# APTS Prerequisites

### Context

APTS aims to be accessible to most starting PhD students in statistics. What follows is a list of prerequisites for APTS itself. Students comfortable with the items in the list below should be able to engage actively with the preliminary material provided for the individual modules and to benefit fully from attending those modules.

Individual APTS modules also have preliminary material which is supplied in advance of the APTS weeks themselves and may take students with non-standard backgrounds a little longer, but it should be approachable on the basis of only these prerequisites.

The intention of APTS is that most students will attend all of the modules in a given year and the later modules may build to some extent on material from the earlier modules. Students attending only a subset of the modules should bear this in mind and be prepared to do a little additional preparation if necessary

These prerequisites are not intended to be onerous and we believe that the vast majority of first year PhD students in statistics should be adequately equipped to benefit fully from APTS. Students who are concerned that they might not be adequately prepared to take the APTS course during the first year of their studies should discuss this with their PhD supervisor (or their local APTS Academic Contact). For some students with unusual backgrounds it may be more appropriate to pursue the APTS course in their second year of study.

### Prerequisites

1. Probability:
• basic probability calculations
• random variables
• probability mass functions and probability density functions
• expectation and variance
• independence
• conditional probability and conditional expectation
• Bayes's theorem
• the Central Limit Theorem for sample means of IID random variables of finite variance
2. Elementary Statistics
• simple statistical models including the IID Normal model
• parameters and estimators
• the maximum likelihood estimate (MLE)
• confidence intervals of the form MLE ± 2se
• hypothesis tests and rejection regions
• the Type 1 error level
3. Linear Algebra
• basic matrix operations
• vector spaces and bases
• linear systems, row and column spaces, rank
• orthogonality
• eigenvalues and vectors
4. Statistical Modelling and Regression
• linear models
• least squares and ML
• BLUE (Best Linear Unbiased Estimators) and the Gauss-Markov Theorem
• ANOVA (Analysis of Variance)
• basic residual diagnostics
5. Computer programming
• some experience of coding with the R programming language

#### Resources

Students who are unfamiliar with any of these topics can find details in the following references:

• J.A. Rice, 1999, Mathematical Statistics and Data Analysis, 2nd edn, Duxbury Press
• Morris H, DeGroot, and Mark J Schervish, 2010, Probability & Statistics, Addison Wesley, 4th edn. Prof. Schervish maintains a list of errata.
• G.A. Young and R.L. Smith, 2005, Essential of Statistical Inference, Cambridge University Press.
• A.C. Davison, 2003, Statistical Models, Cambridge University Press. This book also contains a wealth of applications. Prof. Davison maintains a list of errata.
• G. Casella and R.L. Berger, 2002, Statistical Inference, Thomson.
• G.R. Grimmett and D.R. Stirzaker, 2001, Probability and Random Processes, Oxford University Press.

Students who need to gain some experience of coding should work through the online course R Programming for APTS Students.