The Relation Among the Likelihood Ratio-, Wald-, and Lagrange Multiplier Tests and Their Applicability to Small Samples

摘要

In this paper, we show that the Lagrange multiplier test, the likelihood ratio test and the Wald test are monotonic functions of each other. This implies that they have the same power characteristics. If the critical values are specified such as to equate the probabilities of type I error, the probabilities of type II error will be equal as well, and conflicting results are impossible. The critical values of the three tests are related to each other by the same monotonic functions as the three test statistics. The 'conflicts' pointed out in the literature arise only when the exact critical values for each test are replaced by the asymptotically justified critical value of a Chi-square distributions. The extent to which this critical value differs from the exact one varies from test ot test. Determining the exact critical values for finite samples is somewhat difficult, because the exact finite sample distributions of the LM and LR test are generally not known. However, under the assumption of normality, the W test can be transformed into an F-test by applying the standard degrees of freedom correction. Futhermore, since the LR and LM tests are functionally related to the W test, they can be transformed into the same test statistic. We can thus conduct an exact finite sample test, based either on the LM, LR or W statistic without running the risk of obtaining conflicting results.