## MAS223 Statistical Inference and Modelling

 Both semesters, 2021/22 20 Credits Lecturer: Dr Jonathan Jordan Timetable Reading List Aims Outcomes Assessment Full Syllabus

This unit develops methods for analysing data, and provides a foundation for further study of probability and statistics at Level 3. It introduces some standard distributions beyond those met in MAS113, and proceeds with study of continuous multivariate distributions, with particular emphasis on the multivariate normal distribution. Transformations of univariate and multivariate continuous distributions are studied, with the derivation of sampling distributions of important summary statistics as applications. The concepts of likelihood and maximum likelihood estimation are developed. Data analysis is studied within the framework of linear models. There will be substantial use of the software package R.

Prerequisites: MAS113 (Introduction to Probability and Statistics)
Corequisites: MAS211 (Advanced Calculus and Linear Algebra)

The following modules have this module as a prerequisite:

 MAS31001 Generalised Linear Models MAS352 Stochastic Processes and Finance MAS360 Practical and Applied Statistics MAS361 Medical Statistics MAS362 Financial Mathematics MAS364 Bayesian Statistics MAS369 Machine Learning MAS370 Sampling Theory and Design of Experiments MAS371 Applied Probability MAS372 Time Series MAS452 Stochastic Processes and Finance MAS462 Financial Mathematics MAS464 Bayesian Statistics MAS469 Machine Learning MAS61001 Generalised Linear Models MAS61002 Medical Statistics MAS61003 Sampling Theory and Design of Experiments MAS61007 Machine learning

## Outline syllabus

• Univariate distribution theory
• Multivariate distribution theory
• Likelihood
• Likelihood case studies
• Linear models

## Office hours

Tuesday, 15:30 to 16:30, online. Slots can be booked using Google Calendar.
If you want a face to face meeting email the lecturer to arrange a time.

## Aims

• extend students' familiarity with standard probability distributions;
• give practice in handling discrete and continuous distributions, especially continuous multivariate ones;
• instil an understanding of the rationale and techniques of likelihood exploration and maximisation;
• consider linear regression models in detail;
• extend the comparison of means from two to several groups through ANOVA models;
• give students experience in the use of R for fitting linear models;

## Learning outcomes

• handle a wide range of standard distributions, including the multivariate normal;
• calculate joint, marginal and conditional continuous distributions;
• manipulate multivariate means, variances and covariances;
• transform univariate and multivariate continuous random variables;
• derive, manipulate and interpret likelihood functions, and find maximum likelihood estimators;
• understand regression and ANOVA models as examples of linear models;
• estimate parameters in a linear model;
• make inferences about model parameters through appropriate model comparisons;
• develop a 'best-fitting' model in a systematic and pragmatic way;
• undertake model checking procedures through the use of residuals;
• use R to implement methods covered in the course;
• prepare a structured word processed report of the statistical analysis of an open-ended problem.

44 lectures, 8 tutorials, 3 practicals

## Assessment

80% exam in Spring

20% is based on coursework. This is made up of:
In semester 1: 5 homework assignments and one online test (10%).
In semester 2: a practical project (10%).

## Full syllabus

Univariate distribution theory
Revision of sample spaces, events and random variables; distribution functions, probability functions, probability density functions; moments; random variables without a mean (Cauchy as example); discrete standard distributions: hypergeometric, negative binomial; revision of normal; gamma and beta functions; gamma (χ2 as special case) and beta distributions; visualising distributions in R; transformations of univariate random variables including monotonic case and non-monotonic examples.

Multivariate distribution theory
Random vectors; multivariate p.d.f.s for continuous random vectors; p.d.f.s of marginal and conditional distributions; covariance and correlation; independence; conditional expectation and variance; transformations of multivariate p.d.f.s using Jacobians; applications of transformations including the t distribution and Box-Muller simulation of normal r.v.s; covariance matrices; linear transformations and their effect on covariance matrices; multivariate normal including matrix form of p.d.f.; linear transformations of the multivariate normal; conditional distributions of multivariate normal components.
Likelihood
Data and random samples; models and parameters; definition of likelihood and examples; introduction to maximum likelihood estimation; log likelihood; one parameter MLE; two parameter MLE using Hessian, including MLE for Normal with unknown mean and variance; interval estimation using likelihood; hypothesis tests using likelihood.
Likelihood case studies
Two or three applications of likelihood inference to case studies.
Linear models (Semester 2)
• Matrix representation of a linear model: linear regression, polynomial regression, multiple regression and ANOVA models as examples of linear models.
• Least squares estimation: least squares estimators in matrix notation; distributional properties of least squares estimators and the residual sum of squares.
• Hypothesis testing: the F-test for comparing nested linear models; t-tests.
• Prediction: confidence intervals and prediction intervals
• Model checking: diagnostics using standardized residuals; transformations; R2.
• Factor independent variables: ANCOVA and one-way and two-way ANOVA.
• Fitting and analysing linear models using R.

Type Author(s) Title Library Blackwells Amazon
B Draper and Smith Applied Regression Analysis 519.536 (D) Blackwells Amazon
B Faraway Linear Models with R 519.538 (F) Blackwells Amazon
B Freund, Miller and Miller John E. Freund’s Mathematical Statistics with Applications 519.5 (F) Blackwells Amazon
B Kleinbaum, Kupper, Muller and Nizam Applied Regression Analysis and Other Multivariable Methods 519.536 (A) Blackwells Amazon
B Mood, Graybill and Boes Introduction to the Theory of Statistics 519.5 (M) Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.

## Timetable (semester 1)

 Wed 09:00 - 09:50 lecture Hicks Lecture Theatre 7 Wed 11:00 - 11:50 lecture Hicks Lecture Theatre 7 Fri 14:00 - 14:50 tutorial (group A) (weeks 1,3,5,8,10,12) Richard Roberts Room A87 Fri 14:00 - 14:50 tutorial (group O) Blackboard Online Fri 15:00 - 15:50 tutorial (group B) (weeks 1,3,5,8,10,12) Richard Roberts Room A87 Fri 16:00 - 16:50 tutorial Hicks Lecture Theatre 10 Fri 16:00 - 16:50 tutorial (group C) (weeks 1,3,5,8,10,12) Hicks Lecture Theatre D