2009 8 Chinese Journal of Applied Probability and Statistics Vol.25 No.4 Aug. 2009,, 541004;,, 100124),, 100190), Zhang 2002). λ qt), Kolmogorov-Smirov, Berk and Jones 1979). λ qt).,,, λ qt),. λ qt) 1,. n,,,.,. : :,,. O211.3, O212.7. 1. X F, X 1,, X n X i.i.d., F n. F 0. K λ t, s) = H 0 : F = F 0 H 1 : F F 0. 1.1) 1 λλ + 1) [ t λ+1 s λ ] 1 t)λ+1 + 1 s) λ 1, 1.2) λ, ), t, s [0, 1], λ, t, s, 1.2), K λ t, s) 1.2). 1.1), [1], sup K λ F n x), F 0 x)) /qf 0 x)) λ > 1; <x< R n,λ q) = 1.3) sup K λ F n x), F 0 x)) /qf 0 x)) λ 1, X 1) x<x n) 10661003, 10371126) 0832102). 2007 1 23, 2009 1 4.
422 X 1) X n) X 1,, X n. λ = 0, K 0 F n x), F 0 x)) [2]), R n,λ q), ). λ qt),,., λ = 1, qt) = [t1 t)] 1, 2R n,λ q) Kolmogorov-Smirov KS) ; λ = 1, qt) 1, 2R n,λ q) [3, 4] ; λ = 0, qt) 1, R n,λ q) [5] BJ ; λ = 1, qt) 1, R n,λ q) BJ [6]., [7] λ qt),.,,. 1.3), λ = 1, 0, 1. λ = 1, KS. [8] [9] ) KS. λ = 0, 1, [6] BJ BJ.,.,,,.,. λ qt),. λ qt) 1, 1.3),. qt) 1, 1.3) ;, ;. 2., T = F 0 X). T 0, 1), H 0, T = F 0 X) U0, 1)., F 0 [0, 1] U0, 1)., : H 0 : F U0, 1). 2.1) 1.3) sup K λ F n t), t) /qt) λ > 1; 0<t<1 R n,λ q) = sup K λ F n t), t) /qt) λ 1. X 1) t<x n) 2.2)
: 423 qt) 1 λ, R n,λ 1).. qt), qt) λ,,. 2.1 2.1), i) λ > 1, R n,λ 1) = R n,λ n 1) PR n,λ z) = Pa i X i) b i, 1 i n). 2.3). i a i = min x K λ n, x z, b i = 1 a n i+1, i = 1,, n. ii) λ 1, R n,λ n 2) PR n,λ z} = Pã i X i) b i, 1 i n}. 2.4) i ã i = min x K λ n, x z, i = 1,, n 1; ã n = ã n 1. bi = 1 ã n i+1, i = 1,, n. 2.1. 1 λ = 0 1, 2.3) 2.4) [6] ; λ = 1, 2.3) [8, 9] n, 2.3) 2.4). 2.1 2.1), i) λ > 0, R 1,λ n = 1) PR 1,λ z) = 1 2[1 + λλ + 1)z] 1/λ, z 2 λ 1)/[λλ + 1)]. ii) λ = 0, R 1,λ n = 1) PR 1,0 z) = 1 2e z, z log 2. iii) 1 < λ < 0, R 1,λ n = 1) 0 z < 0; PR 1,λ z) = 2 2[1 + λλ + 1)z] 1/λ 0 z 2 λ 1)/[λλ + 1)]; 1 z > 2 λ 1)/[λλ + 1)].
