2013 4 Chinese Journal of Applied Probability and Statistics Vol.29 No.2 Apr. 2013 (,, 550004) IEC1123,,,., IEC1123 (SMT),,,. :,,, IEC1123,. : O212.3. 1. P.,,,, [1 5]. P, : H 0 : P = P 0 vs H 1 : P = P 1 (P 0 > P 1 ). (1.1), Wald (1947) (SPRT), [1] L 1 : S = h 1 + sn (h 2 < h 1 ), (1.2) L 2 : S = h 2 + sn, h 1, h 2, s, n, X i i, X i = 1, X i = 0, S n = i=n X i, n = 1, 2, 3,... n. i=1 SPRT : S n h 2 + sn, H 0, S n h 1 + sn, H 0, h 2 + sn < S n < h 1 + sn,. SPRT α, β (, (α, β) ), P 0 P 1. SPRT, [6, 7], ( J [2011]2094 ). 2012 3 16, 2012 7 9.
202,., IEC1123 (1991), GB8051 (2002) [4, 5],., IEC1123,, : 1) IEC1123 N t ( ). P 0 = 0.9995, P 1 = 0.9993, (α, β) = (0.05, 0.05), N t = 72574 [4]. [6 8], IEC1123 N t. 2) IEC1123 [9, 10]. (α, β) N t, IEC1123,,,. 3) IEC1123, α, β (α, β) [6 8],, (α, β). 4) IEC1123 α β, α β,, IEC1123 P 0, P 1, (α, β) 160, P 0, P 1, (α, β). IEC1123,,, (1.1), IEC1123. 2. [1 10], P : 2.1 L 1, L 2,..., L Nt U 1, U 2,..., U Nt, 0 L i+1 L i 1, 0 U i+1 U i 1, i = 1, 2,..., N t 1; L i + 2 U i, i = 1, 2,..., N t 1; (2.1) L Nt + 1 = U Nt. M = inf{n S n U n S n L n, n = 1, 2,..., N t }, S M L M, H 0, S M U M, H 0, L n < S n < U n (n = 1, 2,..., N t 1),., N t, U Nt ( N t, H 0 ), T (N t, U Nt ) = (U 1, U 2,..., U Nt ; L 1, L 2,..., L Nt ) (2.2) N t, U Nt (TST), (U 1, U 2,..., U Nt ), (L 1, L 2,..., L Nt ).
: 203 D n = {S n L n, L i < S i < U i, i = 1, 2,..., n 1}, Dn = {S n U n, L i < S i < U i, i = 1, 2,..., n 1}, n = 1, 2,... N t, (1.1) T α, β α (T ) = n=nt P{D n P 0, T }; β (T ) = n=nt P{ D n P 1, T }. T E(M P, T ) E(M P 0, T ) = n=nt n(p{d n P 0, T } + P{ D n P 0, T }); 2.2 E(M P 1, T ) = n=nt n(p{d n P 1, T } + P{ D n P 1, T }). (2.3) (2.4) (α, β) T, (1.1) α (T ) α, β (T ) β, T (1.1) (α, β).,, [11],, (α, β) T (N t, U Nt ),. 2.3 T LA (N t, U Nt ) (α, β), (α, β) T (N t, U Nt ), E(M P 0, T LA ) E(M P 0, T ); E(M P 1, T LA ) E(M P 1, T ), (2.5) T LA (N t, U Nt ) (1.1) N t, U Nt, (α, β) (LOANTST). 2.4 T A N t, (α, β), N t, (α, β) T, E(M P 0, T A ) E(M P 0, T ); E(M P 1, T A ) E(M P 1, T ), (2.6) T A (1.1) N t, (α, β) (OANTST). 2.3 2.4 T LA T A : E(M P 0, T A ) = Min {E(M P 0, T LA (N t, U Nt )}; 0 U Nt N t E(M P 1, T A ) = Min {E(M P 1, T LA (N t, U Nt )}. 0 U Nt N t (2.7)
