Likelihood and Asymptotics

After Fisher's contributions starting from the twenties, likelihood is the central concept of parametric statistical inference, both in the Bayesian and in the classical approaches. Likelihood methods have become since then increasingly popular. In frequency-based inference the main reason for their widespread use is that sampling distributions of statistics such as the maximum likelihood estimator and likelihood ratio tests have simple and well understood (first-order) asymptotic approximations in many relevant models.

Much of the work on likelihood inference over the past two decades aimed at refining first-order asymptotic approximations, moving into higher-order asymptotics. This work has proved beneficial not only for the perhaps limited goal of having reliable approximations in small-sample inference. Many crucial theoretical and practical issues in frequency-based inference have been successfully tackled. Among these: the role of conditioning on ancillary statistics, elimination of nuisance parameters, choosing among first-order equivalent procedures, higher-order matching between frequentist and Bayesian procedures, prediction, geometrical interpretation of likelihood inference, semiparametric and nonparametric inference, robust procedures. These developments together form a unitary vision in modern statistics, referred to as neo-Fisherian.

Until recently, there was a crucial gap between most of this theory and potential applications. A considerable effort has been devoted to the development of software for widespread use. Fairly complex models can now be routinely analised. Among these: generalised linear models, nonlinear normal regression, nonnormal linear regression. Thus, the time is ripe for applied workers to become increasingly aware of the theoretical tools now available, as well as for the theoreticians to face the challenges posed by the complex models used in practice.

The purpose of this site is to contribute to this ongoing process, making new ideas and tools readily accessible both on the side of theory and on the side of applications.