Statistics
ST236: Statistical Inference (5 ECTS)
An introduction to the ideas of statistical inference from a mathematical perspective. Topics covered include: populations and samples, properties of estimators, likelihood functions, principles and methods of point estimation, interval estimates, hypothesis testing and construction of tests.
Taught in Semester(s) II. Examined in Semester(s) II.
Workload: 36 hours (24 Lecture hours, 12 Tutorial hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
1 Construct a full sampling distribution for a simple, small sample probability model and calculate the properties of standard estimators such as the sample mean and variance;
2 Derive a likelihood function for random samples from a probability model and under more complex sampling schemes, eg mixed populations, censoring;
3 Calculate simple unbiased estimators and calculate optimal combinations of estimators;
4 Find maximum likelihood estimators by solving the score equation and obtain an estimate of precision based on observed and expected information;
5 Find confidence intervals for simple problems using pivotal quantities;
6 Calculate the size and power function for a given test procedure;
7 Obtain a most powerful test of two simple hypotheses using the Neyman Pearson lemma and extend this to a uniformly most powerful test of one-sided alternatives;
8 Use the likelihood ratio procedure to derive a test of nested hypotheses for some simple statistical models.
Indicative Content
This course provides and introduction to the ideas and mathematics of statistical inference. Topics covered include: 1. Basic notions: populations and samples, sampling distributions, estimates and estimators, the likelihood function.
Point estimation: general concepts, criteria including consistency, unbiasedness, minimum variance; methods of constructing estimators, unbiased estimation and MVUE, method of moments, maximum likelihood.
Interval estimation: confidence intervals, likelihood intervals.
Hypothesis testing: simple and composite hypotheses, type I and type II error, size and power, most-powerful tests, Neyman Pearson Lemma, uniformly most powerful tests, Likelihood ratio tests,
Module Resources
Statistical Inference (2nd Ed) by Casella & Berger, Duxbury.
Introduction to the Theory of Statistics, Mood, Graybill & Boes, McGraw Hill
Probability and statistical inference by Robert V. Hogg Elliot A Tanis, MacMillan
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