Nonparametric Bayesian estimation under shape constrained

Research summary:

This project aims derive estimators for shape constrained problems with Bayesian avor.The problem of estimating a decreasing density is encountered in various statistical applica-tions. However, the well-known Maximum Likelihood Estimator is inconsistent at zero. Variousconsistent alternative estimators exist, but these need appropriate choices of tuning parame-ters. The mixture representation of decreasing densities gives a natural way to estimate sucha density in a Bayesian framework. The main diculty is how to choose the prior that ensureconsistency of the posterior, especially also at zero. A natural prior on the class of all distri-bution functions is the distribution of the Dirichlet Process with parameters base measure andconcentrate rate. The properties of the base measure play a crucial role to drive the consistentresult. To avoid spiking problem, we expect the base measure induces enough penalization toachieve consistency of the posterior distribution. The Bayesian estimator de nitely takes moree ort to compute estimator. However, ecient iterative algorithms using MCMC methods canbe derived and also provide a big advantage to obtain the credible sets.

Lixue Pang Joint work with Geurt Jongbloed and Frank van der Meulen