Gwybodaeth Modiwlau
Course Delivery
Assessment
| Assessment Type | Assessment length / details | Proportion |
|---|---|---|
| Semester Assessment | Class Test Class Test: short questions on essential semester 1 material. 50 Minutes | 30% |
| Semester Exam | 2 Hours Semester Exam Written examination. | 70% |
| Supplementary Exam | 2 Hours Supplementary Exam Written supplementary examination. | 100% |
Learning Outcomes
On successful completion of this module students should be able to:
understand the basic properties and the construction of the likelihood function for random samples in the context of classical parametric statistical inference;
find maximum likelihood estimators (MLE), minimum variance unbiased estimators, sufficient statistics, as well as likelihood and confidence intervals using the pivotal method and the asymptotic Normality of the MLE;
carry out hypothesis testing based on critical regions and the likelihood ratio, and construct and interpret tests for a population proportion, mean and variance;
describe affine transformations of multivariate random vectors in matrix form, identify a Multivariate Normal distribution and distributions of quadratic forms, and relate these to Student's t and Fisher's F distributions;
formalise a suitable statistical experiment as a linear model in matrix form, understand the Ordinary Least Squares estimator in the context of general likelihood inference, identify linear unbiased estimators of minimum variance, and construct confidence intervals and regions for linear combinations of parameters;
carry out simple linear hypothesis tests based on critical regions using analysis of variance (ANOVA), and calculate leverages and residuals for outlier detection;
apply the basic principles of Bayesian inference, including prior, posterior, and predictive distributions and the concept of prior ignorance, in the context of standard conjugate families (Gamma, Beta, Normal);
construct Bayesian credible intervals and compare them with confidence and likelihood intervals of classical inference.
Brief description
The module begins by re-examining the ideas of likelihood, parameter estimation, confidence intervals and hypothesis testing in classical (frequentist) statistical inference, and considering their interpretation and theoretical aspects more deeply, in the context of general statistical models and the Cramer-Rao theory of minimum error estimators. Combined with the matrix formalism of multivariate probability theory, these principles are subsequently applied to Linear Statistical Models, including detailed theoretical consideration of Analysis of Variance (ANOVA) in simple situations. Bayesian inference is then introduced as an alternative approach to classical inference, incorporating subjective knowledge into the information obtained from observed data. Applications to inference about a population proportion, mean, variance, and other parameters are discussed in detail using simple examples. Finally, the two approaches to inference are compared, mainly in the context of interval estimation.
Aims
To introduce the basic ideas and concepts of classical statistical inference based on the likelihood function, apply it to the case of linear statistical models, and compare this approach to the alternative Bayesian inference.
Content
2. MULTIVARIATE PROBABILITY THEORY: Random vectors, mean vector, dispersion matrix. Multivariate Normal distribution and its identification. Affine transformations, quadratic forms, and distributions of random vectors. Independence. Relation to chi-squared, Student's t, and Fisher's F distributions.
3. APPLICATION TO LINEAR MODELS: Formulation of the general linear statistical model. Ordinary Least Squares estimator and normal equations. The assumption of independent homoscedastic errors. The Best Linear Unbiased Estimator and the minimum variance bound. Confidence intervals and regions for known variance by pivotal method. The hat matrix, ANOVA, variance estimation. Linear hypothesis testing based on critical regions with ANOVA, leading to F-tests and t-tests. Consideration of outliers: residuals and leverage.
4. BAYESIAN INFERENCE: The core principle based on the Bayes Theorem. Prior and posterior distributions with standard conjugate families (Gamma, Beta, Normal). Prior ignorance. Predictive distributions. Interval estimation with credible intervals and highest density intervals.
5. COMPARATIVE STATISTICAL INFERENCE: Comparisons between classical and Bayesian approaches using likelihood and confidence/credible intervals.
Module Skills
| Skills Type | Skills details |
|---|---|
| Adaptability and resilience | Good understanding of the contents requires considerable intellectual effort over an extended period of time. |
| Co-ordinating with others | Discussing the theory and solving problems together during the module is encouraged. |
| Creative Problem Solving | Problem sessions based on problem sheets to be solved independently. This is crucial to prepare for the problems in the exam. |
| Critical and analytical thinking | Theory is developed rigorously and compared with real world situations. |
| Digital capability | Insights are provided into interpretation of statistical data. |
| Professional communication | Discussing the theory and solving problems together during the module is encouraged. |
| Real world sense | Theory is compared with real world application. |
| Reflection | Intuitive ideas need to be translated into mathematical reasoning. |
| Subject Specific Skills | The ability to solve problems in statistical inference and linear models is tested in the exam. |
Notes
This module is at CQFW Level 6