Question

Show that the likelihood ratio principle leads to the same test when testing a simple hypothesis \(H_{0}\) against an alternative simple hypothesis \(H_{1}\), as that given by the Neyman- Pearson theorem. Note that there are only two points in \(\Omega\).

Short answer

Both the Neyman-Pearson theorem and the likelihood ratio principle are fundamentally based on calculating a test statistic for making a decision about the hypothesis. In both the tests, the hypothesis is rejected if the calculated test statistic (or the likelihood ratio) is greater than a certain critical value. Hence, the Neyman-Pearson theorem and likelihood ratio principle yield the same test for the simple hypothesis testing.

Step-by-step solution

Understanding of Likelihood Ratio Principle

The likelihood ratio for a given set of observations is the ratio of the likelihoods under two different hypotheses. It is a fundamental tool for statistical inference. The likelihood ratio test is a statistical test used for comparing the goodness of fit of two statistical models - the null model against an alternative model.

Understanding of Neyman-Pearson Theorem

The Neyman-Pearson (NP) fundamental lemma is a result in testing hypothesis that says a test with size \(\alpha\) that has maximum power among all tests of size \(\alpha\) is a likelihood-ratio test. In other words, it is a method for testing a simple hypothesis vs another simple hypothesis that maximizes the power of the test.

Comparing the two principles

When using the Neyman-Pearson theorem for hypothesis testing, we utilize the test statistic, set up the critical region, then the test is done. The acceptance or rejection of the hypothesis is based on whether the test statistic lies in the critical region. On the other hand, the likelihood ratio principle forms a ratio of probabilities under the two hypotheses, and the decision on acceptance or rejection of the hypothesis is based on whether the likelihood ratio exceeds a certain threshold. From these definitions, it can be inferred that both the Neyman-Pearson theorem and the likelihood ratio principle are essentially methods that utilize a test statistic to make a decision about a hypothesis.

Demonstrating the Equivalence

The Neyman-Pearson theorem and likelihood ratio principle are equivalent for testing a simple hypothesis, if the data set consists of n independent and identically distributed random variables. In the Neyman-Pearson test, the test statistic is the likelihood ratio statistic, and we reject \(H_0\) if likelihood ratio statistic is greater than a certain threshold. Similarly, in the likelihood ratio test, we reject \(H_0\) if the calculated likelihood ratio is greater than the critical value calculated using a desired significance level. Hence, the two tests are equivalent for simple hypothesis test.