To introduce some of the concepts and principles which provide theoretical underpinning for the various statistical methods, and, thus, to consolidate the theory behind the other second year and third year statistics options.

After completing the module students should have a good understanding of the classical approach to, and especially the likelihood methods for, statistical inference.  The students should also gain an appreciation of the blossoming area of Bayesian approach to inference.

Convergence of random variables: convergence in probability and distribution.   Chebyshev''s inequality.  Central Limit Theorem.

Order Statistics.  Distribution of order statistics.

Properties of estimators

The sample, parametric models, definition of a statistic;

Estimators: unbiasedness, consistency, sufficiency, the factorisation criterion, mean squared error.

Minimum variance unbiased estimators, Cramer-Rao lower bound without proof, attainment by the exponential family.

Maximum likelihood estimation

The likelihood function for one and two parameters.

Finding MLE''s, the Newton-Raphson methods.

General properties: uniqueness, sufficiency, turning points are maxima for exponential family.

Asymptotic properties without proof: consistency,unbiasedness, efficiency, normality.

Hypothesis testing and confidence intervals

Hypotheses, significance, power.

Neyman-Pearson lemma.

Uniformly most powerful tests, two-sided tests.

Confidence Intervals

Calculation of Confidence Intervals - The Pivotal Quantity Method.

Asymptotic Confidence Intervals.

Relationship between tests and Confidence intervals

Bayesian Inference

Bayes'' theorem for one or more parameters.

Comparison of Normal means.

Prior distribution and their specification.  Non-informative and Improper Priors.  Subjectively assessed priors.  Conjugate Priors.

MCMC Method.