204 (1.1) C(N t, U Nt ) (N t, S Nt U Nt, H 0,, H 0 ), N t 1 U Nt, N t, H 0, N t 2, N t 3,..., U Nt ; H 0., C(N t, U Nt ),. 2.5 (1.1) C(N t, U Nt ), T E 21 3 2 U NtUNt = 1 U NtUNt U NtUNt +1 U NtNt (2.8) 1 1 1 2 1 Nt UNt 0 Nt UNt +1 1 Nt UNt +2 (U Nt 1) Nt C(N t, U Nt ) (ETST),, 1 Nt UNt N t U Nt 1 H 0, N t U Nt 1,, H 0, 1 1, 1 2,..., 1 Nt UNt 2 1, 3 2,..., U NtUNt 1. 0 Nt UNt +1 N t U Nt + 1 0, H 0,, 0 Nt UNt +1, 1 Nt UNt +2,..., (U Nt 1) Nt U NtUNt,..., U NtNt. 1 C(15, 13) T E 2 3 4 5 6 7 8 9 10 11 12 13 13 13 13 T E =. (2.9) 1 1 0 1 2 3 4 5 6 7 8 9 10 11 12,. 2.1 (1.1) C(N t, U Nt ), P{S Nt <U Nt P 0 }=α, P{S Nt U Nt P 1 } = β, T E C(N t, U Nt ), n=n t n=n t P{D n P 0, T E } = α ; P{ D n P 1, T E } = β, P (0, 1), E(M T E, P ) N t. : T E P (0, 1), P{D n P, T E } = 0, n = 1, 2,..., N t U Nt ; P{ D n P, T E } = 0, n = 1, 2,..., U Nt 1; (2.10) D n {S Nt < U Nt }, n = N t U Nt + 1, N t U Nt + 2,..., N t ; D n {S Nt U Nt }, n = U Nt, U Nt + 1,..., N t.
: 205 {S Nt < U Nt } = n=nt D n, {S Nt U Nt } = n=nt (i j), n=n t P{D n P 0, T E } = P{S Nt < U Nt P 0 } = α ; n=n t P{ D n P 1, T E } = P{S Nt U Nt P 1 } = β. D n D i D j = φ, Di D j = φ P (0, 1), P{D n P, T E } 0, P{ D n P, T E } 0 n=nt (P{D n P, T E } + P{ D n P, T E }) = 1, E(M P, T E ) = n=nt n(p{d n P, T E } + P{ D n P, T E }) N t. 2.1 T E (N t, U Nt ) C(N t, U Nt ),,. 2.2 (α 1, β 1 ), (α 2, β 2 ). T 1 (N t, U Nt ) T 2 (N t, U Nt ) i) i 0 {1, 2,..., N t 1}, L i0 (T 1 ) L i0 (T 2 ), i i 0, L i (T 1 ) = L i (T 2 ); j {1, 2,..., N t 1}, U j (T 1 ) = U j (T 2 ), α 1 α 2, β 1 β 2, E(M P 0, T 1 ) E(M P 0, T 2 ); E(M P 1, T 1 ) E(M P 1, T 2 ). (2.11) ii) j 0 {1, 2,..., N t 1}, U j0 (T 1 ) U j0 (T 2 ), i j 0, U j (T 1 ) = U j (T 2 ); i {1, 2,..., N t 1}, L i (T 1 ) = L i (T 2 ), α 1 α 2, β 1 β 2, E(M P 0, T 1 ) E(M P 0, T 2 ); E(M P 1, T 1 ) E(M P 1, T 2 ). : i), ii). γ 1 = n=nt P{ D n D i0 = L i0 (T 1 ), P 0, T 2 },, γ 1 0, α 1 α 2. n=i 0 +1 α 1 = n=nt P{D n P 0, T 1 } = α 2 + γ 1 α 2, (2.12) E(M P 0, T 1 ) = E(M P 0, T 2 ) n=nt n=i 0 +1 + P{ D n D i0 = L i0 (T 1 ), P 0, T 2 }) E(M P 0, T 2 ), (n i 0 )(P{D n D i0 = L i0 (T 1 ), P 0, T 2 }
