Module Directory

This module introduces distribution theory, estimation and Maximum Likelihood estimators, hypothesis testing ending by exploring basic linear regression and multiple linear regression implemented in R.

This module uses the R software environment for statistical computing and graphics.

The aims of this module are:

• To cover part of the CS1 IFOA syllabus.

By the end of the module, students will be expected to be able to:

1. Define and be familiar with the discrete distributions: binomial, Poisson and uniform and be familiar with the continuous distributions: normal, exponential, chi-square, t, F and uniform.


2. Use the one-to-one correspondence between an mgf and a pdf for sums of RVs.


3. Handle bivariate distributions, understanding the relations between joint, marginal, conditional distributions and independence.


4. Understand the uses of the central limit theorem.


5. Determine maximum likelihood and least squares estimates of unknown parameters. Be able to define the terms: bias and mean squared error. Determine efficiency w.r.t. the Cramer-Rao lower bound for unbiased estimators.


6. Determine confidence intervals for means, variances and differences between means.


7. Concepts of random sampling, statistical inference and sampling distribution, Hypothesis tests. Null and alternative hypotheses, type I and type II errors, test statistic, critical region, level of significance, probability-value and power of a test. Use tables of the t-, F-, and chi-squared distributions.


8. Use R to implement the methods discussed in (1)-(7), [R Core Team (2017), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, https://www.R-project.org/] for the data analysis examples of the module.

The module follows the Graduate (Level 6) standards in Statistics of the Royal Statistical Society.

Indiciative Syllabus:

Distribution theory

1. Sums of IID random variables, strong and weak law of large numbers, central limit theorem.


2. Concept of Monte Carlo simulation.


3. Joint, marginal and conditional distributions. Independence. Covariance and correlation.


4. Convolution formula and change of variable formula.


5. Moment generating functions to find moments of the PDF and distributions of sums of random variables.

Estimation

1. Sampling distributions.


2. Bias in estimators and mean squared error, efficiency and the Cramer-Rao lower bound for unbiased estimators.


3. Maximum likelihood estimation and finding estimators analytically.


4. The mean and variance of a sample mean.


5. The distribution of the t-statistic for random samples from a normal distribution. - The F distribution for the ratio of two sample variances from independent samples taken from normal distributions.


6. Chi Square distributions for the sum of squared standard normal variates

Hypothesis testing and Confidence intervals

1. Confidence intervals for means, variances and differences between means.


2. Hypothesis tests concerning means and variances.


3. Null and alternative hypotheses, type I and type II errors, test statistic, critical region, level of significance, probability-value and power of a test.


4. Use R to implement methods discussed above.

Linear models

1. Linear relationships between variables using regression analysis.


2. The correlation coefficient for bivariate data and the coefficient of determination. Response and explanatory variables and the least squares estimates of the slope and intercept parameters in a simple linear regression model.


3. Multiple linear regression with IID normal errors, implemented in R.


4. Use R to implement methods discussed above.

Teaching in the School will be delivered using a range of face-to-face lectures, classes, and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.