Bayesian Semiparametic estimation of densities with unknown support
Van Der Molen Moris, Johan
We address the nonregular semiparametric problem of estimating a boundary point of the support of an unknown density, under local asymptotic exponentiality. The aim is to find the limiting marginal posterior distribution of the nonregular parameter and the rate of concentration for the density. Here we investigate two approaches. The first consists in extending the results found for parametric models to the case where the dimension of the regular nuisance parameter grows to infinity along with the number of observations. We used a Log-Spline prior to obtain the local concentration result for the marginal posterior of the lower support point; a Bernstein - von Mises type theorem with exponential limiting distribution. We also obtained contraction for the density at minimax rate up to a log factor. In the second approach, 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 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. This is important since the rate parameter of the limiting Exponential distribution is equal to the value of the density at the lower support point. 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 implement this model using two different representations of the prior process; illustrate performance of this approach on simulated data, and apply it to model distribution of bids in procurement auctions.