Please note that this module can be taken by some combined degree 3rd year students with the title Statistical Theory 1 and the module code 0590015

Aims

The aim of the module is to extend your knowledge of statistical theory and its applications to introduce you to some of the basic methods of statistical inference and estimation, which are needed in data analysis. The aim will be achieved through a mixture of theory and practice.

Learning objectives

At the end of the module you should be able to...

• Use and manipulate standard density and distribution functions of both discrete and continuous random variables.
• Be acquainted with the concepts of joint and conditional distributions and the distribution of functions of random variables.
• Understand the use of moment generating functions and to see this in use with, for example, Poisson and exponential random variables.
• Understand the use of characteristic functions.
• Comprehend the usefulness of the normal distribution in aiding the calculation of the distribution of certain functions of the data, for instance sample means and variances, and the t-ratio.
• Describe and understand different notions of convergence of random variables and apply various limit theorems.
• Be able to perform basic statistical inference on the parameters of standard distribution functions, including estimation of parameters using several methods, construction of confidence intervals.

Syllabus

Background.  Probability, random variables and probability distributions, moment generating function and transformations of random variable. 

Joint distributions.  Discrete and continuous random variables, conditional distributions, expectation and variance, conditional expectations and transformations of random variables. Bivariate normal distribution, chi-square, t- and F-distributions. 

 Limit theorems.  Modes of convergence (in distribution, in probability, almost surely), Central Limit Theorem, Weak Law of Large Numbers, Strong Law of Large Numbers. 

Estimation of parameters.  Point estimation: generalities. Unbiased and consistent estimators, MSE distance, efficient and sufficient estimators, Factorisation Theorem, the method of moments, the method of maximum likelihood, Cramer-Rao bound, efficiency of ML estimators under Gaussianity.

Recommended texts

Students should obtain copies of the following tables:

• H R Neave, Statistics Tables for mathematicians, engineers, economists and the behavioural and management sciences, Routledge (paperback) (SF 0.83 NEA and REF SF 0.83 NEA).
• ***P G Hoel Introduction to Mathematical Statistics (5th edn), Wiley (SF HOE).
• *** R V Hogg and A T Craig, Introduction to Mathematical Statistics (5th edn), Prentice-Hall (SF HOG).
• ***R V Hogg and E A Tanis, Probability and Statistical Inference (4th edn), Macmillan (S 9 HOG).
• **J A Rice, Mathematical Statistics and Data Analysis (2nd edn), Wadsworth and Brooks/Cole (SF RIC).
• *G Cassella and R L Berger, Statistical Inference (2nd edn), Duxbury Press. (SF 2 CAS)
• *W Mendenhall, R L Scheaffer, and D D Wackerly, Mathematical Statistics with applications (4th edn), Duxbury Press. (SF MEN)
• *R V Hogg, J W McKean and A T Craig, Introduction to Mathematical Statistics (6th edn), Prentice-Hall (SF HOG). 

Teaching

• Autumn Term
• 2 lectures per week
• 1 seminar per week

Assessment

One and a half hour closed examination in week 1 Spring Term.

Elective information

Introduction to distributions arising in inference from normally distributed samples (and some others). Central limit theorem. Sample mean and variance.

Please check prerequisites carefully before asking to take this module as an elective. In choosing this module as an elective it will be assumed that you are familiar with all the material taught in the first year courses Calculus and Core Algebra, or are willing to learn the material if necessary.

Prerequisites

• Probability Theory I (0590032 if taken in 09/10) or Introduction to Probability (0521003 if taken in 10/11) or Introduction to Probability and Statistics (MAT0004C if taken in 10/11)).  The Economics Department module: Introdulction to Statistical Theory can also serve as a pre-requisite provided students familiarise themselves with the Probability concepts taught in Introdcution to Probability and Statistics.