The likelihood function plays a central role in the definition of many inferential procedures which, under some regularity conditions and mainly asymptotically, as the sample size increases, possess desirable optimality properties for a wide variety of statistical models. Nevertheless, in many applications in which high dimensional models with complex dependence structures are needed, the applicability of the traditional likelihood-based inferential procedures, such as, maximum likelihood estimation, is challenged and often encounters non-neglectable difficulties. This is frequently due to the fact that an analytic closed form, or a numerical approximation, of the likelihood function is not available since it would require the solution of multidimensional integrals. In many of these cases, the problem can be overtaken by resorting to some Monte Carlo simulation strategies which allow the likelihood function, or the maximum likelihood estimate (MLE), to be computationally evaluated. In particular, the (marginal) likelihood surface can sometimes be approximated by resorting to some straight Monte Carlo or Markov chain Monte Carlo (MCMC) methods, by replacing the analytic integration required in the computation of the (marginal) likelihood with a conditional simulation of the unobserved components with respect to the observed data. Analogously, the maximum likelihood estimate can be
numerically found by adopting some Monte Carlo versions of the EM algorithm (MCEM) in which the analytic evaluation required in the E-step of the EM algorithm is replaced by a Monte Carlo evaluation.
The duty of the Verona Research Unit is the development of Monte Carlo likelihood-based inferential procedures for high dimensional models involving latent components, or accounting for measurement errors in the variables, of great interest in today applications in economics, finance, environmetrics, epidemiology, medicine and biology. In particular, the Research Unit will focus on:
1) Monte Carlo likelihood for multivariate ultra-high-frequency models in finance;
2) Monte Carlo likelihood for multivariate non-Gaussian geostatistical models;
3) likelihood methods in measurement error problems.