Uniformly Hyper-Efficient Bayes Inference in a Class of Nonregular Problems

Daniel J. Nordman Daniel J. Nordman is Assistant Professor, Department of Statistics, Iowa State University, Ames, IA 50011 . Stephen B. Vardeman is University Professor, Department of Statistics and Department of Industrial and Manufacturing Systems Engineering (IMSE), Iowa State University, Ames, IA 50011 . Melissa A. Bingham is Assistant Professor, Department of Mathematics, University of Wisconsin–La Crosse, La Crosse, WI 54601 . The authors thank Professor John Stufken, an Associate Editor, and a referee for helpful comments that improved an earlier draft of this article. This research was supported by NSF grant DMS 0502347 EMSW21-RTG awarded to the Statistics Department, Iowa State University.

Stephen B. Vardeman Daniel J. Nordman is Assistant Professor, Department of Statistics, Iowa State University, Ames, IA 50011 . Stephen B. Vardeman is University Professor, Department of Statistics and Department of Industrial and Manufacturing Systems Engineering (IMSE), Iowa State University, Ames, IA 50011 . Melissa A. Bingham is Assistant Professor, Department of Mathematics, University of Wisconsin–La Crosse, La Crosse, WI 54601 . The authors thank Professor John Stufken, an Associate Editor, and a referee for helpful comments that improved an earlier draft of this article. This research was supported by NSF grant DMS 0502347 EMSW21-RTG awarded to the Statistics Department, Iowa State University.

Melissa A. Bingham Daniel J. Nordman is Assistant Professor, Department of Statistics, Iowa State University, Ames, IA 50011 . Stephen B. Vardeman is University Professor, Department of Statistics and Department of Industrial and Manufacturing Systems Engineering (IMSE), Iowa State University, Ames, IA 50011 . Melissa A. Bingham is Assistant Professor, Department of Mathematics, University of Wisconsin–La Crosse, La Crosse, WI 54601 . The authors thank Professor John Stufken, an Associate Editor, and a referee for helpful comments that improved an earlier draft of this article. This research was supported by NSF grant DMS 0502347 EMSW21-RTG awarded to the Statistics Department, Iowa State University.

Abstract

We present a tractable class of nonregular continuous statistical models where 1) likelihoods have multiple singularities and ordinary maximum likelihood is intrinsically unavailable, but 2) Bayes procedures achieve convergence rates better than n⁻¹ across the whole parameter space. In fact, for every p>1, there is a member of the class for which the posterior distribution is consistent at rate n^−p uniformly in the parameter.

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Uniformly Hyper-Efficient Bayes Inference in a Class of Nonregular Problems

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