The differential prior gives second order inference for scalar and vector parameters

Jeffreys (1946) proposed a root-information prior for the likelihood analysis of a statistical model. Then for an exponential model with scalar parameter Welch & Peers (1963) showed that this leads to second-order inference, or more specifically that posterior intervals have second order confidence. But for an exponential model with vector-parameter Jeffreys (1961) found that the root information prior gave unsatisfactory results, even for Normal location-scale and Normal regressionscale models. And for these non-compliant models he proposed a modified prior that evolved (Hartigan, 1964; Fraser, 1968) to the right-invariant prior widely preferred for many regular models. In this paper likelihood asymptotic methods are used to develop a locally based prior that produces full second-order inference for linear (Fraser et al, 2010a) parameters; such parameters admit second-order posterior inference without individual targeting and thus avoid the marginalization paradoxes (Dawid et al, 1973; Fraser & Reid, 2002; Fraser, 2011a; Fraser et al, 2010c). The new prior has the form of a modified Jeffreys that accommodates ∗Address for correspondence: Department of Statistical Sciences, University of Toronto, 100 St. George Street, 6th floor, Toronto, ON, Canada M5S 3G3