##
**On large deviations and choice of ancillary for \(p^*\) and \(r^*\).**
*(English)*
Zbl 0894.62026

Summary: 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, \(a^0\). 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, \(a^0\) 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 \(a^0\) 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, \(a^0\) 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, \(a^0\) 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 J. L. Jensen [Biometrika 79, No. 4, 693-703 (1992; Zbl 0764.62021)] and the results of this paper are discussed and clarified.

On a first impression, the last result is a little surprising because, in a repeated sampling framework, \(a^0\) 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, \(a^0\) 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 J. L. Jensen [Biometrika 79, No. 4, 693-703 (1992; Zbl 0764.62021)] and the results of this paper are discussed and clarified.