On the Existence of Unbiased Hypothesis Tests: An Algebraic Approach

In hypothesis testing problems the property of strict unbiasedness describes whether a test is able to discriminate, in the sense of a difference in power, between any distribution in the null hypothesis space and any distribution in the alternative hypothesis space. In this work we examine conditions under which unbiased tests exist for discrete statistical models. It is shown that the existence of an unbiased test can be reduced to an algebraic criterion; an unbiased test exists if and only if there exists a polynomial that separates the null and alternative hypothesis sets. This places a strong, semialgebraic restriction on the classes of null hypotheses that have unbiased tests. The minimum degree of a separating polynomial coincides with the minimum sample size that is needed for an unbiased test to exist, termed the unbiasedness threshold. It is demonstrated that Gr\"obner basis techniques can be used to provide upper bounds for, and in many cases exactly find, the unbiasedness threshold. Existence questions for uniformly most powerful unbiased tests are also addressed, where it is shown that whether such a test exists can depend subtly on the specified level of the test and the sample size. Numerous examples, concerning tests in contingency tables, linear, log-linear, and mixture models are provided. All of the machinery developed in this work is constructive in the sense that when a test with a certain property is shown to exist it is possible to explicitly construct this test.