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, ao. 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, ao 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 ao 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, ao 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, ao 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.