ ] [ ]} 4j + 2 4j + 2 {I I 4 6S 4 6S where I 4, I 4 modified Bessel function I z = ( z +2k 2) z < arg z < k + k + + k= and statistic (3) follows the pdf (Bolshev and Smirnov, 983) of the form a2s = 2 S ( ) j + j 2 4j + ( ) exp { 4j + } j + 8S { S exp 8y 2 + 4j + } 2 2 y 2 dy 8S j=

4 462 Lemeshko et al. Figure. The Kolmogorov statistic () pdf s for testing composite hypotheses with calculating MLE of two law parameters. 2. Statistic Distributions of the Tests in Testing Composite Hypotheses In composite hypotheses testing, the conditional distribution law of the statistic GS H is affected by a number of factors: the form of the observed law Fx corresponding to the true hypothesis H ; the type of the parameter estimated and the number of parameters to be estimated; sometimes it is a specific value of the parameter (e.g., in the case of gamma-distribution and beta-distribution families); the method of parameter estimation. The distinctions in the limiting distributions of Figure 2. The Anderson Darling statistic (3) pdf s for testing composite hypotheses with calculating MLE of two law parameters.

5 Statistic Distribution Models 463 Figure 3. The Cramer Mises Smirnov statistic (2) pdf s for testing composite hypotheses with calculating MLE of Weibull distribution law parameters. the same statistics in testing simple and composite hypotheses are so significant that we cannot neglect them. For example, Fig. shows pdf s of the statistic () when testing different composite null hypotheses using maximum likelihood estimators (MLEs) of two parameters, and Fig. 2 represents pdf s of the statistic (3) in similar situation. Figure 3 illustrates statistic distribution dependence (2) of the Cramer Mises Smirnov test upon the type of parameter estimated by the example of Weibull law. Kac et al. (955) was a pioneer in investigating statistic distributions of the nonparametric goodness-of-fit tests with composite hypotheses. Then, for the solution to this problem, various approaches were used (Chandra et al., 98; Durbin, 976; Martinov, 978; Pearson and Hartley, 972; Stephens, 97, 974; Tyurin, 984; Tyurin and Savvushkina, 984). Table Random variable distribution Random variable Density function Random variable Density function distribution fx distribution fx Exponential Seminormal Rayleigh Maxwell Logistic Extreme-value (maximum) Extreme-value (minimum) Weibull e x/ 2 2 e x2 /2 2 x e x2 / x 2 3 e x2 /2 2 2 Laplace Normal Log-normal Cauchy Density function fx 3 exp { x }/[ { x 3 + exp }] 2 3 exp { x exp ( x )} exp { x exp ( x )} x exp { ( ) x } e x / 2 2 e x x 2 e ln x 2 / x 2

6 Table 2 Upper percentage points of statistic distribution of the nonparametric goodness-of-fit tests for the case of MLE usage Cramer Von Mises Kolmogorov s test Smirnov s test Anderson Darling s test Random variable Parameter distribution estimated Exponential & Scale Rayleigh Seminormal Scale Maxwell Scale Laplace Scale Shift Two parameters Normal & Scale Log-normal Shift Two parameters Cauchy Scale Shift Two parameters Logistic Scale Shift Two parameters Extreme-value & Scale Weibull Shift Two parameters Note. we estimated the Weibull distribution shape parameter; 2 the Weibull distribution scale parameter. 464

