Semiparametric Bernstein-von Mises theorem for a nonregular model under a mixture prior

Natalia Bochkina (University of Edinburgh)

Friday 8th November 15:00-16:00 Maths 311B

Abstract

We consider the problem of density estimation where the density has unknown lower support point, under local asymptotic exponentiality, from a Bayesian perspective. We state general sufficient conditions for the local concentration of the marginal posterior of the lower support (Bernstein - von Mises type theorem) which has a faster 1/n rate and exponential distribution with a random shift, under an adaptive estimation of the unknown density.

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 a 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. Consistent estimation of the unknown density at the lower support point is important, as it is the scale parameter of the limiting shifted exponential distribution.

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 don't 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.

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