Example The strength of concrete depends, to some extent on the method used for drying it. Two different drying methods were tested independently on specimens. The strength using each of the methods follow a normal distribution with mean μ x and μ y respectively and the same variance. The results are…. Do the methods appear to produce concrete with different mean strength?

Likelihood Ratio Tests - Introduction Neyman-Pearson lemma provides a method of constructing most powerful tests for simple hypothesis when the distribution of the observations is known except for the value of a single unknown parameter. Sometimes it can be utilized to find uniformly most powerful test for composite hypothesis that involve a single parameter. In many cases, the distribution of interest has more than one unknown parameter. Likelihood ratio test is a general method used to derive tests of hypothesis for simple or composite hypotheses.

Likelihood Ratio Test The null hypothesis specifies that the parameter (possibly a vector) lies in a particular set of possible values denoted by Ω 0 and the alternative hypothesis specifies another set of possible values denoted by Ω a, which does not overlap with Ω 0. Examples… A likelihood ratio test has a test statistic Λ defined by For a fixed size α test the decision rule is: reject H 0 if Λ ≤ k where k is determined such that P(Λ ≤ k | H 0 ) = α.

Translation of the Likelihood Ratio Test Small value of Λ indicates that the likelihood of the sample is smaller under H 0 and therefore the data suggest that H 0 is false. Large value of Λ indicates no evidence against H 0.

Distribution of the Likelihood Ratio Statistic In many cased the distribution of the test statistic Λ is known and can be used to find k and the rejection region. If the distribution of Λ is unknown we use the fact that where r is the number of parameters specified in H 0. This result is true for large n. The critical region in this case is: reject H 0 if

Examples