Moments based inference in small samples

Abstract

In this work we propose a nonparametric estimator for parameters which are embodied in given moment conditions. Here we are interested in small samples problems. We analyze conditions under which it is possible to represent the distribution of data points conditional on their relative frequencies as a multinomial distribution. We derive the posterior distribution for the relative frequencies of the observed data points via Bayes’ Law and specifying a Dirichlet objective prior, the latter is obtained matching the prior modes for unknown relative frequencies with the optimal weights computed under a special form of the Empirical Likelihood estimator for the parameters of interest. The prior specification proposed has an interesting interpretation in terms of information-theoretic arguments. The estimators we construct in this paper share the general idea with the Bayesian Bootstrap by Rubin (1981), however its derivation starts from a different point o view, and also the prior specification over the distribution function of the data is objectively derived via Empirical Likelihood methods. We propose a Monte Carlo algorithm to derive a distribution for the parameters of interest based on the derived posterior for the relative frequencies of data points. A simulated toy example shows that in small sample the estimation proposed is more accurate than alternative non-bayesian nonparametric methods. Future works will include an asymptotic analysis of the proposed estimator, as well as more complex simulated validations for small samples.