206 E(M P 0, T 1 ) E(M P 0, T 2 )., γ 2 = n=nt P{ D n D i0 = L i0 (T 1 ), P 1, T 2 }, γ 2 0,, n=i 0 +1 β 1 = n=nt P{ D n P 1, T 1 } = β 2 γ 2 β 2, β 1 β 2. E(M P 1, T 1 ) = E(M P 1, T 2 ) n=nt n=i 0 +1 + P{ D n D i0 = L i0 (T 1 ), P 1, T 2 }) E(M P 1, T 2 ), E(M P 1, T 1 ) E(M P 1, T 2 ). (n i 0 )(P{D n D i0 = L i0 (T 1 ), P 1, T 2 } 2.2,,, α, β ;,.,. 2.3 (α, β) LOANTST. T LA (1.1) N t, U Nt, i) T LA T, α (T ) > α. ii) T LA T, β (T ) > β. : ( ) i) α (T ) α,, T T LA, 2.2, α (T ) α, β (T ) β, E(M P 0, T ) E(M P 0, T LA ); E(M P 1, T ) E(M P 1, T LA ). LOANTST, T LA (1.1) N t, α, β LOANTST,. ii). 2.3 (α, β) LOANTST T LA.,,. 3. 2.1 2.2, (1.1) N t, U Nt LO- ANTST T LA, C(N t, U Nt ) T E,
: 207.,,., C(N t, U Nt ) T E T 0 : U1 U 2 U 3 U Nt 1 U Nt T 0 =. (3.1) L 1 L 2 L 3 L Nt 1 L Nt, T 0 i=nt (U i L i + 1), 2N t, T 0 i=1. T 1, : i) U i, T 1,Ui, U1 U 2 U i 1 U i 1 U i+1 U Nt T 1,Ui =. (3.2) L 1 L 2 L i 1 L i L i+1 L Nt ii) L i, T 1,Li, U1 U 2 U i 1 U i U i+1 U Nt T 1,Li =. (3.3) L 1 L 2 L i 1 L i + 1 L i+1 L Nt U i L i? T 0 (3.2) (3.3) T 1 2.1,. 1) U i L i 2, i = 1, 2,..., N t, U i L i, T 1 N t. 2) T 0 L i, L i = 1, L i = 1 L i+1 = 0; T 0 U i, U i > i, U i > i U i+1 = i + 1. 3) T 0 L i, L i = L i+1, L i ; T 0 U i, U i = U i 1, U i. 3.1 T 0., {L i, i Θ 0 } {U i, i Θ 1 }, Θ 0, Θ 1. (2.3), (2.4), T 0 T 1 α (T 0 ) = n=nt P{D n P 0, T 0 }; β (T 0 ) = n=nt P{ D n P 1, T 0 }, α (T 1 ) = n=nt P{D n P 0, T 1 }; β (T 1 ) = n=nt P{ D n P 1, T 1 }, (3.4) (3.5)
208 E(M P 0, T 0 ) = n=nt n(p{d n P 0, T 0 } + P{ D n P 0, T 0 }); E(M P 1, T 0 ) = n=nt n(p{d n P 1, T 0 } + P{ D n P 1, T 0 }), E(M P 0, T 1 ) = n=nt n(p{d n P 0, T 1 } + P{ D n P 0, T 1 }); E(M P 1, T 1 ) = n=nt n(p{d n P 1, T 1 } + P{ D n P 1, T 1 }). (3.6) (3.7) T 0,., T 0 T 1 α (T 1 ) α (T 0 ), β (T 1 ) β (T 0 ), P 0, P 1 E(M P 0, T 0 ) E(M P 0, T 1 ), E(M P 1, T 0 ) E(M P 1, T 1 ), T 0 α (T 0 ), β (T 0 ) (α, β). T 0, g(l i ) = [E(M P 0, T 0 ) E(M P 0, T 1,Li )] [α α (T 0 )] [α (T 1,Li ) α (T 0 )] [β (T 0 ) β, (T 1,Li )] i Θ 0, (3.8) g(u j ) = [E(M P 1, T 0 ) E(M P 1, T 1,Uj )] [β β (T 0 )] [β (T 1,Uj ) β (T 0 )] [α (T 0 ) α, (T 1,Uj )] j Θ 1. (3.9), W 1 : i) α (T 0 ) < α β (T 0 ) β, W 1, g(w 1 ) = Max{g(L i ), i Θ 0 }. (3.10) ii) α (T 0 ) α β (T 0 ) < β, W 1, g(w 1 ) = Max{g(U j ), j Θ 1 }. (3.11) iii) α (T 0 ) α β (T 0 ) β, W 1 g(w 1 ) = Max{Max{g(L i ), i Θ 0 }, Max{g(U j ), j Θ 1 }}. (3.12) W 1, T 0 W 1 T 1. T 0, T 1 W 2, T 1 W 2 T 3,,,, T 0, T 1,..., T n,.... m = sup{n β (T n ) < β and α (T n ) < α}. (3.13) 2.2 2.3, T m (1.1) N t, U Nt, (α, β) LOANTST T LA. N t OANTST T A, (2.7), U Nt 0 N t, LOANTST T LA,, T A.