424 iv) λ 1, R 2,λ n = 2) 0 z < 0; PR 2,λ z} = 1 2l z ) 2 0 z 2 λ+1) 1)/[λλ + 1)]; 1 z > 2 λ+1) 1)/[λλ + 1)]. l z 0 l z 1/2 K λ 1/2, l z ) = z; λ = 1, λ 0. : 2.1. 2.1, n = 1 n = 2, 2.3) 2.4). n, 2.3) 2.4). [10] a i b i.,. 1. n, λ, z; 2. 2.1 a i, b i, i = 1,, n; 3. [10] 2.3) 2.4) PR n,λ z)., n, λ z, PR n,λ z). R n,λ,. 1 n, λ) R n,λ 0.95. 1 R n,λ 0.95 ) n λ 10 20 30 50 70 100 200 300-5 8.6965 7.6847 6.2166 4.3594 3.3302 2.4518 1.3004 0.8843-4 3.2070 2.5456 1.9865 1.3535 1.0212 0.7450 0.3909 0.2648-3 1.3486 0.9651 0.7275 0.4821 0.3596 0.2600 0.1351 0.0912-2 0.6779 0.4402 0.3214 0.2078 0.1534 0.1101 0.0567 0.0382-1 0.4323 0.2574 0.1831 0.1165 0.0856 0.0613 0.0317 0.0215-1/2 0.6784 0.3572 0.2427 0.1481 0.1066 0.0752 0.0380 0.0254 0 0.4729 0.2479 0.1692 0.1042 0.0756 0.0538 0.0276 0.0187 1/2 0.7736 0.3922 0.2632 0.1590 0.1141 0.0802 0.0403 0.0270 1 2.0645 1.0368 0.6922 0.4159 0.2972 0.2081 0.1041 0.0694 2 27.484 13.769 9.1858 5.5144 3.9398 2.7583 1.3795 0.9197 3 539.38 269.89 179.97 108.01 77.154 54.012 27.008 18.010 4 12748. 6373.0 4248.5 2549.0 1820.7 1274.5 637.18 424.85 5 335353 167515 111642 66969. 47829. 33480. 16738. 11158.
: 425 3., n, λ), R n,λ., n,, ).,,. 3.1 λ = 5, 4, 3, 2, 1, 1/2, 0, 1/2, 1, 2, 3, 4, 5, n = 10, 20, 30, 50, 70, 100, 200, 300. n, λ), R n,λ 0.95 1 ). 10 5 n U0, 1), R n,λ, R n,λ 10 5 0.95, 2., R n,λ, 0.95 3. 2 R n,λ 0.95 ) n λ 10 20 30 50 70 100 200 300-5 8.7127 7.5383 6.2436 4.3818 3.3746 2.4152 1.3260 0.8763-4 3.2124 2.5590 1.9922 1.3623 1.0114 0.7530 0.3893 0.2641-3 1.3540 0.9687 0.7248 0.4894 0.3621 0.2619 0.1340 0.0903-2 0.6784 0.4416 0.3194 0.2088 0.1542 0.1097 0.0578 0.0379-1 0.4337 0.2553 0.1828 0.1165 0.0861 0.0609 0.0317 0.0215-1/2 0.6829 0.3584 0.2436 0.1492 0.1064 0.0753 0.0379 0.0255 0 0.4728 0.2469 0.1689 0.1051 0.0756 0.0540 0.0276 0.0187 1/2 0.7831 0.3892 0.2598 0.1598 0.1146 0.0804 0.0402 0.0267 1 2.0753 1.0221 0.6733 0.4131 0.2934 0.2067 0.1042 0.0704 2 28.368 13.638 9.2875 5.4013 3.9854 2.6193 1.3519 0.9575 3 516.27 285.39 173.66 102.58 78.036 57.212 25.465 18.240 4 13162. 7015.9 4163.4 2565.8 1758.4 1240.7 675.90 453.25 5 375696 149859 119355 68965. 50002. 36951. 16326. 11174. 3,., ). n,,. 4. PentiumR) 4, CPU 3.06GHz, 480MB, R 2.2.0).
