5 200 2 Vol 5 No JOURNAL OF HARBIN UNIVERSITY OF SCIENCE AND TECHNOLOGY Feb 200 2 2 50080 2 30024 O2 A 007-2683 200 0-0087- 05 A Goodness-of-fit Test Based on Empirical Likelihood and Application ZHOU yan WANG guo-chang School of Apolied Science Harbin University of Science and Technology Harbin 50080 China 2 School of Mathematics & Statistics of Northeast Normal University Changchun 30024 China Abstract For the goodness-of-fit test the paper concerns how to construct an appropriate statistics to control the type I error at the same tim reduce the type II error as far as possible The experience likelihood in goodness of fit s application using the data which obey the null hypothesis distributed condition to look for the estimating e- quation and constructs the test statistics from this Experience likelihood of goodness-of-fit tests can be applied to two overall examinations Key words experience likelihood goodness-of-fit test statistics 2 2 933 Kolmogorov Kolmogorov 979 Freeman - Turkey Fienberg 979 Moore H 0 F = F 0 H F F 0 986 2 Pearson χ 2 3 984 Cresse Read 4 H 0 F L 0 H F L 0 A B Owen 988 990a 5-9 L 0 N μ σ 2 μ σ 2008-0 - 06 982 E-mail zhouy06@ 63 com 983 -
88 5 p - p 2 - p p 2 - p p k - p R F = L F = var b X i = 2 p - p 2 p k = L F n - p k p p k - p 2 k R μ 0 = max { n nw i n w i X i - μ 0 = 0 w i 0 n w } χ 2 k - 95% c k - = χ 2 k - 95 P - 2logR F 0 > c k - = 0 05-2logR F 0 c k - 2 c H 0 F F 0 H F F k - 0 05 H 0 0 F 0 K - 2logR F 0 λ w i A A 2 A k p j = p A j = Aj df 0 x - 2logR F 2 k P = p p 2 p k T 0 = - 2 n lognw i p n n 2 n k A lim I n n Xi A j = lim 2 2 A 2 A k H 0 F L 0 H F L 0 n j n = p j 2 k E I Xi A j = p j 2 k b X i = I Xi A I Xi A 2 I Xi A k T Eb X i = P R F 0 = max { n nw i = i n w i b X i - i = P = 0 w i 0 n 2 X w } i = X X 2 X n F 0 Eb X i = lim n n b X i = p p 2 p k T = P rank ( ) diag P - PP T = k - R F 0-2logR F 0 0 05 H 0-2logR F 0 R F 0 θ = max { n nw i n w i b X i - P θ = 0 n w } 2 P θ = p θ p k θ T θ s θ R F 0 θ θ^ - 2logR F 0 θ - 2logR F 0 θ X 2 X n F 0 θ θ s b X i θ = I Xi A θ I Xi A k θ T 2 n b X i = I Xi A I Xi A k T 2 n Eb X i θ = P θ k Eb X i = P k - θ θ^ n - n -2logR F 0-2logR F 0 θ^ χ 2 k - χ 2 k - s - k 0 A X X 2 X n 95% c k - s - = χ 2 k - s - 95 b X i = I Xi A I Xi A k T - 2logR F 0 θ^ c k - s - 0 05 H 0-2logR F 0 θ^ c k - s - 0 05 H 0
89 3 K t = - logr F t t = n log m + n n n n + m + m + n n - n X X 2 X n - n log n { n n + m - n - m } + F x m Y Y 2 Y m G x F log m + n m { m n + m } + x G x m + n m - m m - m log { n n + m - n - m } 8 H 0 F x G x H x R F x G x m + n n n = 0 n log { Wilcoxon n n + m } = 0 n = m + n n - n n n - n log { 2006 jin zhang n n + m - n - m } = 0 F x G xt R F x = G x m = 0 m log m + n m { m n + m } = 0 m = m m + n m - m t R m - m log { n n + m - n - m } = 0 EI Xi t = F t EI Yj t = G t = F t u i v j X i Y j T = sup K t t = X X 2 X 3 X n Y Y 2 Y 3 Y m n u i I Xi t - F t = 0 9 m v j I Yj t - F t = 0 b X i = I Xi t b Y i = I Yi t R F t t = max { n nu i = i m m v j = j n u i b X i - F t = 0 m v j 0 u i 0 n u m T = λ 2 λ sup - < t < λ = n i = v j b Y j - F t = 0 - logr F t n - nf t F t - F t m - mf t F t - F t v } 3 4 5 F t λ λ 2 u i v j - logr F t t { } T T 3 Wilcoxon 4 4 Monte Carlo Pearsonχ 2 Neymanχ 2 Kolmogorov-Smirnov Ks Cramer-Von Mise W 2 λ 2 = Anderson - Darling A 2 Jin Zhang Z A 6 m Z C Z K Pearsonχ 2 n = n I Xi t m = m Neymanχ 2 I Yj t F t G t i = j = S-plus 000 F x G x X X 2 X n Y Y 2 Y m F t α = 0 05 0 F^ t = ( m + n n I Xi t + m n = 6 8 20 25 I Yj ) 7 t 30 40 50 70 00 50 200 300
