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1 20104 Chinese Journal of Applied Probability and Statistics Vol.26 No.2 Apr P (,, ) P P. Wang (2006)P, P, P,. : P,,,. : O212.1, O , (). : X 1, X 2,, X n N(θ, σ 2 ), σ 2. H 0 : θ = θ 0 H 1 : θ θ 0. θ : P(θ = θ 0 ) = π 0, θ θ 0 π 1 g 1 (θ), π 1 = 1 π 0, g 1 (θ),. nz = n x θ 0 /σ P, H 0 α 0 1. Lindley [1][2],. BergerSellke (1987):, π 0 = 0.5, n, g 1 (θ), α 0 P, Berger-Sellke. P, H 0, α 0,.,.,, π 0, g 1 (θ)α 0. α 0 α 0. Lindley (1997)Lavine(1999)α 0. P (),. P H 0., H 0 : θ = 50 H 1 : θ 50. σ = 1, n = 2500, x = 50.1, : z = 5, P = 2[1 Φ(5)] = , P ,

2 172,,,,. P,. Box (1980)P, Guttman (1967)Rubin (1984)P, Meng (1994), Gelman(1996)De la HorraRodriguez-Bernal (2003). BayarriBerger (2000)P P.. : CasellaBerger (1987), Gomez-VillegasSanz (1998, 2000), OhDasGupta (1999), Gomez-Villegas(2002), MicheasDcy (2003), De la Horra (2005), MicheasDey (2007). P., P, P..,.., ( ),,,,., k(k4 ),.,,.,,,,.,,. Wang (2006) (2007)P, P, Wang (2006)P, P,, P. 2. P 2.1 P X 1, X 2,, X n N(θ, σ 2 ), σ 2. H 0 : θ = θ 0 H 1 : θ θ 0. x = 1 n n n(x θ) n(x θ0 ) x i, z =, z 0 =, σ σ i=1

3 : P 173 Z. P 2P(Z > z 0 ), z 0 > 0; P = 2P(Z < z 0 ), z 0 < 0, P = P( Z > z 0 ) = 2[1 Φ( z 0 )], Φ( )., θ 0 = 0σ = 1, Y i = (X i θ 0 )/σ. : z 0 = nx, z = z 0 nθ. : H 0 : θ Θ 0 = {0}H 1 : θ Θ 1 = ( a, 0) (0, b), a, b > 0.,, θ 0, a = b., a b,. a = b, a b,. Θ = Θ 0 Θ Wang (2006)P P W P W P W = P θ=θ0 ( Z > z ) inf P( Z > z ). P( Z > z ) sup P( Z > z ) inf P W (a) = P W, z = z 0 nθ, P W (a) = P W (a) = = sup 2.3 P P F P F 2 P F = P( Z > z 0 ) inf P( Z > z 0 nθ ) P( Z > z 0 nθ ) inf P( Z > z 0 nθ ). P( Z > z 0 ) P( Z > z 0 + na) P( Z > max( z 0 na, 0)) P( Z > z 0 + na) Φ[ z 0 + na] Φ[ z 0 ] Φ[ z 0 + na] Φ[max( z 0 na, 0)]. (2.1) 2 P θ=θ0 (Z > z) inf P(Z > z) sup sup P(Z > z) inf P(Z > z), z 0 > 0; P θ=θ0 (Z < z) inf P(Z < z) P(Z < z) inf P(Z < z), z 0 < 0.

