Title: Semi-parametric Bernstein-von Mises theorem for linear models with one-sided error
Authors: Natalia Bochkina - University of Edinburgh (United Kingdom) [presenting]
Judith Rousseau - University of Oxford (United Kingdom)
Jean-Bernard Salomond - Universite Paris Dauphine (France)
Johan van der Molen Moris - University of Cambridge (United Kingdom)
Abstract: The problem of linear regression with one-sided errors whose density is unknown is considered from a Bayesian perspective. We state general sufficient conditions for the local concentration of the marginal posterior of the parameters in the linear regression model (Bernstein - von Mises type theorem), which have a faster $1/n$ contraction rate and a constrained multivariate exponential distribution with random constraint, under an adaptive estimation of the unknown density. Consistent estimation of the unknown density of errors at zero is important, as this value is the scale parameter of the limiting constrained exponential distribution. In particular, to ensure that the error density is asymptotically consistent pointwise in a neighbourhood of zero, instead of usual Dirichlet mixture weights, we consider a non-homogeneous Completely Random Measure mixture. We illustrate the performance of this approach on simulated data, and apply it to model the distribution of bids in procurement auctions.