Likelihood ratio, score, and Wald tests in a constrained parameter space

Likelihood ratio, score, and Wald tests statistics are asymptotically equivalent. This statement is widely known to hold true under standard conditions. But what if the parameter space is constrained and the null hypothesis lies on the boundary of the parameter space, such as, for example, in variance component testing? Quite a bit is known in such situations too, but knowledge is scattered across the literature and considerably less well known among practitioners. Motivated from simple but generic examples, we show there is quite a market for asymptotic one-sided hypothesis tests, in the scalar as well as in the vector case. Reassuringly, the three standard tests can be used here as well and are asymptotically equivalent, but a somewhat more elaborate version of the score and Wald test statistics is needed. Null distributions take the form of mixtures of chi(2) distributions. Statistical and numerical considerations lead us to formulate pragmatic guidelines as to when to prefer which of the three tests.