Dominance of likelihood ratio tests under cone constraints (Q1208662)

Let \(Y\) be a \(k\)-dimensional random vector having a normal distribution \(N_ k(\theta,\Gamma)\) with \(\theta\) restricted to belong to a closed convex cone \(C\) of \(\mathbb R^ k\). Let \(L\) be a linear subspace and consider the problem of testing \(H_ 0: \theta\in C\cap L\) against \(H_ a-H_ 0\) where \(H_ a: \theta\in C\). Under suitable conditions the authors prove the dominance of the LRT (likelihood ratio test), and also provide conditions which guarantee the equivalence of the LRT and a simple test. The cases of known and unknown variances are considered and examples are discussed to illustrate the results.