《数理统计》课程教学资源（参考资料）Likelihood Ratio, Wald, and（Rao）Score Tests

Likelihood Ratio,Wald,and (Rao)Score Tests Stat 543 Spring 2005 There are common large sample alternatives to likelihood ratio testing.These are the "Wald tests"(the testing version of the confidence ellipsoid material considered earlier)and the (Rao)"score tests.These are discussed in Section 6.3.2 of B&D and,for example,on pages 115-120 of Silvey's Statistical Inference. Consider cRand I("genuinely different"/"independent")restrictions on 0 g1(8)=92(8)=·=g(8)=0, (like,for instance,0=026=...=00=0).Define g0)÷(g1(0),92(0),9(0)'. We consider testing Ho:g(0)=g2(0)=...=g()=0,that is Ho:g()=0.The obvious likelihood ratio test statistic for this hypothesis is supf(X 0) 入n= sup f(X 0) 0s.t.g(0)=0 and standard theory suggests(at least in iid cases)that 2InAnx (which leads to setting critical values for likelihood ratio tests). Suppose that on is an MLE of 0.Then if Ho:g(0)=0 is true,g()ought to be near 0,and one can think about rejecting Ho if it is not.The questions are how to measure "nearness"and how to set a critical value in order to have a test with size approximately a.The Wald approach to doing this is as follows. We expect (under suitable conditions)that under Po the (k-dimensional)estimator On has √元(an-8)9Nk(0,I-1(8) Then if G(0)÷ 0g:(θ) kx ∂8， the delta method suggests that V元(g(an)-g(0)）9N(0,G(8)I(0)G(a. Abbreviate G()I()G(0)'as B().Since g(0)=0 if Ho is true,the above suggests that ng(0)'B()-1g(0)x under Ho. Now ng(n)'B()-1g(n)can not serve as a test statistic,since it involves 0,which is not completely specified by Ho.But it is plausible to consider the statistic Wm÷ng(an)'B(0n)-1g(an)） and hope that under suitable conditions WnSX好under Ho· If this can be shown to hold up,one may reject for Wn >(1-a)quantile of the x?distribution.This is the use of "expected Fisher information"in defining a Wald statistic.With Hn()the matrix of second partials of the log-likelihood evaluated at 0,an "observed Fisher information"version of the above is to let -1 B(0)=c(0)-=Hn() G()

Likelihood Ratio, Wald, and (Rao) Score Tests Stat 543 Spring 2005 There are common large sample alternatives to likelihood ratio testing. These are the "Wald tests" (the testing version of the confidence ellipsoid material considered earlier) and the (Rao) "score tests." These are discussed in Section 6.3.2 of B&D and, for example, on pages 115-120 of Silvey’s Statistical Inference. Consider Θ ⊂ (1 − α) quantile of the χ2 l distribution. This is the use of "expected Fisher information" in defining a Wald statistic. With Hn (θ) the matrix of second partials of the log-likelihood evaluated at θ, an "observed Fisher information" version of the above is to let B∗ n(θ) = G(θ) µ − 1 n Hn (θ) ¶−1 G(θ) 0 1

and use the test statistic Wi =ng(on)'Bi(on)-1g(on) The "(Rao)score test"or "x2test"is an alternative to the LR and Wald tests.The motivation for it is that on occasion it can be easier to maximize the loglikelihood In()subject to g(0)=0than to simply maximize In(0)without constraints.Let On be a "restricted"MLE (i.e.a maximizer of In(0) subject to g()=0).One might expect that if Ho:g()=0 is true,then On ought to be nearly an unrestricted maximizer of l(0)and the partial derivatives of In()should be nearly 0 at en.On the other hand,if Ho is not true,there is little reason to expect the partials to be nearly 0.So (again/still with ()In f())one might consider the statistic or using the observed Fisher information. Rn= （品aol)(a》'(0la） What is not obvious is that Rn (or R)can typically be shown to differ from 2In An by a quantity tending to 0 in Po probability.That is,this statistic can be calibrated using x2 critical points and can form the basis of a test of Ho asymptotically equivalent to the likelihood ratio test.In fact,all of the test statistics mentioned here are asymptotically equivalent to the LRT (Wald and Rao score alike). Observe that 1.an LRT requires computation of an MLE,0n,and a restricted MLE,, 2.a Wald test requires only computation of the MLE,On,and 3.a Rao score test requires only computation of the restricted MLE,On. Any of the tests discussed above (LRT,Wald,score)can be inverted to find confidence regions for the values of l parametric functions u(),u2(),...,u().That is,for any vector of potential values for these functions c=(c1,c2,...,c),one may define ge.i()=ui(0)-ci and a test of some type above for Ho:g(0)=0.The set of all c for which an approximately a-level test does not reject constitutes an approximately (1-a)x 100%confidence set for the vector (u1(0),u2(0),...,u())'.When Wald tests are used in this construction,the regions will be ellipsoidal. 2

and use the test statistic W∗ n . = ng(θ ˆn) 0 B∗ n(θ ˆn) −1g(θ ˆn) The “(Rao) score test” or “χ2 test” is an alternative to the LR and Wald tests. The motivation for it is that on occasion it can be easier to maximize the loglikelihood ln(θ) subject to g(θ) = 0 than to simply maximize ln(θ) without constraints. Let θ ˜n be a “restricted” MLE (i.e. a maximizer of ln(θ) subject to g(θ) = 0). One might expect that if H0:g(θ) = 0 is true, then θ ˜n ought to be nearly an unrestricted maximizer of ln(θ) and the partial derivatives of ln(θ)should be nearly 0 at θ ˜n. On the other hand, if H0 is not true, there is little reason to expect the partials to be nearly 0. So (again/still with ln(θ) = Pn i=1 ln f(Xi|θ)) one might consider the statistic Rn . = 1 n à ∂ ∂θi ln(θ) ¯ ¯ ¯ ¯ θ=θ ˜n !0 I1(θ ˜n) −1 à ∂ ∂θi ln(θ) ¯ ¯ ¯ ¯ θ=θ ˜n ! or using the observed Fisher information, R∗ n . = à ∂ ∂θi ln(θ) ¯ ¯ ¯ ¯ θ=θ ˜n !0 ³ −Hn ³ θ ˜n ´´−1 à ∂ ∂θi ln(θ) ¯ ¯ ¯ ¯ θ=θ ˜n ! What is not obvious is that Rn (or R∗ n) can typically be shown to differ from 2 ln λn by a quantity tending to 0 in Pθ probability. That is, this statistic can be calibrated using χ2 critical points and can form the basis of a test of H0 asymptotically equivalent to the likelihood ratio test. In fact, all of the test statistics mentioned here are asymptotically equivalent to the LRT (Wald and Rao score alike). Observe that 1. an LRT requires computation of an MLE, θ ˆn, and a restricted MLE, θ ˜n, 2. a Wald test requires only computation of the MLE, θ ˆn, and 3. a Rao score test requires only computation of the restricted MLE, θ ˜n. Any of the tests discussed above (LRT, Wald, score) can be inverted to find confidence regions for the values of l parametric functions u1(θ), u2(θ),...,ul(θ). That is, for any vector of potential values for these functions c = (c1, c2,...,cl) 0 , one may define gc,i (θ) = ui (θ) − ci and a test of some type above for H0:gc(θ) = 0. The set of all c for which an approximately α-level test does not reject constitutes an approximately (1 − α) × 100% confidence set for the vector (u1(θ), u2(θ),...,ul(θ))0 . When Wald tests are used in this construction, the regions will be ellipsoidal. 2