# MATH651: Likelihood Inference

**Credits: 10**

**Tutor:**** Emma Eastoe**

**The module presents the key tools for statistical inference, stressing the fundamental role of the likelihood function. It will cover:**

Outline:

Outline:

- Definition of the likelihood function for single and multi-parameter models, and how it is used to calculate point estimates (maximum likelihood estimates).
- Asymptotic distribution of the maximum likelihood estimator, and the profile deviance, and how these are used to quantify uncertainty in estimates.
- Inter-relationships between parameters and the definition and use of orthogonality.
- Generalised Likelihood Ratio Statistics and their use for hypothesis tests.
- Calculating likelihood functions for non-iid data.
- Simple use of computational methods to calculate maximum likelihood estimates and confidence intervals and to perform hypothesis tests.
- Model criticism through qq-plots for residuals.

**Objectives: **Statistical theory is the theory of extracting information about the unknown parameters of an underlying probability model from observed data. This underpins all practical statistical applications, such as those considered in later MRes modules.

This course considers the idea of statistical models, and how the likelihood function, the probability of the observed data viewed as a function of unknown parameters, can be used to make inference about those parameters. This inference includes both estimates of the values of these parameters, and measures of the uncertainty surrounding these estimates. We consider single and multi-parameter models, and models which do not assume the data are independent and identically distributed. We also cover basic computational aspects of likelihood inference that are required in many practical applications.

**Learning Outcomes: **On successful completion of this module students will:

- understand how to construct statistical models for simple applications;
- appreciate how information about the unknown parameters is obtained and summarised via the likelihood function;
- be able to calculate the likelihood function for independent and identically distributed data;
- be able to calculate the likelihood function for some statistical models which do not assume independent identically distributed data;
- be able to evaluate point estimates and make statements about the variability of these estimates;
- understand about the inter-relationships between parameters, and the concept of orthogonality;
- be able to perform hypothesis tests using the generalised likelihood ratio statistic;
- use computational methods to calculate maximum likelihood estimates;
- use computational methods to construct confidence intervals and perform hypothesis tests;
- be able to look at residuals to judge how appropriate a model is;
- be able to find maximum likelihood estimators using the statistical package R.

**Core texts:**

- A Azzalini (1996) Statistical Inference: Based on the Likelihood. Chapman and Hall.
- Y Pawitan (2001) In All Likelihood: Statistical Modeling and Inference Using Likelihood. OUP.

**Assessment: **Assessment will be through a combination of summer exam (80%) and end of module test (20%).

**Contact hours:** There will be a mixture of lectures, tutorials and computer workshops totalling approximately 25 hours contact. In addition, private study will make up the majority of the learning hours.