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Intermediate•Statistics
Hypothesis Testing
p-values, Z-tests, and T-tests.
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Research-Level Deep Dive & Equations
Statistical hypothesis testing is formally structured as a decision-theoretic problem. Let represent a random sample drawn from a probability space where the distribution is indexed by a parameter vector . The parameter space is partitioned into two disjoint subsets: the null parameter space and the alternative parameter space , such that and . We define the null hypothesis and the alternative hypothesis .
A non-randomized statistical decision rule (or test) is a mapping , where denotes the rejection of (acceptance of ) and denotes a failure to reject . The critical region is the set of outcomes for which the null hypothesis is rejected: .
The performance of any test is fully characterized by its power function .
- For , represents the Type I Error Rate (false positive probability), which is the probability of rejecting a true null hypothesis. The significance level (or size) of the test is defined as the supremum of this error rate: .
- For , represents the Power of the test, while is the Type II Error Rate (false negative probability), which is the probability of failing to reject a false null hypothesis.
The **Neyman-Pearson Lemma** provides the fundamental theoretical justification for constructing optimal tests. For simple hypotheses versus , the most powerful test of size rejects if the likelihood ratio falls below a threshold :
where is chosen such that .
For composite hypotheses, Uniformly Most Powerful (UMP) tests are generally unavailable unless the family of distributions possesses a Monotone Likelihood Ratio (MLR). In the general case, we construct tests using the Likelihood Ratio Test (LRT) statistic:
Wilks' Theorem provides the asymptotic distribution under regular conditions. As , the test statistic converges in distribution to a chi-squared distribution under :
where the degrees of freedom represents the number of constraints imposed by the null hypothesis.
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