: 209 4. IEC1123 SMT IEC1123, [6] (SMT) IEC1123,,, IEC1123 SMT. 2, (α, β) = (0.2, 0.2), : H 0 : P 0 = 0.9, H 1 : P 1 = 0.7. (4.1) N t = 15, U Nt = 13, C(15, 13) α = 0.1841 < 0.2, β = 0.1268 < 0.2., C(15, 13) 15. T E C(15, 13) ETST, 3.1, T E 1 (2.9)., T E α = 0.1841, β = 0.1268 E(M 0.9, T E ) = 13.4525 < 15; (4.2) E(M 0.7, T E ) = 9.3867 < 15. C(15, 13), ETST T E,. (4.1) N t = 15, U Nt = 13, (0.2, 0.2) LOANTST T LA, (2.9) T E, LO- ANTST T LA 2 3 4 5 6 6 7 8 9 10 10 11 12 13 13 T LA =. (4.3) 1 0 1 2 3 4 5 5 6 7 8 9 10 11 12, T LA α = 0.1983, β = 0.1894, E(M 0.9, T LA ) = 7.7656; (4.4) E(M 0.7, T LA ) = 6.1795. (4.2) (4.4),, T LA T E,. N t = 15, U Nt = 0, 1, 2,..., 15 LOANTST,, (4.3) T LA N t = 15, (0.2, 0.2) OANTST T A. (4.1), IEC1123 N t = 15, U Nt = 13 2 3 4 5 6 6 7 8 9 10 10 11 12 13 13 T IEC =. (4.5) 1 0 1 2 3 3 4 5 6 7 8 9 10 11 12
210 T IEC α = 0.1704, β = 0.1990, E(M 0.9, T IEC ) = 8.1684; E(M 0.7, T IEC ) = 6.8102. (4.6) (4.4) (4.6),, IEC1123. IEC1123 SMT, [6 8]. 3 (α, β) = (0.2, 0.2), : H 0 : P 0 = 0.9, H 1 : P 1 = 0.8. (4.7) IEC1123 N t = 49, U Nt = 43 T IEC LOANTST T LA 1: 1 IEC1123 α β E(M 0.9) E(M 0.8) IEC1123 T IEC 0.1977 0.1990 23.8815 21.0264 T LA 0.1991 0.1996 23.5968 20.6080 1, (α, β) = (0.2, 0.2), T LA T IEC. U Nt, N t = 49 OANTST T A U Nt = 43 LOANTST T LA. N t U Nt, (4.7) N t = 37, U Nt = 32., α = 0.1993 < 0.2, β = 0.1999 < 0.2, E(M 0.9, T A ) = 29.5141; (4.8) E(M 0.8, T A ) = 21.3090. N t = 37 [6] SMT 37,, [6] N t = 38, U Nt = 33 SMT α = 0.1749 < 0.2, β = 0.2039 > 0.2, β = 0.2039 > 0.2,, SMT (4.7) (0.2, 0.2). [8], SMT N t = 39, U Nt = 34,,, (4.7) SMT N t = 39 N t = 37., N t = 39, U Nt = 34 α, β, 2.
