Vector Exponential Models and Second Order Inference

SUMMARY For an exponential model with scalar parameter, Welch & Peers (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.