Statistics
ST411: Mathematical Statistics: Point Estimation (5 ECTS)
NOT RUNNING IN ACADEMIC YEAR 2013-2014.
This module provides a mathematical approach to data reduction and estimation in statistics. Primary focus is on the development of optimal theory of estimation in parametric models using various criteria.
Taught in Semester(s) I. Examined in Semester(s) I.
Workload: 34 hours (24 Lecture hours, 10 Tutorial hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- Appreciate different approaches to statistics and the role of Probability in Statistical Inference.
- Use Probability Theory in developing properties of estimators of parameters in Statistics.
- Derive complete and sufficient statistics by a number of methods.
- Derive theorems relating to minimum variance unbiased estimators, and implement this theory in practical examples; be able to give examples where such estimators do not exist.
- Understand, develop properties of, and implement maximum likelihood estimation.
- Derive lower bounds on the variance of unbiased estimators, and be able to prove when such a bound equals the variance of the minimum variance unbiased estimator when such an estimator exists.
- Derive a number of longer and shorter theorems relating to data reduction and point estimation
- Prove theorems relating to point estimation methods, be able to prove relationships between procedures, be able to give examples where methods do not give the same estimators, and be able to apply all procedures.
Indicative Content
This module provides a mathematically rigorous approach to mathematical statistics with primary emphasis on optimal estimation in parametric models of random phenomena. Topics covered are:
- A brief overview of approaches to Statistics, including data analytic, frequentist, Bayesian, robust, non-parametric, structural and fiducial.
- The role of probability in the frequentist approach to statistical inference.
- Brief review of random variables, their distributions and moments, and sampling distributions and limiting theory.
- Sufficiency as a data reduction method.
- Parameter estimation methods including minimum variance unbiased estimation and maximum likelihood estimation.
Some theorems include those of Basu, Rao and Blackwell, Lehmann-Scheffé, results relating to the Cramér-Rao lower bound, and relationships between procedures. Emphasis is placed on the mathematical and statistical theory, relationships between various estimation criteria and implementation of procedures on given parametric families.
Module Resources
Roussas, G.G., A Course in Mathematical Statistics, Third Edition. Academic Press, 2014.
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