90 5 H 0 X i U 0 H X i Beta p q Beta p q p = q = Beta U 0 06 04 02 50 Neyman χ 2 200 00 50 Neyman χ 2 a Z C Pearson χ 2 Kolmogorov-Smirnov Cramer- Von Mise b Z A Z A Z C 0 08 06 04 02 00 0 08 06 04 02 00 50%%%%%%%%%%00%%%%%%%%50%%%%%%%%200%%%%%%%%250%%%%%%%%300 (a)beta(08,08) 分布 50%%%%%%%%00%%%%%%%%50%%%%%%%200%%%%%%%%250%%%%%%%300 (b)beta(6,6) 分布 2 Pearsonχ 2 3 Neymanχ 2 4 Z k 5 Z A 6 Ks 7 A 2 8 W 2 9 Z C H 0 F = U 0 H F = Beta p q 2 H 0 X i N μ σ 2 H X i t k 2 a 2 b t 0 t 3 μ = 0 δ 2 = - k / k - 2 k 3 2 b Z A Z C Z A Z C 0 08 0 08 06 04 02 00 50%%%%%%%%00%%%%%%%%50%%%%%%%200%%%%%%%%250%%%%%%%300 (a)t(0) 的分布 50%%%%%%%%00%%%%%%%%50%%%%%%%200%%%%%%%%250%%%%%%%300 (b)t(3) 的分布 2 Pearsonχ 2 3 Neymanχ 2 4 Z k 5 Z A 6 Ks 7 A 2 8 W 2 9 Z C 2 5 H 0 F = N μ σ 2 H F = t k Jin Zhang Z A Z C Z C Neymanχ 2 20 25 Pearsonχ 2 FIENBERG S E The Use of Chi-squad Statistics for Categorical data Problems J Journal of the RoyalStatistical Society Series B 979 4 54-64
9 2 MOORE D S Tests of Chi-squared type J In Goodness of fit Techniques 986 68 63-95 3 M 993 4 KEITH A BAGGERLY Empirical Likelihood as a Goodness-offit Measure J Biometrika 998 85 3 535-547 5 OWEN A B Computing Empirical Likelihoods C / /in Computer Science and Statistics Proceedings of the 20th Symposium on the Interface 988 442-447 6 OWEN A B Empirical Likelihood Ratio Confidence Intervals for a single Functional J Biometrika 988b 75 2 237-249 7 OWEN A B Empirical Likelihoods and Small Samples J in Computer Science and Statistics Proceedings of the 20th Symposium on the interface 990 79-88 8 OWEN A B Empirical Likelihood ratio Confidence Regions J the annals of Statistics 990 8 90-20 9 OWEN A B Empirical Likelihood M CHAPMAN \ &HALL / CRC 200 0 JIN Zhang Powerful Two-sample Tests Based on the Likelihood ratio J Journal of the Royal Statistical Society Series B 2002 64 2 28-294 A + = diagp [ - 2 I - diagp - 2 PP T diagp ] - 2 diagp - 2 = diagp - - diagp - PP T diagp - = diagp - - Ⅱ T I T b X - P = I T b X - I T P = - = 0 n b X - P T + b X - P = n b X - P T diagp - - Ⅱ T b X - P = n b X - P T diagp - b X - P = n i - np i 2 np i - 2logR F 0 n i - np i 2 np i θ P θ = p θ p 2 θ p k θ - 2logR F 0 = n b X - P T S + b X - P + o p n b X - P T + b X - P = n i - np i θ 2 np i θ θ^n θ θ^n θ ~ - 2logR F 0-2logR F 0 θ^ = -2max { logr F 0 θ + o p } = min n i - np i θ 2 np i θ + o p = n i - np i θ ~ 2 np i θ ~ θ ~ χ 2 θ χ 2 θ = n ( i ) p i θ 2 p i θ = 0 2 s - 2logR F 0 χ 2 θ ~ n χ 2 k - s - Pearsonχ 2-2logR F 0 = n b X - P T S + b X - P + o p Θ R s 2 θ θ Θ θ θ 2 n b X - P T + b X - P + o p P θ P θ 2 = diagp - PP T + 3 p i θ Θ 2 k 3 A A 0 2 s A = A + 4 A θ = P θ ( θ ) = p i θ ( θ ) ranka θ = s θ Θ P L 0 χ 2 θ^n k - s - χ 2 993 3 2 3 j - 2logR F 0 θ^n χ 2 θ^n - 2logR F 0 θ^n χ 2 k - s - k s