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.

(LO1) To acquire a good understanding of the classical approach to, and especially the likelihood methods for, statistical inference.

(LO2) To acquire an understanding of the blossoming area of Bayesian approach to inference.

(S1) Problem solving skills

(S2) Numeracy

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 Conf idence Intervals - The Pivotal Quantity Method. 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.