7 Table 3 Models of limiting statistic distributions of nonparametric goodness-of-fit when MLE are used Test Random variable distribution Estimation of scale parameter Estimation of shift parameter Estimation of two parameters Kolmogorov s Exponential & Rayleigh (5.92;.86;.295) Seminormal (4.5462;.;.3) Maxwell (5.4566;.794;.287) Laplace (3.395;.426;.345) (5.92;.86;.295) (6.2949;.624;.263) Normal & Log-normal (3.569;.4;.3375) (7.534;.58;.24) (6.472;.58;.262) Cauchy (3.987;.463;.335) (5.986;.78;.2528) (5.3642;.654;.26) Logistic (3.4954;.4;.3325) (7.6325;.53;.2368) (7.542;.45;.2422) Extreme-value & Weibull (3.685;.355;.335) (5.294;.848;.292) 2 ( 6.62;.563;.2598) Cramer Von Exponential & Rayleigh Sb(3.3738;.245;.792;.) Mises Smirnov s Seminormal Sb(3.527;.55;.5527;.2) Maxwell Sb(3.353;.22;.9786;.8) Laplace Sb(3.2262;.946; 2.73;.5) Sb(2.9669;.2534;.6936;.) Sb(3.768;.2865;.8336;.3) Normal & Log-normal Sb(3.53;.9448; ;.6) Sb(3.243;.35;.6826;.95) Sb(4.395;.4428;.95;.9) Cauchy Sb(3.895;.934; 2.69;.3) Sb((2.359;.732;.595;.29) Sb(3.4364;.678;.;.) Logistic Sb(3.264;.958; 2.746;.4) Sb(4.26;.2853;.;.22) Sb(3.237;.362;.36;.5) Extreme-value & Weibull Sb(3.343;.987; 2.753;.5) Sb(3.498;.2236;.632;.) 2 Sb(3.3854;.4453;.4986;.7) Anderson Darling s Exponential & Rayleigh Sb(3.8386;.3429; 7.5;.9) Seminormal Sb(4.29;.298;.5;.) Maxwell Sb(3.959;.3296; 7.8;.) Laplace Sb(4.326;.982; 27.;.) Sb(3.56;.3352; ;.96) Sb(3,87;,353; 5,89;,) Normal & Log-normal Sb(4.327;.895; 28.;.2) Sb(3.385;.443; ;.8) Sb(3.56;.4846; 3.987;.8) Cauchy Sb(3.783;.678; 8.;.) Sb(3.484;.2375; 7.8;.) Sb(3.29;.29; 5.837;.99) Logistic Sb(3.56;.54; 4.748;.7) Sb(5.36;.568;.;.65) Sb(3.49;.434; 2.448;.95) Extreme-value & Weibull Sb(3.52;.64; 4.496;.25) Sb Sb(3.483;.538; 3.;.7) Note. we estimated the Weibull distribution shape parameter; 2 the Weibull distribution scale parameter. 465

8 466 Lemeshko et al. In our research (Lemeshko and Postovalov, 998, 2; Lemeshko and Maklakov, 24), statistic distributions of the nonparametric goodness-of-fit tests are investigated by the methods of statistical simulating, and for constructed empirical distributions approximate models of law are found. The results obtained were used by us to develop recommendations for standardization (R , 22). 3. Improvement of Statistic Distribution Models of the Nonparametric Goodness-of-Fit Tests In this article, we present more precise results (tables of percentage points and statistic distribution models) for the nonparametric goodness-of-fit tests in testing composite hypotheses using the MLE. Table contains a list of distributions relative to which we can test composite fit hypotheses using the constructed approximations of the limiting statistic distributions. Upper percentage points are presented in Table 2, and constructed statistic distribution models are presented in Table 3. Percentage points in Table 2 and models in Table 3 do not depend on values of unknown parameters of the law Fx. Distributions GS H of the Kolmogorov statistic are best approximated by gamma-distributions family (see Table 3) with the density function 2 = x 2 e x 2/ and distributions of the Cramer Mises Smirnov and Anderson Darling statistics are well approximated by the family of the Sb-Johnson distributions with the density function Sb = { 2 x x exp [ ] x 2 } 2 ln x The tables of percentage points and statistic distributions models were constructed by modeled statistic samples with the size N = 6 (N is the number of runs in simulation). This number ensures the deviation of the empirical pdf G N S H from the theoretical (true) to be less than 3. In this case, the samples of pseudorandom variables, belonging to Fx, were generated with the size n = 3. For such value of n, statistic pdf GS n H almost coincides with the limit pdf GS H. An accuracy of constructed percentage points can be indirectly assessed by the closeness of obtained results to the known results. For example, for = 5, percentage points obtained by the analytical methods (see Martinov, 978) for testing null hypothesis about the normal law when two parameters are unknown, are equal to.35,.26,.788, correspondingly. In this article (see Table 2), we have obtained:.34,.257,.777, correspondingly.

9 Statistic Distribution Models 467 Figure 4. The Kolmogorov statistic () distributions for testing composite hypotheses with calculating MLE of scale parameter depend on the shape parameter value of gammadistribution. 4. Improvement of Statistic Distribution Models of the Nonparametric Goodness-of-Fit Tests in the Case of Gamma Distribution In composite hypotheses testing subject to gamma distribution with the density function ) fx = x ( exp x limiting statistics distributions of the nonparametric goodness-of-fit tests depend on values of the shape parameter. For example, Fig. 4 illustrates dependence of the Kolmogorov statistic distribution upon the value in testing a composite hypothesis in the case of calculating MLE for the scale parameter of gamma-distribution only. Upper percentage points constructed as a result of statistical modeling are presented in Table 4, and statistics distributions models are given in Table 5. In this case, statistics distributions are well approximated by the family of the III. Type beta-distributions with the density function B = ) ( ) ( x 4 x [ ] 3 B + 2 x The results presented in R (22) are made considerably more precise by the upper percentage points and statistics distributions models given in Tables 4 and 5.

