Third order efficient tests in exponential lattice models

In full rank multivariate exponential families of lattice distributions, one sided testing problems are considered. For these testing problems the uniformly most powerful unbiased tests are not uniformly efficient of order o(n-1) in the class 2* of all tests which are asymptotically similar of order o(n-1) ([2]. J. Multivariate Anal. 13 67-108). In this note, better asymptotically similar tests are constructed which--for one specified value of the nuisance parameter--have maximal power in the class 2*. The power of tests is computed up to terms of order o(n-1), at contiguous alternatives, and for the specified value of the nuisance parameter.