4 174 P F (a) = P F, Φ[z 0 + na] Φ[z 0 ] 2 P F Φ[z 0 + na] Φ[z 0 na], z 0 > 0; (a) = Φ[ na z 0 ] Φ[ z 0 ] 2 Φ[ na z 0 ] Φ[ z 0 na], z 0 < P F (a)p W (a) 2.1 : P F Φ( z 0 + na) Φ( z 0 ) (a) = 2 Φ( z 0 + na) Φ( z 0 na). (2.2) z 0 > na x > a, P F (a) = 2P W (a). z 0 > 0, z 0 < 0, (2.1) P W (a) = (2.2): P F (a) = 2P W (a),. 2.2 (1) dp F (a)/da < 0. (2) lim P F (a) = P, lim P F (a) = 1. a a 0 (3) sup P F (a) = 1, inf P F (a) = P. : (1) Φ[ z 0 + na] Φ[ z 0 ] Φ[ z 0 + na] Φ[ z 0 na]. P F Φ[ z 0 + na] Φ[ z 0 ] (a) = 2 Φ[ z 0 + na] Φ[ z 0 na]. F (a) = z > 0, F (a) < 0. F (a) = = Φ(z + a) Φ(z) Φ(z + a) Φ(z a), φ(z + a)[φ(z) Φ(z a)] φ(z a)[φ(z + a) Φ(z)] [Φ(z + a) Φ(z a)] 2 φ(z + a) [Φ(z) Φ(z a)] e 2az [Φ(z + a) Φ(z)] 2π [Φ(z + a) Φ(z a)] 2, φ( ). G(a) = [Φ(z) Φ(z a)] e 2az [Φ(z + a) Φ(z)],

5 : P 175 G (a) = φ(z a) e 2az φ(z + a) 2ze 2az [Φ(z + a) Φ(z)] = 2ze 2az [Φ(z + a) Φ(z)] < 0. G(0) = 0, a > 0G(a) < 0, F (a) < 0dP F (a)/da < 0. > 0. (2) lim (a) a = lim Φ( z 0 + na) Φ( z 0 ) a Φ( z 0 + na) Φ( z 0 na) = 2[1 Φ( z 0 )] = P lim P F (a) = lim a 0 (3) (1)(2), Φ( z 0 + na) Φ( z0 ) a 0 Φ( z 0 + na) Φ( z 0 na) = 2 lim a 0 = 1. nφ( z0 + na) nφ( z0 + na) + nφ( z 0 na) sup P F (a) = lim P F (a) = 1, a 0 inf P F (a) = lim P F (a) = P. a 2.3 (1) lim P W (a) = P, lim P W (a) = 1. a a 0 (2) z 0 > na, x >a, dp W (a)/da<0. z 0 < na, x <a, dp W (a)/da (3) sup P W (a) = 1, ( inf P W (a) = P W z 0 ) = Φ(2 z 0 ) Φ( z 0 ) < P. n Φ(2 z 0 ) 0.5 (4) z 0 < na, x < a, P W (a) < P. : (1) (2.1), lim P W (a) = 1 Φ[ z 0 ] = 2[1 Φ[ z 0 ]] = P a 1 0.5

6 176 lim P W Φ[ z 0 + na] Φ[ z0 ] (a) = lim a 0 a 0 Φ[ z 0 + na] Φ[max( z 0 na, 0)] Φ[ z 0 + na] Φ[ z0 ] = lim a 0 Φ[ z 0 + na] Φ[ z 0 ] = 1. (2) z 0 > na, P F (a) = 2P W (a), z 0 < na, dp W (a)/da > 0. (3) (1)(2) (2)(2.1) dp W (a) da = 1 dp F (a) < 0; 2 da P W (a) = Φ[ z 0 + na] Φ[ z 0 ] Φ[ z 0 + na] 0.5, sup P W (a) = lim P W (a) = 1. a 0 ( inf P W (a) = P W z 0 ) = Φ(2 z 0 ) Φ( z 0 ) < lim n Φ(2 z 0 ) 0.5 P W (a) = P. a (4) z 0 < na, (2), dp W (a)/d; (4), lim P W (a)=p. P W (a)<p. a. 2.5 P F (a)p W (a) (1) P W (a), a < z 0 / n, a; a > z 0 / n, a.,!, 2.3 (4), a > z 0 / n, P W (a) < P. a > z 0 / n,., P W (a)p., P H 0, P W (a) P. (2) P F (a) P F (a)a,, a, H 0, P F (a)a. sup P F (a) = 1, inf P F (a) = P P F (a) H 0. inf P F (a) = P P F (a).