10 Table 4 Upper percentage points of statistic distribution of the nonparametric goodness-of-fit tests when MLE are used in the case of gamma-distribution Cramer Von Mises Kolmogorov s test Smirnov s test Anderson Darling s test Value of the shape parameter Parameter estimated Scale Shape Two parameters Scale Shape Two parameters Scale Shape Two parameters Scale Shape Two parameters Scale Shape Two parameters Scale Shape Two parameters Scale Shape Two parameters

11 Table 5 Models of limiting statistic distributions of the nonparametric goodness-of-fit when MLE are used in the case of gamma-distribution Value of the shape Test parameter Estimation of scale parameter Estimation of shape parameter Estimation of two parameters Kolmogorov s.3 B 3 (6.687; ; 4.447;.944;.28) B 3 (6.4536; 5.759; 3.399;.653;.28) B 3 (6.975; ; ;.57;.27).5 B 3 (6.9356; 5.8; ;.847;.28) B 3 (6.386; ; 3.228;.654;.28) B 3 (6.483; ; 3.263;.4483;.2774). B 3 (6.787; 5.374; ;.6875;.282) B 3 (6.76; 6.474; ;.55;.28) B 3 (5.63; 6.293; 2.765;.367;.293) 2. B 3 (5.8359; ; 2.92; 4.;.282) B 3 (6.387; ; 2.62;.484;.28) B 3 (5.8324; 6.446; ;.328;.2862) 3. B 3 (5.955; ; 2.996; 4.;.282) B 3 (6.22; 6.63; ;.459;.28) B 3 (6.393; 6.276; 2.832;.323;.2827) 4. B 3 (5.949; ;.95; 4.;.282) B 3 (6.827; 6.795; ;.4494;.28) B 3 (6.584; 6.87; ;.37;.287) 5. B 3 (5.8774; 3.692;.799; 4.;.282) B 3 (6.887; ; ;.4432;.28) B 3 (6.957; 6.4; ;.34;.28) Cramer Von.3 B 3 (3.2722;.9595; 6.768;.75;.3) B 3 (3.247; ;.3;.7755;.25) B 3 (2.367; 4.84; 7.66;.689;.45) Mises Smirnov s.5 B 3 (3.2296; 2.984; 4.353;.7;.3) B 3 (3.43; 3.354;.95;.724;.25) B 3 (2.726; ; ;.5372;.3). B 3 (3.2; 2.546;.2;.6;.3) B 3 (2.9928; 3.476; ;.6346;.25) B 3 (3.; ; ;.458;.2) 2. B 3 (2.9463; 3.24; 9.6;.6;.3) B 3 (2.999; ; 8.2;.5786;.25) B 3 (3.533; 3.942; 7.73;.4246;.8) 3. B 3 (2.884; ; ;.6;.3) B 3 (2.9737; ; ;.5549;.25) B 3 (3.73; 3.968; 7.34;.463;.7) 4. B 3 (2.8522; ; 8.44;.6;.3) B 3 (2.9677; ; ;.548;.25) B 3 (3.967; ; 7.64;.422;.6) 5. B 3 (2.8249; 3.628; ;.6;.3) B 3 (2.9638; ; ;.5334;.25) B 3 (4.4332; ;.552;.498;.84) Anderson.3 B 3 (3.3848; ; 4.684; 6.46;.88) B 3 (3.73; 3.739; 8.677; ;.2) B 3 (4.5322; 4.6;.78; 2.922;.78) Darling s.5 B 3 (5.45; ; ; ;.77) B 3 (3.4; ; 8.678; 4.32;.2) B 3 (5.79; 4.56;.292; ;.73). B 3 (5.34; 3.848; ; 4.384;.77) B 3 (3.49; 3.799; 7.483; 3.677;.2) B 3 (5.34; 4.93; 9.6; ;.73) 2. B 3 (4.9479; ; 3.426; 3.834;.77) B 3 (3.434; 4.62; 7.56; 3.85;.2) B 3 (4.9237; 4.29; ; ;.73) 3. B 3 (5.367; 3.429; 2.93; ;.77) B 3 (3.565; 3.992; ; ;.2) B 3 (4.9475; 4.27; ; 2.252;.73) 4. B 3 (4.9432; 3.538; 2.224; 3.632;.77) B 3 (3.53; ; 6.769; ;.2) B 3 (4.9274; ; ; 2.239;.73) 5. B 3 (4.88; ;.7894; 3.65;.77) B 3 (3.52; 3.964; 6.75; 3.424;.2) B 3 (4.927; ; 8.488; 2.234;.73) 469

