# APTS module: Statistical Asymptotics

Module leader: A T A Wood

Please see the full Module Specifications document for background information relating to all of the APTS modules, including how to interpret the information below.

Aims: This module has the twin aims of introducing students to asymptotic theory and developing their practical skills in using asymptotic approximations.

Learning outcomes: After taking this module, students will have a basic understanding of the asymptotic properties of parametric likelihoods and posterior distributions, and the knowledge and skills to derive and implement first-order Laplace and saddlepoint density approximations in simple examples.

Prerequisites: Preparation for this module should establish:

• basic knowledge of likelihood methods, exponential families and Bayesian inference, to the level developed in a typical third-year undergraduate inference course;
• knowledge of limit theorems in the univariate IID case (laws of large numbers and CLT);
• familiarity with different modes of convergence (convergence in distribution, in probability, almost sure and Lp);
• familiarity with Taylor expansions in the multivariable case;
• familiarity with o(.), O(.), o_P(.) and O_P(.) notation.

Topics:

• Multivariate central limit theorem, (a gentle introduction to) the continuous mapping theorem, the delta method;
• Stochastic asymptotic expansion;
• Likelihood asymptotics (including asymptotic properties of MLEs);
• Asymptotic normality of posterior distributions (parametric case);
• Laplace's approximation (univariate and multivariate);
• Introduction to Edgeworth expansions and saddlepoint density approximations (via tilting);
• Saddlepoint approximations to tail probabilities.

Assessment: A mini-project which ideally has both a theoretical component (e.g., discussion of conditions for asymptotic normality in a particular set-up, or derivation of a suitable approximation in particular examples) and a computational component (e.g., numerical implementation of a Laplace or saddlepoint approximation).