Statistical inference in possibly misspecified nonregular models

Natalia Bochkina, University of Edinburgh

Finite dimensional statistical models are called nonregular if it is possible to construct an estimator with the rate of convergence that is faster than the parametric root-n rate. I will give an overview of such models with the corresponding rates of convergence in the frequentist setting under the assumption that they are well-specified. In a Bayesian approach, I will consider a special case where the “true” value of the parameter for a well-specified model, or the parameter corresponding to the best approximating model from the considered parametric family for a misspecified model, occurs on the boundary of the parameter space. I will show that in this case the posterior distribution (a) asymptotically concentrates around the ``true’’ value of the parameter (or the best approximating value under a misspecified model), (b) has not only Gaussian components as in the case of regular models (the Bernstein–von Mises theorem) but also Gamma distribution components whose form depends on the behaviour of the prior distribution near the boundary, and (c) has a faster rate of convergence in the directions of the Gamma distribution components. One implication of this result is that for some models, there appears to be no lower bound on efficiency of estimating the unknown parameter if it is on the boundary of the parameter space. I will discuss how this result can be used for identifying misspecification in regular models. The results will be illustrated on a problem from emission tomography. This is joint work with Peter Green (University of Bristol).