Θ 0 τ v 1 n ˆ θ n θ 0 χ 2 kα parenrightbig 1

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θ 0 ) τ V 1 n ( ˆ θ n θ 0 ) 2 k,α parenrightBig 1 . Example 6.27. Let X 1 ,...,X n be i.i.d. random variables from a symmetric c.d.f. F having finite variance and positive F . Consider the problem of testing H 0 : F is symmetric about 0 versus H 1 : F is not symmetric about 0. Under H 0 , there are many estimators satisfying (6.92). We consider the following five estimators: (1) ˆ θ n = ¯ X and θ = E ( X 1 ); (2) ˆ θ n = ˆ θ 0 . 5 (the sample median) and θ = F 1 ( 1 2 ) (the median of F ); (3) ˆ θ n = ¯ X a (the a -trimmed sample mean defined by (5.77)) and θ = T ( F ), where T is given by (5.46) with J ( t ) = (1 2 a ) 1 I ( a, 1 a ) ( t ), a (0 , 1 2 ); (4) ˆ θ n = the Hodges-Lehmann estimator (Example 5.8) and θ = F 1 ( 1 2 ); (5) ˆ θ n = W/n 1 2 , where W is given by (6.83) with J ( t ) = t , and θ = T ( F ) 1 2 with T given by (5.53). Although the θ ’s in (1)-(5) are different in general, in all cases θ = 0 is equivalent to that H 0 holds. For ¯ X , it follows from the CLT that (6.92) holds with V n = σ 2 /n for any F , where σ 2 = Var( X 1 ). From the SLLN, S 2 /n is a consistent estimator of V n for any F . Thus, the test having rejection region (6.93) with ˆ θ n = ¯ X and V n replaced by S 2 /n is asymptotically correct. This test is asymptotically equivalent to the one-sample t-test derived in § 6.2.3. From Theorem 5.10, ˆ θ 0 . 5 satisfies (6.92) with V n = 4 1 [ F ( θ )] 2 n 1 for any F . A consistent estimator of V n can be obtained using the bootstrap method considered in § 5.5.3. Another consistent estimator of V n can be obtained using Woodruff’s interval introduced in § 7.4 (see Exercise 86 in § 7.6). The test having rejection region (6.93) with ˆ θ n = ˆ θ 0 . 5 and V n replaced by a consistent estimator is asymptotically correct. It follows from the discussion in § 5.3.2 that ¯ X a satisfies (6.92) for any F . A consistent estimator of V n can be obtained using formula (5.110) or the jackknife method in § 5.5.2. The test having rejection region (6.93) with ˆ θ n = ¯ X a and V n replaced by a consistent estimator is asymptotically correct. From Example 5.8, the Hodges-Lehmann estimator satisfies (6.92) for any F and V n = 12 1 γ 2 n 1 under H 0 , where γ = integraltext F ( x ) dF ( x ). A
454 6. Hypothesis Tests consistent estimator of V n under H 0 can be obtained using the result in Exercise 102 in § 5.6. The test having rejection region (6.93) with ˆ θ n = the Hodges-Lehmann estimator and V n replaced by a consistent estimator is asymptotically correct. Note that all tests discussed so far are not of limiting size α , since the distributions of ˆ θ n are still unknown under H 0 . The test having rejection region (6.93) with ˆ θ n = W/n 1 2 and V n = (12 n ) 1 is equivalent to the one-sample Wilcoxon signed rank test and is shown to have limiting size α ( § 6.5.1). Also, (6.92) is satisfied for any F ( § 5.2.2). Although Theorem 6.12 is not applicable, a modified proof of Theorem 6.12 can be used to show the consistency of this test (exercise).

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