Strong matching of frequentist and Bayesian parametric inference

Abstract

We define a notion of strong matching of frequentist and Bayesian inference for a scalar parameter, and show that for the special case of a location model strong matching is obtained for any interest parameter linear in the location parameters. Strong matching is defined using one-sided interval estimates constructed by inverting test quantities. A brief survey of methods for choosing a prior, of principles relating to the Bayesian paradigm, and of confidence and related procedures leads to the development of a general location reparameterization. This is followed by a brief survey of recent likelihood asymptotics which provides a basis for examining strong matching to third order in general continuous statistical methods. It is then shown that a flat prior with respect to the general location parameterization gives third-order strong matching for linear parameters; and for nonlinear parameters the strong matching requires an adjustment to the flat prior which is based on the observed Fisher information. A computationally more accessible approach then uses full dimensional pivotal quantities to generate default priors for linear parameters; this leads to second-order matching. A concluding section describes a confidence, fiducial, or default Bayesian inversion relative to the location parameterization. This provides a method to adjust interval estimates by means of a personal prior taken relative to the flat prior in the location parameterization.