MATH20802 - 2010/2011

 

Unit specification

Aims

To introduce estimation and hypothesis testing methods based on likelihood and other approaches.

Brief description

Statistical methodology is concerned with taking the numerical information contained in a sample (the data) and using it to make statements (or inferences) about the population from which the sample is drawn. In that the sample provides incomplete information about the entire population, there is inevitably some uncertainty relating to any inferences made. The methods developed in this course unit not only acknowledge this uncertainty but seek to model it in a meaningful way.

Intended learning outcomes

On successful completion of this unit students will:

• have an understanding of the underlying theory;
• be able to use these techniques on simple data sets.

Future topics requiring this course unit

Third and fourth level Statistics course units.

Syllabus

1. Point estimator, point estimate, sampling distribution; unbiased estimator, bias, MSE, asymptotic unbiasedness, consistency, relative efficiency and their relationships; Properties given with some proofs [2].
2. Basic definition of order statistics, pdf and cdf of order statistics; order statistics of discrete and continuous distributions, examples; moments of order statistics, examples [6].
3. Method of moments estimation; Generalized method of moments estimation; Minimum chi squared method estimation; Weighted least squares estimation; maximum likelihood estimation: likelihood function, ML estimators for single and multi parameter cases, ML estimators for discrete and continuous models; Properties of ML estimation including invariance principle and asymptotic confidence intervals (without going into details about the Fisher information); Many examples of ML estimation given, including simple linear regression and ANOVA [8].
4. Simple hypotheses, composite hypotheses, null hypotheses, alternative hypotheses; test statistic, acceptance region, rejection region, type I error, type II error, level of significance, power, power function; N-P lemma; N-P lemma illustrated using one-sample and two-sample problems; Uniformly most powerful tests, examples; Confidence intervals for one--sample, two--sample and multi--sample problems [8].

Textbooks

• J.E. Freund, Mathematical Statistics with Applications, 7th edition, Pearson Prentice Hall 2004.
• W. Mendenhall, D.D. Wackerly and R.L. Scheaffer, Mathematical Statistics with Applications, PWS-Kent 1990.
• J.A. Rice, Mathematical Statistics and Data Analysis, 2nd edition, Duxbury Press 1995.

Learning and teaching processes

Two lectures and one examples class each week. In addition students are expected to do at least four hours private study each week on this course unit.

Assessment

Coursework; Weighting within unit 20%

2 hours end of semester examination; Weighting within unit 80%

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Arrangements