426 3 R n,λ 0.95 ) 0.95 10 4 n λ 10 20 30 50 70 100 200 300-5 1.198-10.761 2.292 2.602 6.631-7.499 9.470-4.391-4 1.343 3.604 1.887 4.147-6.142 6.671-2.526-1.680-3 4.085 3.390-3.176 12.422 5.799 5.912-5.966-7.741-2 1.067 3.741-7.600 5.580 5.850-4.300 21.612-10.963-1 5.510-13.833-2.936 1.343 10.651-12.067-3.377 5.984-1/2 13.607 6.950 7.505 14.720-4.706 3.373-3.433 6.281 0-0.276-9.395-4.080 19.907-0.624 9.641-1.360 2.390 1/2 11.962-8.309-13.846 5.356 5.195 3.377-3.433-12.245 1 2.707-7.446-14.702-3.503-6.812-3.680 0.540 7.597 2 7.967-2.450 2.792-5.302 2.920-13.332-5.164 10.150 3-7.380 9.247-6.013-8.700 1.900 9.546-9.948 2.160 4 3.976 11.860-2.534 0.822-4.364-3.363 7.307 8.027 5 11.194-11.220 6.612 2.918 4.407 9.738-2.482 0.148 4 R n,λ 0.95 : ) n λ 10 50 100 110 120 140 150 200 300-5 0.96 54.46 390.63 520.22 736.24 1030.53 1410.08 3013.49 9249.16 81.36 335.96 653.11 709.64 768.55 896.89 961.44 1264.77 1880.47-3 0.99 51.97 355.10 462.95 591.54 922.03 1257.25 2509.36 8025.30 78.74 334.05 647.78 703.09 764.64 892.54 949.47 1265.21 1878.28-1 1.11 47.24 319.00 415.04 533.26 832.31 1150.39 2309.59 7541.06 78.28 330.03 640.50 696.70 759.99 883.33 943.49 1249.09 1859.97-1/2 1.23 55.28 386.95 424.06 544.94 851.67 1168.52 2717.00 8762.45 82.73 338.11 657.80 715.77 779.44 913.86 966.23 1266.69 1884.95 0 1.61 49.01 312.28 422.17 541.68 842.72 1145.76 2334.75 7818.53 81.32 337.44 647.75 702.86 764.25 888.77 949.17 1259.69 1877.45 1/2 1.90 55.24 345.38 446.26 569.83 880.18 1189.48 2433.66 7783.12 82.82 340.83 657.25 706.05 772.15 901.47 957.55 1272.12 1897.19 1 2.31 67.23 414.19 537.80 780.21 1020.79 1352.06 2744.67 8468.02 82.18 335.77 646.99 703.81 763.00 889.84 955.86 1262.02 1873.00 3 2.74 71.02 479.89 640.00 906.31 1283.95 1691.20 3604.07 11805.36 82.53 336.06 646.85 707.61 769.99 890.53 959.57 1238.80 1886.82 5 3.06 72.48 484.25 638.54 925.51 1285.75 1692.10 3604.71 11902.67 81.62 336.06 646.96 710.81 768.42 891.72 958.78 1261.82 1878.18
: 427 4, λ, n,,. n 110, ; n 150,. n = 200, 2. n = 300 3.,, R n,λ,., : n 110, ; 110 < n < 150, ; n 150,. 3.2,. Bliss 1967) [11]. 20 21 leghorn chicks) X : ), : 156, 162, 168, 182, 186, 190, 190, 196, 202, 210, 214, 220, 226, 230, 230, 236, 236, 242, 246, 270.,, 200, 35., H 0 : X N200, 35 2 ). 5 λ R n,λ p p p -5 0.39125280 0.46449 0.4641790 0.0019832-4 0.21758770 0.48799 0.4894907 0.0620025-3 0.12956640 0.54129 0.5399682 0.3107841-2 0.09243243 0.58576 0.5878658 0.5426022-1 0.07519322 0.65470 0.6533871 0.6685316-1/2 0.21445050 0.24530 0.2437218 0.0658822 0 0.11020660 0.53159 0.5309238 0.4220073 1/2 0.07750332 0.75805 0.7588946 0.6515868 1 0.08307234 0.72482 0.7229935 0.6106335 2 0.09764782 0.70607 0.7083852 0.5057546 3 0.11835960 0.71972 0.7192991 0.3723125 4 0.14787230 0.73376 0.7341621 0.2274757 5 0.19019300 0.74803 0.7476744 0.1048004