: 211 2 (SMT) α β E(M 0.9) E(M 0.8) T SMT 0.1966 0.1876 27.53 23.83 T LA 0.1984 0.1996 25.2318 21.2443 2,, SMT,. IEC1123, IEC1123 160 16, 1, 1, 4 IEC1123, IEC1123., IEC1123 SMT, N t,,. 3, α, β,, 1 IEC1123 4. LOANTST, OANTST,,, 2, 3 1, IEC1123 SMT. 1 IEC1123 IEC1123 T LA P 0 P 1 α = β N t U Nt α β E(M P 0 ) E(M P 1 ) α β E(M P 0 ) E(M P 1 ) 1 0.8 0.4 0.3 4 3 0.2576 0.1984 2.0880 1.7360 0.2576 0.1984 2.0880 1.7360 2 0.8 0.4 0.2 5 4 0.1962 0.1907 2.7808 2.7488 0.1962 0.1907 2.7808 2.7488 3 0.8 0.4 0.1 12 8 0.0820 0.0828 6.3538 5.8481 0.0973 0.0907 5.8768 5.2512 4 0.8 0.4 0.05 17 11 0.0499 0.0485 8.6603 7.6213 0.0496 0.0465 8.8610 7.6527 5 0.8 0.6 0.3 10 8 0.2502 0.3042* 4.8245 4.6394 0.2746 0.2755 5.0621 4.5402 6 0.8 0.6 0.2 20 15 0.2031* 0.1878 9.9265 8.8915 0.1946 0.1992 9.6539 8.8266 7 0.8 0.6 0.1 44 32 0.1027* 0.0964 20.2280 18.1636 0.0990 0.0996 19.9791 18.2917 8 0.8 0.65 0.3 13 10 0.2779 0.2987 7.4480 6.7235 0.2878 0.2957 7.1565 6.4346 9 0.8 0.65 0.2 36 27 0.1890 0.1968 16.7856 15.4135 0.1994 0.1987 15.9813 14.3444 10 0.8 0.7 0.3 28 22 0.2963 0.2908 14.9407 14.1196 0.3000 0.2978 13.9172 13.0088 11 0.85 0.55 0.3 6 5 0.2109 0.3251* 2.2563 2.0497 0.2528 0.2117 3.2576 2.6043 12 0.85 0.55 0.2 9 7 0.1435 0.1876 5.4164 4.7874 0.1829 0.1944 4.7193 3.9737 13 0.85 0.55 0.1 19 14 0.0872 0.0958 9.7077 7.8732 0.0992 0.0993 9.1386 7.2362 14 0.85 0.55 0.05 31 9 0.0442 0.0487 13.6573 11.8797 0.0499 0.0487 13.4066 11.1969 15 0.85 0.7 0.3 13 11 0.2616 0.2973 7.0684 6.4395 0.2954 0.2812 6.5344 5.7211 16 0.85 0.7 0.2 19 14 0.1816 0.1966 15.2792 13.6536 0.1998 0.1976 14.1851 12.2821 *.
212 [1] Wald, A., Sequential Analysis, Wiley and Sons, New York, 1947. [2],,,,, 30(3)(2008), 577 580. [3],,,, 1(2001), 9 16. [4] IEC1123, Reliability Testing-Compliance Test Plans for Success Ratio, 1991, 1 61. [5] GB/T 8051-2002, 1 43. [6],,,, ( ), 125(2006), 67 71. [7],,,,, 23(1)(2007), 77 83. [8],,,, 30(1)(2010), 91 95. [9],,,,, 34(3)(2010), 287 290. [10] Hu, S.G., Optimum truncated sequential test of binomial distribution, 2011 ICRMS, Guiyang, China, Vol.I: 293 298. [11],, :, 2005. Truncated Sequential Test for a Proportion and it s Sample Space Ordering Method Hu Sigui (College of Basic Medical Science, Guiyang Medical University, Guiyang, 550004 ) This paper proposes a new sampling plan, the sample space ordering method, to compute the optimum truncated sequential test in order to overcome the disadvantages of the widely use sequential sampling methods that IEC1123 has presented. The main ideal of this new method is to establish an order at the truncated sequential sample space, and optimize point by point to arrive the optimal truncated sequential test. The paper presents in detail how to realize the new plan, and shows that this new plan has most powerful to control the sample number and least average sample number comparing with the methods which IEC1123 and SMT have proposed. Keywords: Success/failure test, sequential analysis, truncated sequential test, IEC1123, sample space ordering method. AMS Subject Classification: 90B25.