2.4 Testing Hypotheses

2.4.1 Null hypothesis vs. alternative hypothesis

https://en.wikipedia.org/wiki/Test_statistic

2.4.2 The Neyman-Pearson Lemma

Consider a hypothesis test with null hypothesis \(H_0\) and alternative hypothesis \(H_1\). Let \(X\) be a random variable with probability density function (or probability mass function) \(f(x; \theta_0)\) under \(H_0\) and \(f(x; \theta_1)\) under \(H_1\). Let \(T\) be a test statistic, and let \(C\) be the critical region (rejection region) of the test. The Neyman-Pearson Lemma states that the most powerful test of size \(\alpha\) is given by the likelihood ratio test, which rejects \(H_0\) if

\[ \frac{f(x; \theta_1)}{f(x; \theta_0)} > k \]

where \(k\) is a constant chosen such that the probability of type I error is equal to \(\alpha\), i.e.,

\[ P(X \in C | \theta = \theta_0) = \alpha. \]

Equivalently, the most powerful test can also be expressed as rejecting \(H_0\) if

\[ \log \frac{f(x; \theta_1)}{f(x; \theta_0)} > \log k = c \]

where \(c\) is a constant.

If the likelihood ratio is such that \(\frac{f(x; \theta_1)}{f(x; \theta_0)} = k\) on a set of probability zero under \(H_0\) and \(H_1\), then the test is still most powerful. If the likelihood ratio is always constant, then no test is more powerful than any other.