Abstract
In this paper, we consider the well known problem of estimating a density function under qualitative assumptions. More precisely, we estimate monotone non-increasing densities in a Bayesian setting and derive concentration rate for the posterior distribution for a Dirichlet process and finite mixture prior. We prove that the posterior distribution based on both priors concentrates at the rate $(n/\log(n))^{-1/3}$, which is the minimax rate of estimation up to a $\log(n)$ factor. We also study the behaviour of the posterior for the point-wise loss at any fixed point of the support of the density and for the sup-norm. We prove that the posterior distribution is consistent for both loss functions.
Highlights
The non-parametric problem of estimating monotone curves, and monotone densities in particular, has been well studied in the literature both from theoretical and applied perspectives
Monotone densities appear in a wide variety of applications such as survival analysis, where it is natural to assume that the uncensored survival time has a monotone non-increasing density
Concentration rates of the posterior distributions have been well studied in the literature and some general results link the rate to the prior distribution (see Ghosal et al (2000))
Summary
The non-parametric problem of estimating monotone curves, and monotone densities in particular, has been well studied in the literature both from theoretical and applied perspectives. We use a modified version of their results and obtain for the two families of prior a concentration rate of order (n/ log(n))−1/3 which is the minimax estimation rate up to a log(n) factor under the L1 or Hellinger distance We extend these results to densities with support on R+ and prove that under some conditions on the tail of the distribution, the posterior still concentrates at an almost optimal rate. We study the behaviour of the posterior distribution for the sup-norm when the density has a compact support This problem has been addressed recently in the frequentist literature by Durot et al (2012). We derive concentration rates for the posterior of the density taken at a fixed point and for the sup-norm on subsets of [0, L] for L < ∞.
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