428 F 0 N200, 35 2 ), F 0 X i ), X i i ; 5. 5 λ; λ ; p ; p ; p. : 1) p p,, p. 0.05, p 0.05, H 0,, N200, 35 2 ), Bliss 1967) [11, 12]. 2) p, λ, p 0.05,.,. 4.,. 4.1 0 < t 1 < t 2 < 1, z > 0, : 0 < t 1 < t 2 < 1, x K λ t 1, x) z, K λ t 2, x) z}, maxx K λ t 1, x) z, K λ t 2, x) z} = maxx K λ t 1, x) z}, 4.1) minx K λ t 1, x) z, K λ t 2, x) z} = minx K λ t 2, x) z}. 4.2) 1.2), t, λ, K λ t, x) x t, x t, t = x K λ t, t) = 0. maxx K λ t, x) z} [t, 1], minx K λ t, x) z} [0, t]. 0 < t 1 < t 2 < 1, x [t 1, t 2 ], K λ t 1, x) K λ t 1, t 1 ) = 0; K λ t 2, x) K λ t 2, t 2 ) = 0. K λ t 1, x) K λ t 2, x) x [t 1, t 2 ], c, K λ t 1, c) = K λ t 2, c). x [c, t 2 ], K λ t 1, x), K λ t 2, x), K λ t 2, x) K λ t 1, x), x [c, t 2 ]., x, t 0, 1) λ, ) K λ t, x) = 1 [ 1 t) λ+1 ] tλ+1 x λ + 1 1 x) λ+1 x λ+1, 2 K λ t, x) x t = tλ x λ+1 1 t)λ < 0. 1 x) λ+1 K λ t, x)/ x) t 0, 1). x 0, 1) K λ t 1, x) x > K λt 2, x). x
: 429 x [c, 1], K λ t 2, x) K λ t 1, x). 4.3) z > K λ t 1, c), maxx K λ t 1, x) z, K λ t 2, x) z} > c. 4.3) maxx K λ t 1, x) z, K λ t 2, x) z} = maxx K λ t 1, x) z}. z = K λ t 1, c), maxx K λ t 1, x) z, K λ t 2, x) z} = c = maxx K λ t 1, x) z}. 0 < z < K λ t 1, c),. 4.1). 4.2). 4.2 : maxx K λ t 1, x) z, K λ t 2, x) z} =, z > 0 t 0, 1), maxx K λ t, x) z} = 1 minx K λ 1 t, x) z}. 4.4) 1.2), t, x 0, 1), K λ t, x) = K λ 1 t, 1 x). 4.5) λ 0. t, x 1, K λ t, x). 4.1,, z > 0 t 0, 1), x [t, 1], x = maxx K λ t, x) z}, K λ t, x ) = z. K λ 1 t, 1 x ) = K λ t, x ) = z 1 x [0, 1 t], 1 x = minx K λ 1 t, x) z}. λ < 0, t, x 1, K λ t, x) [t λ+1 1]/[λλ + 1)];, x 0, K λ t, x) [1 t) λ+1 1]/[λλ + 1)]. 0 z < [t λ+1 1]/[λλ + 1)], x [t, 1], x = maxx K λ t, x) z}, K λ t, x ) = z, 4.5) 1 x = minx K λ 1 t, x) z}, 4.4). z [t λ+1 1]/[λλ+1)], maxx K λ t, x) z} = 1, minx K λ 1 t, x) z} = 0, 4.4). 2.1 : i) λ > 1. K λ t, x), x 0, 1), K λ t, x). i 1 ) i )} } PR n,λ z} = P max K λ 1 i n n, X i) K λ n, X i) z. 4.6)
430 4.6) : 1 A = K λ 0, X 1) ) K λ n, X 1) z ; 4.7) i 1 ) i )} } B = max K λ 2 i n 1 n, X i) K λ n, X i) z ; 4.8) n 1 ) } C = K λ n, X n) K λ 1, X n) ) z. 4.9) i 1 ) a i = min x K λ n, x b i = max x 4.7) 4.9) A, B, C A = a 1 X 1) b 1 }, i K λ n, x z } i 1 ) i ) K λ n, x K λ n, x z, i = 1,, n,, i = 1,, n, B = n 1 a i X i) b i }, C = a n X n) b n }. 4.1 i a i = min x K λ n, x z, i = 2,, n 1, i 1 b i = max x K λ n, x z, i = 2,, n 1. i=2 4.1 1 a 1 = min x K λ n, x z, b 1 = maxx K λ 0, x) z}; n 1 a n = minx K λ 1, x) z}, b n = max x K λ n, x z. 4.2 1 i n, i a i = min x K λ n, x z = 1 b n i+1. 2.3). ii) λ 1. λ > 1, n > 2, i ) i )} } R n,λ z} = max K λ 1 i n 1 n, X i) K λ n, X i+1) z 1 n 1 = K λ n, X 1) z K λ n, X n) z n 1 i 1 ) i K λ i=2 n, X i) K λ n, X i) z = ã i X i) b i, 1 i n}.
