Density Estimation under Qualitative Assumptions in Higher Dimensions

We study a method for estimating a density f in Rd under assumptions which are of qualitative nature. The resulting density estimator can be considered as a generalization of the Grenander estimator for monotone densities. The assumptions on f are given in terms of shape restrictions of the density contour clusters [Gamma]([lambda]) = (x : f(x) >= [lambda]). We assume that for all [lambda] >= 0 the sets [Gamma]([lambda]) lie in a given class of measurable subsets of Rd. By choosing appropriately it is possible to model for example monotonicity, symmetry, or multimodality. The main mathematical tool for proving consistency and rates of convergence of the density estimator is empirical process theory. It turns out that the rates depend on the richness of measured by metric entropy.

Rates of Convergence for a Bayesian Level Set Estimation

We are interested in estimating level sets using a Bayesian non-parametric approach, from an independent and identically distributed sample drawn from an unknown distribution. Under fairly general conditions on the prior, we provide an upper bound on the rate of convergence of the Bayesian level set estimate, via the rate at which the posterior distribution concentrates around the true level set. We then consider, as an application, the log-spline prior in the two-dimensional unit cube. Assuming that the true distribution belongs to a class of Hölder, we provide an upper bound on the rate of convergence of the Bayesian level set estimates. We compare our results with existing rates of convergence in the frequentist non-parametric literature: the Bayesian level set estimator proves to be competitive and is also easy to compute, which is of no small importance. A simulation study is given as an illustration. Copyright 2005 Board of the Foundation of the Scandinavian Journal of Statistics..

Multi-step forecasts from threshold ARMA models using asymmetric loss functions

Nonlinear prediction, General loss functions, SETARMA, Linex,