12/12/2022 0 Comments Statplus paired t test![]() ![]() The other reason is that the Wald test uses two approximations (that we know the standard error or Fisher information and the maximum likelihood estimate), whereas the likelihood ratio test depends only on the ratio of likelihood functions under the null hypothesis and alternative hypothesis.Non-invariance: As argued above, the Wald test is not invariant under reparametrization, while the likelihood ratio tests will give exactly the same answer whether we work with R, log R or any other monotonic transformation of R.There are several reasons to prefer the likelihood ratio test or the Lagrange multiplier to the Wald test: Although they are asymptotically equivalent, in finite samples, they could disagree enough to lead to different conclusions. Engle showed that these three tests, the Wald test, the likelihood-ratio test and the Lagrange multiplier test are asymptotically equivalent. There exist several alternatives to the Wald test, namely the likelihood-ratio test and the Lagrange multiplier test (also known as the score test). ![]() For example, asking whether R = 1 is the same as asking whether log R = 0 but the Wald statistic for R = 1 is not the same as the Wald statistic for log R = 0 (because there is in general no neat relationship between the standard errors of R and log R, so it needs to be approximated). ![]() The fact that one uses an approximation of the variance has the drawback that the Wald statistic is not-invariant to a non-linear transformation/reparametrisation of the hypothesis: it can give different answers to the same question, depending on how the question is phrased. This result is obtained using the delta method, which uses a first order approximation of the variance. Under the Wald test, the estimated θ ^ is the derivative of c evaluated at the sample estimator.
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