On large deviations and choice of ancillary for p* and r*

Abstract

The large deviation properties of p^*, the approximation to the conditional density of the maximum likelihood estimator, and r^*, the modified directed likelihood, are studied. Attention is restricted to curved exponential models. Various specifications of an approximate ancillary, which are required in the construction of p^*and r^*, are considered, including: a modified directed likelihood ancillary, a^*, and an unmodified directed likelihood ancillary, a^o. It is shown that if a^* is used then p^* and r^* achieve saddlepoint accuracy on both normal and large deviation regions; if, on the other hand, a^o is used in the construction of p^*, then saddlepoint accuracy is not achieved, though the relative error still stays bounded on large deviation regions. It is also shown that if a^o rather than a^* is held fixed in the sample space differentiations needed to calculate r^*, then saddlepoint accuracy is still attained in both normal and large deviation regions. On a first impression, the last result is a little surprising because, in a repeated sampling framework, a^o is only ancillary to order O(n^-1/2). However, this finding is also of direct practical interest because, from the point of view of calculation, a^o is often substantially easier to work with than a^*. An important aspect of our approach is the development of guidelines, referred to as Laplace-spa calculus, for the construction of invariant saddlepoint-style approximations to marginal and conditional densities. Finally, connections between recent work by Jensen and the results of this paper are discussed and clarified.