Nonparametric Bayesian Estimation of Probability Density Function

Estimation of a probability density function based on parametric statistical mod- els can be highly imprecise and misleading when data are sparse and irregular. In these cases a semiparametric or nonparametric model is preferable and can better capture the data structure. We propose a Bayesian hierarchical model for the estimation of the probability density function. We use a Polya tree (Lavine 1992, 1994) as a nonparametric prior for a random probability measure. The binary partition of the Polya tree is obtained through the quantiles of a Generalized Gamma density function whose parameters are themselves Gamma-distributed random variables. The estimation technique is based on a MCMC sampler using Metropolis-Hastings within Gibbs sampling. We then apply the presented model to the interevent times between strong earthquakes which have occurred between 1600 and 1992 in Italy