On the Resolution of Statistical Hypotheses

Abstract Let ω0 be the space of a parameter θ. Let ω i be a subset of ω i–1, i = 1, 2, ···, k, We test θ∈ω k against θ∈ω0 – ω k by testing iteratively the following hypotheses: θ∈ωi against θ∈ω i–1 – ω i , i = 1, 2, ···, k. The hypothesis θ∈ω k is accepted if and only if all of the intermediate hypotheses are accepted. If the test statistic for each intermediate hypothesis is based on the corresponding likelihood ratio λ i , we demonstrate why, under fairly general conditions, these test statistics are mutually stochastically independent. This argument is based on an independence theorem which deals with complete sufficient statistics. A number of illustrative examples are given; these include the equality of means and variances, the analysis of variance, the independence of p variates, and a regression problem.