Accurate Directional Inference for Vector Parameters in Linear Exponential Families

We consider inference on a vector-valued parameter of interest in a linear exponential family, in the presence of a finite-dimensional nuisance parameter. Based on higher order asymptotic theory for likelihood, we propose a directional test whose p-value is computed using one-dimensional integration. For discrete responses this extends the development of Davison et al. (2006), and some of our examples concern testing in contingency tables. For continuous responses the work extends the directional test of Cheah et al. (1994). Examples and simulations illustrate the high accuracy of the method, which we compare with the usual likelihood ratio test and with an adjusted version due to Skovgaard (2001). In high-dimensional settings, such as covariance selection, the approach works essentially perfectly, whereas its competitors can fail catastrophically. ∗Anthony Davison is Professor of Statistics, EPFL-FSB-MATHAA-STAT, Station 8, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland, [email protected]. Nancy Reid and Don Fraser are Professors of Statistics, Department of Statistics, University of Toronto, Toronto, Canada M5S 3G3, [email protected], [email protected]. Nicola Sartori is Associate Professor of Statistics, Dipartimento di Scienze Statistiche, Università degli Studi di Padova, Via Cesare Battisti 241, 35121 Padova, Italy, [email protected]. This research was partially supported by the Swiss National Science Foundation, the Canadian Natural Sciences and Engineering Research Council, the Senior Scholars funding from York University, Canada, and the Cariparo Foundation Excellence Grant 2011/2012. We thank the reviewers for their cordial and constructive comments.