: 431 1 ã 1 = min x K λ n, x 1 z, b1 = max x K λ n, x z ; i 1 ) i ã i = min x K λ n, x K λ n, x z, i = 2,, n 1, i 1 ) i bi = max x K λ n, x K λ n, x z, i = 2,, n 1; n 1 ã n = min x K λ n, x n 1 z, bn = max x K λ n, x z. 4.1 PR n,λ z} = Pa i X i) b i, 1 i n}. i ã i = min x K λ n, x z, i = 2,, n 1, i 1 bi = max x K λ n, x z, i = 2,, n 1. 4.2 1 i n, ã i = 1 b n i+1. 2.4). [1] Zhang, J., Powerful goodness-of-fit tests based on the likelihood ratio, J. R. Statist. Soc. B, 642) 2002), 281 294. [2] Owen, A.B., Nonparametric likelihood confidence bands for a distribution function, J. Amer. Statist. Assoc., 901995), 516 521. [3] Eicher, F., The asymptotic distribution of the suprema of the standardized empirical processes, Ann. Statist., 71)1979), 116 138. [4] Jaeschke, D., The asymptotic distribution of the supremum of the standardized empirical distribution function on subintervals, Ann. Statist., 71)1979), 108 115. [5] Berk, R.H. and Jones, D.H., Goodness-of-fit statistics that dominate the Kolmogorov statistics, Z. Wahrscheinlieitstheorie Verw. Gebiete., 471979), 47 59. [6] Jager, L. and Wellner, J.A., A new goodness of fit test: the reversed Berk-Jones statistic, Technical Report 443, Department of statistics, University of Washington, http://www.stat.washington.edu/ www/research/reports/2004/tr443.ps, 2004. [7],,, 2006. [8] Niederhausen, H., Scheffer polynomials for computing exact Kolmogorov-Smirnov and Renyi type distributions, Ann. Statist., 91981), 923 944. [9] Shorack, G.R. and Wellner, J.A., Empirical Processes with Applications to Statistics, New York: Wiley, 1986.
432 [10] Noe, M., The calculation of distributions of two-sided Kolmogorov-Smirnov type statistics, Ann. Math. Statist., 431972), 58 64. [11] Bliss, C., Statistics in Biology: Statistical Methods for Research in the Natural Sciences, New York: McGraw-Hill, 1967. [12] Stephens, M.A., Tests based on EDF statistics, in Goodness-of-Fit Techniques eds. D Agostino, R.B., Stephens, M.A.), New York: Marcel Dekker, 1986, 97 193. On Exact Distribution of a Class of Supremum-type Statistics for Goodness of Fit Zhang Junjian College of Mathematical Sciences, Guangxi Normal University, Guilin, 541004 ) College of Applied Sciences, Beijing University of Technology, Beijing 100124 ) Li Guoying Zhao Zhiyuan Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100190 ) For goodness of fit tests with simple null hypothesis, Zhang 2002) constructed a classes of supremumtype tests. Different parameter λ and different weighted function qt) result in different tests, including the Kolmogorov-Smirov test, Berk and Jones 1979) test and so on. So far, only a few tests corresponding to particular λ and qt) have been studied in the literature. However, for different problems, the best tests are different. It is necessary to discuss the tests for all λ and the general qt). In this paper, the exact distributions of the test statistics for all λ and qt) 1 are derived. When sample size n is large, it takes a long time to get the exact quantile. So we give some advice on the computation methods for different sample size by simulation studies, and a real example to simply illustrate the above methods. Keywords: Exact distribution, generalized nonparametric likelihood ratio, goodness-of-fit test. AMS Subject Classification: 62E15, 62G10.