Математическая статистика: Критерии согласия для простых и сложных гипотез

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  • 1. 6. . .., .. CS Center -, 2014 .., .. (CS Center) ... -, 2014 1 / 26

2. C 1 -- - 2 - 3 - - .., .. (CS Center) ... -, 2014 2 / 26 3. ( 2 ) . . , F, P, X[n] = (X1, . . . , Xn). H0 : F = F0, , F0(x) . : H1 : F = F0. .., .. (CS Center) ... -, 2014 3 / 26 4. r = a0 < a1 < . . . < ar = , i = (ai1, ai ], r = 1, , r, 2: 2 (X[n]) = r i=1 (ni np (0) i )2 np (0) i . p (0) i = F0(ai ) F0(ai1). H0 , 2 (X[n]) d n , - r 1 . H1, 2 .. n . .., .. (CS Center) ... -, 2014 4 / 26 5. (0, 1). (C(r 1, 1 ), ), C(r 1, 1 ) 1 2 r 1 , H0. 2(X[n]) > C(r 1, 1 ), H0 , 2(X[n]) C(r 1, 1 ), . p value = PH0 { > 2 (X[n])} = 1 F2 r1 (2 (X[n]) .., .. (CS Center) ... -, 2014 5 / 26 6. , F, P, X[n] = (X1, . . . , Xn). H0 : F = F0, H1 : F = F0. : F(x) R. : Dn(X[n]) = sup xR |F n (x) F0(x)| . (1) H0, Dn(X[n]) .. n 0. H1, . . F G = F0, Dn(X[n]) .. n sup xR |G(x) F0(x)| > 0. .., .. (CS Center) ... -, 2014 6 / 26 7. H0 Dn(X[n]) F0. 1 H0 , F0(x) R, Dn = sup xR F n (x; x[n]) F0(x) . .., .. (CS Center) ... -, 2014 7 / 26 8. : P{sup xR F n (x, X[n]) F0(x) z} = P{ sup y[0,1] F n (y, Y[n]) y z}. z. , F0(x). z1 . (z1, 1]. (1) , H0 H1. Dn [0, z1), H0. .., .. (CS Center) ... -, 2014 8 / 26 9. n . 1 (.. ) H0 , F0(x) R, : P{ nDn(X[n]) z} n K(z) = 1 + 2 m=1 (1)m e2m2z2 . d1 : K(d1) = 1 . . nDn(X[n]) (d1, ), H0 , nDn(X[n]) / (d1, ), H0 . .., .. (CS Center) ... -, 2014 9 / 26 10. Dn(X[n]) : Dn(X[n]) = max 1 i n i n F0(X(i)), F0(X(i)) i 1 n , X(1) < . . . < X(n) , X[n]. .., .. (CS Center) ... -, 2014 10 / 26 11. -- 2 ( --) F X[n] = (X1, . . . , Xn) . H0 : F = F0, H1 : F = F0. : 2 n = 1 12n + n i=1 F0(X(i)) 2i 1 2n 2 , X(1) < . . . < X(n) , X[n]. H0 F0 - n F0. .., .. (CS Center) ... -, 2014 11 / 26 12. -- n , n ( n ) 2 n. , Dn 2 n: F = F0. H0 H1, H0, H1, 1 . .., .. (CS Center) ... -, 2014 12 / 26 13. - - F X[n] = (X1, . . . , Xn) . H0 : F = F0, H1 : F = F0. S = n 2 n i=1 2i 1 2n ln(F0(x(i))) + 1 2i 1 2n ln(1 F0(x(i))) , S > S, S a2. . .., .. (CS Center) ... -, 2014 13 / 26 14. - - - . : H0: F(x) F(x) . H1: F(x) = F(x) . , , , , : F(/) : = (1, . . . , l ) Rl . , : H0 : F F(/) : Rl . H1 : H0 . .., .. (CS Center) ... -, 2014 14 / 26 15. - k : 1, . . ., k , i i = R, i j = , i = j. : n1, . . ., nk, k i=1 ni = n. 1, . . ., k : p1(), . . ., pk(). 2 ( ) Rl . : 1 : k i=1 pi () = 1. 2 : pi () > c > 0 i = 1, k. 3 : pi ()/j , 2pi ()/(uv ) i = 1, . . . , k, u, v, j = 1, . . . , l. 4 pi () j i,j=1,k l. .., .. (CS Center) ... -, 2014 15 / 26 16. - , n1, . . . , nk, . . = arg max L({ni }, ), L({ni }, ) = n! n1! . . . nk! k i=1 pni i (), -: = arg min k i=1 (ni npi ())2 npi () . , H0 , 2 () = k i=1 (ni npi ())2 npi () d n 2 kl1. .., .. (CS Center) ... -, 2014 16 / 26 17. - H0 2() : S = (u1,kl1, ), u1,kl1 1 - k l 1 . . .., .. (CS Center) ... -, 2014 17 / 26 18. , F X[n] = (X1, . . . , Xn). : H0: N(a, 2) H1: . , , . .., .. (CS Center) ... -, 2014 18 / 26 19. .., .. (CS Center) ... -, 2014 19 / 26 20. QQ- - (QQ-plot) . , .., .. (CS Center) ... -, 2014 20 / 26 21. - - . - JB = n 6 Sk2 + 1 4 K2 , Sk = 3 s3 , s2 = 1 n n i=1 (xi x)2 , 3 = 1 n i = 1n(xi x)3 K = 4 s4 3, 4 = 1 n n i=1 (xi x)4 S = (C1, +). .., .. (CS Center) ... -, 2014 21 / 26 22. - , JB 2 2 JB d n 2 (2) JB, C1 2 2 1 . n - 1 . n > 2000. n C1 -. .., .. (CS Center) ... -, 2014 22 / 26 23. - . . D(X[n]) = sup xR F n (x, X[n]) (x) , (x) x s2 " ". , . S = (C1, +), C1 - ( ) .., .. (CS Center) ... -, 2014 23 / 26 24. - - . - W = 1 S2 t i=1 ai (x(ni+1) x(i)) 2 , S2 = n i=1(xi x)2, t i=1 ai (x(ni+1) x(i)) - , ai , t = n/2 n, t = (n 1)/2 n. W . W . .., .. (CS Center) ... -, 2014 24 / 26 25. - W < W, . W H0 z = + ln W 1 W , , , . - , . .., .. (CS Center) ... -, 2014 25 / 26 26. - . ., . . . .: . , 1983. . ., . . . . .. . .: -, 1998. ., . . .: , 1983. 518 . Greenwood P. E., Nikulin M. S.A Guide to Chi-Squared Testing. New York, John Wiley & Sons, Inc., 1996. .. .., .. (CS Center) ... -, 2014 26 / 26