Vector exponential models and second order inference

For an exponential model with scalar parameter,welchp:1963 examined the role of bayesian analysis in statistical inference,more specifically the use of the jeffreys:1946 prior. they determined that bayesian intervals and thus in effect bayesian quantiles had second order confidence accuracy. we use a taylor series expansion of the log-model to develop a second order version of the vector exponential model; this is developed as a contribution to theory in statistics at a time when algorithms are prominent,and it provides a basis for generalizing the welch-peers approach to the vector parameter context.