  Likelihood ratio tests are widely used to test statistical hypotheses about
parametric families of probability distributions. If interest is restricted to
a subfamily of distributions, then it is natural to inquire if the restricted
LRT is superior to the unrestricted LRT. Marden's general LRT conjecture posits
that any restriction placed on the alternative hypothesis will increase power.
The only published counterexample to this conjecture is rather technical and
involves a restriction that maintains the dimension of the alternative. We
formulate the dimension-restricted LRT conjecture, which posits that any
restriction that replaces a parametric family with a subfamily of lower
dimension will increase power. Under standard regularity conditions, we then
demonstrate that the restricted LRT is asymptotically more powerful than the
unrestricted LRT for local alternatives. Remarkably, however, even the
dimension-restricted LRT conjecture fails in the case of finite samples. Our
counterexamples involve subfamilies of multinomial distributions. In
particular, our study of the Hardy-Weinberg subfamily of trinomial
distributions provides a simple and elegant demonstration that restrictions may
not increase power.