7 : P H 0 : θ = 500H 1 : θ 500θ (500 t, t). x = 506, σ = 90, n = 800, z 0 = n(x θ 0 )/σ = 1.886, P = 2[1 2Φ(1.886)] = 0.059, t a P F P W P : (1) a > z 0 / n, x (θ 0 t, θ 0 + t),, P W, P W, P,,.. (2) P F θ 2, P, (Θ), P F P, P F. : P F (a, b)p. [1] Shafer, G., Lindley s paradox, J. Amer. Statist. Assoc., 77(1982), [2] Berger, J.O.,,, [3] Berger, J.O. and Sellke, T., Testing a point null hypothesis: the irreconcilability of P values and evidence (with discussion), J. Amer. Statist. Assoc., 82(1987), [4] Lindley, D.V., Discussion forum: some comments on bayes factors, Journal of Statistical Planning and Inference, 61(1997), [5] Lavine, M. and Schervish, M.J., Bayes factors: what they are and what they are not, The American Statistician, 53(2)(1999), [6] Box, G.E.P., Sampling and Bayes inference in scientific modelling and robustness (with discussion), J. Roy. Statist. Soc. A, 143(1980), [7] Guttman, L., The use of the concept of a future observation in goodness-of-fit problems, J. Roy. Statist. Soc. B, 29(1967), [8] Rubin, D.B., Bayesianly justifiable and relevant frequency calculations for the applied statistician, Ann. Statist., 12(1984), [9] Meng, X.L., Posterior predictive P-values, Ann. Statist., 22(1994), [10] Gelman, A., Meng, X.L. and Stern, H., Posterior predictive assessment of model fitness via realized discrepancies (with discussion), Statistica Sinica, 6(1996),

8 178 [11] De la Horra, J. and Rodriguez-Bernal, M.T., Bayesian robustness of the posterior predictive p-value, Commun. Statist. Theory-Meth., 32(2003), [12] Bayarri, M.J. and Berger, J.O., P-values for composite null models, J. Amer. Statist. Assoc., 95(2000), [13] Casella, G. and Berger, R.L., Reconciling Bayesian and frequentist evidence in the one-sided testing problem, J. Amer. Statist. Assoc., 82(1987), [14] Gomez-Villegas, M.A. and Sanz, L., Reconciling Bayesian and frequentist evidence in the point null testing problem, Test, 7(1998), [15] Gomez-Villegas, M.A. and Sanz, L., ε-contaminated priors in testing point null hypothesis: a procedure to determine the prior probability, Statist. Probab. Lett., 47(2000), [16] Oh, H.S. and DasGupta, A., Comparison of the P-value and posterior probability, Journal of Statistical Planning and Inference, 76(1999), [17] Gomez-Villegas, M.A., Main, P. and Sanz, L., A suitable Bayesian approach testing point null hypothesis: some examples revisited, Commun. Statist. Theor. Meth., 31(2002), [18] Micheas, A.C. and Dey, D.K., Prior and posterior predictive p-values in the one-sided location parameter testing problem, Sankhya, 65(2003), [19] De la Horra, J., Reconciling classical and prior predictive P-values in the two-sided location parameter testing problem, Commun. Statist. Theor. Meth., 34(2005), [20] Micheas, A.C. and Dey, D.K., Reconciling Bayesian and Frequentist Evidence in the One-Sided Scale Parameter Testing Problem, Communications in Statistics-Theory and Methods, 36(2007), , [21] Wang, H., Modified p-values for two-sided test for normal distribution with restricted parameter space, Communications in Statistics-Theory and Methods, 35(2006), [22] Wang, H., Modified p-values for one-sided testing in restricted parameter space, Statistics and Probability Letters, 77(2007), Modification of P -Value of Two-Sided Test with Restricted Parameter Space and Its Reconciliation with Bayesian Evidence Fu Junhe (Department of Information Management, College of International Business Management, Shanghai International Studies University, Shanghai, ) This paper makes research on modification of P -value of two-sided test with restricted parameter space and reconciling Bayesian test with classical test based on modified P -value, which shows that there exist critical defects in modified P -value put forward by Wang in This paper sets forth another modified P -value, which has comparative reasonable characteristics, based on which the conflict of Bayesian test and classical test can be reconciled to some extent. Keywords: Modified P -value, evidence, Bayesian, classical Statistics. AMS Subject Classification: 97K70.

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