This paper considers likelihood ratio tests for testing hypotheses that a collection of parameters satisfy some order restriction. The first problem considered is to test a hypothesis specifying an order restriction on a collection of means of normal distributions. Equality of the means is the subhypothesis of the null hypothesis which yields the largest type I error probability (i.e., is least favorable). Furthermore, the distribution of $T = -\ln$ (likelihood ratio) is similar to that of a likelihood ratio statistic for testing the equality of a set of ordered normal means. The least favorable status of homogeneity is a consequence of a result that if $X$ is a point and $A$ a closed convex cone in a Hilbert space and if $Z \in A$, then the distance from $X + Z$ to $A$ is no larger than the distance from $X$ to $A$. The results of a Monte Carlo study of the power of the likelihood ratio statistic are discussed. The distribution of $T$ is also shown to serve as the asymptotic distribution for likelihood ratio statistics for testing trend when the sampled distributions belong to an exponential family. An application of this result is given for underlying Poisson distributions.
"Likelihood Ratio Tests for Order Restrictions in Exponential Families." Ann. Statist. 6 (3) 485 - 505, May, 1978. https://doi.org/10.1214/aos/1176344195