12 47 Lemeshko et al. 5. Conclusions In this article, we present more precise models of statistic distributions of the nonparametric goodness-of-fit tests in testing composite hypotheses subject to some laws considered in recommendations for standardization (R , 22). Models of statistic distributions of the nonparametric goodness-offit tests in composite hypothesis testing relative to Sb-, Sl-, Su-Johnson family distributions (Lemeshko and Postovalov, 22) and exponential family (Lemeshko and Maklakov, 24) were made more precise earlier and did not improve now. In the case of the Types I, II, III beta-distribution families statistic distributions depend on a specific value of two shape parameter of these distributions. Statistic distributions models and tables of percentage points for various combinations of values of two shape parameters (more than,5 models) were constructed in the dissertation (Lemeshko, 27). The results of comparative analysis of goodness-of-fit tests power (nonparametric and 2 type) subject to some sufficiently close pair of alternative are presented in Lemeshko et al. (27). Acknowledgments This research was supported by the Russian Foundation for Basic Research, project no References Bolshev, L. N., Smirnov, N. V. (983). Tables of Mathematical Statistics. Moscow: Science. (in Russian) Chandra, M., Singpurwalla, N. D., Stephens, M. A. (98). Statistics for test of fit for the extrem value and weibull distribution. Journal of the American Statistical Association 76(375): Durbin, J. (976). Kolmogorov Smirnov test when parameters are estimated. Lecture Notes in Mathematics 566: Kac, M., Kiefer, J., Wolfowitz, J. (955). On tests of normality and other tests of goodness of fit based on distance methods. Annals of Mathematical Statistics 26:89 2. Lemeshko, B. Yu., Postovalov, S. N. (998). Statistical distributions of nonparametric goodness-of-fit tests as estimated by the sample parameters of experimentally observed laws. Industrial laboratory 64(3): Lemeshko, B. Yu., Postovalov, S. N. (2). Application of the nonparametric goodnessof-fit tests in testing composite hypotheses. Optoelectronics, Instrumentation and Data Processing 37(2): Lemeshko, B. Yu., Postovalov, S. N. (22). The nonparametric goodness-of-fit tests about fit with Johnson distributions in testing composite hypotheses. News of the SB AS HS (5): (in Russian) Lemeshko, B. Yu., Maklakov, A. A. (24). Nonparametric test in testing composite hypotheses on goodness of fit exponential family distributions. Optoelectronics, Instrumentation and Data Processing 4(3):3 8. Lemeshko, B. Yu., Lemeshko, S. B., Postovalov, S. N. (27). The power of goodness-of-fit tests for close alternatives. Measurement Techniques 5(2):32 4. Lemeshko, S. B. (27). Expansion of Applied Opportunities of Some Classical Methods of Mathematical Statistics. Ph.D. dissertation, Cand. Tech. Sci. Novosibirsk State Technical University, Novosibirsk. (in Russian) Martinov, G. V. (978). Omega-Quadrate Tests. Moscow: Science. (in Russian)

13 Statistic Distribution Models 47 Pearson, E. S., Hartley, H. O. (972). Biometrica Tables for Statistics. Vol. 2. Cambridge: Cambridge University Press. R (22). Recommendations for Standardization. Applied Statistics. Rules of Check of Experimental and Theoretical Distribution of the Consent. Part II. Nonparametric Goodness-of-Fit Test. Moscow: Publishing House of the Standards. (in Russian) Stephens, M. A. (97). Use of Kolmogorov Smirnov, Cramer Von Mises and related statistics without extensive table. Journal of Royal Statistic Society 32:5 22. Stephens, M. A. (974). EDF statistics for goodness of fit and some comparisons. Journal of the American Statistical Association 69: Tyurin, Yu. N. (984). On the limiting Kolmogorov Smirnov statistic distribution for composite hypothesis. News of the AS USSR Series in Mathematics 48(6): (in Russian) Tyurin, Yu. N., Savvushkina, N. E. (984). Goodness-of-fit test for Weibull Gnedenko distribution. News of the AS USSR. Series in Technical Cybernetics 3:9 2. (in Russian)