- University of Edinburgh and The Alan Turing Institute
Wednesday, September 25, 2019
Polo Santa Marta, Via Cantarane 24, Sala Vaona
Bernstein-von Mises theorem in semi-parametric non-regular mixture model
We consider the problem of density estimation where the density has unknown lower support point, under local asymptotic exponentiality. To obtain the local concentration result for the marginal posterior of the lower support (Bernstein - von Mises type theorem), we constructed an adaptive mixture prior for a decreasing density with the following properties: a) posterior distribution of the density with known lower support point concentrates at the minimax rate, up to log factor, b) the density is estimated consistently, uniformly in a neighbourhood of the lower support point, c) marginal posterior distribution of the lower support point of the density has shifted exponential distribution in the limit. In particular, to ensure that the density is asymptotically consistent pointwise in a neighbourhood of the lower support point, instead of a usual Dirichlet mixture weights, we consider a non-homogeneous Completely Random Measure mixture. The general conditions for the BvM type result we have are different from those by Knapik and Kleijn (2013); the latter do not hold for a hierarchical mixture prior we consider. We illustrate performance of this approach on simulated data, and apply it to model distribution of bids in procurement auctions.
(Joint work with Judith Rousseau, J.-B. Salomond, Johan van der Molen Moris).