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 examine in detail several parametric distributions (Binomial, Poisson, Uniform, Normal, Exponential, Gamma, Chi-square, t, F) and explore their applications for modelling various real data settings.


• To explore the concept of multivariate distribution (e.g., copulas) in modelling the association between multiple variables, and understand the relations between joint, marginal distributions and independence.


• To learn the change of variables formula and know how to derive the distributions of functions of random variables.


• To understand the concepts of moment estimators and maximum likelihood estimators of unknown parameters. Be able to define the bias and mean squared error, and determine the efficiency of unbiased estimators using the Cramer-Rao theorem.


• To implement the linear regression model in studying the relationship between various variables, and determine the least squares estimates of the parameters. Conduct a diagnostic study of the model and its assumptions.


• To understand the concepts of hypothesis test, null and alternative hypotheses, type I and type II errors, test statistic, critical region, level of significance, probability-value and power of a test.


• To conduct Z-test, one-sample t-test, two-samples t-test, and F-test for evaluating hypotheses tests in real data situations.


• To understand the concept of interval estimation, and the use of central limit theorem in deriving confidence intervals.

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

1. Study the properties of discrete distributions (Bernoulli, Binomial, Poisson) and continuous distributions (Uniform, Normal, Exponential, Gamma, Chi-square, t, F), and be familiar with their real applications.


2. Handle bivariate distributions, and understand the relations between joint, marginal, conditional distributions and independence.


3. Employ the change of variables formula in deriving the distributions of functions of random variables.


4. Determine the moments estimators and maximum likelihood estimators of unknown parameters. Be able to define the bias and mean squared error of estimators, and determine the efficiency w.r.t. the Cramer-Rao lower bound for unbiased estimators.


5. Implement the linear regression model in studying the relationship between various variables, and compute the least squares estimates of the parameters. Conduct a diagnostic study of the model and its assumptions.


6. Understand the concepts of hypothesis test, null and alternative hypotheses, test statistic, critical region, type I and type II errors, level of significance, probability-value and power of a test.


7. Conduct Z-test, one-sample t-test, two-samples t-test, and F-test for evaluating hypotheses tests in real data situations.


8. Derive interval estimators of unknown parameters, and understand the relation between hypothesis tests and confidence intervals.


9. Use R to implement the methods discussed in (1)-(8).

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.