Synopsis for Statistics

Overview

The theme is the investigation of real data using the method of maximum likelihood to provide point estimation, given unknown parameters in the models. Maximum likelihood will be the central unifying approach. Examples will involve a distribution with a single unknown parameter, in cases for which the confidence intervals may be found by using the Central Limit Theorem (statement only). The culmination of the course will be the link of maximum likelihood technique to a simple straight line fit with normal errors.

Learning Outcomes

Students will have:

1. an understanding of the concept of likelihood, and the use of the principle of maximum likelihood to find estimators;
2. an understanding that estimators are random variables, property unbiasedness and mean square error;
3. an understanding of confidence intervals and their construction including the use of the Central Limit Theorem;
4. an understanding of simple linear regression when the error variance is known.

Synopsis

Random samples, concept of a statistic and its distribution, sample mean as a measure of location and sample variance as a measure of spread.

Concept of likelihood; examples of likelihood for simple distributions. Estimation for a single unknown parameter by maximising likelihood. Examples drawn from: Bernoulli, binomial, geometric, Poisson, exponential (parametrized by mean), normal (mean only, variance known). Data to include simple surveys, opinion polls, archaeological studies, etc. Properties of estimators—unbiasedness, Mean Squared Error = (bias + variance). Statement of Central Limit Theorem (excluding proof). Confidence intervals using CLT. Simple straight line fit, , with normal independent errors of zero mean and common known variance. Estimators for , by maximising likelihood using partial differentiation, unbiasedness and calculation of variance as linear sums of . (No confidence intervals). Examples (use scatter plots to show suitability of linear regression).

1. F. Daly, D. J. Hand, M. C. Jones, A. D. Lunn, K. J. McConway, Elements of Statistics (Addison Wesley, 1995). Chapters 1–5 give background including plots and summary statistics, Chapter 6 and parts of Chapter 7 are directly relevant.

Further Reading

1. J. A. Rice, Mathematical Statistics and Data Analysis (Wadsworth and Brooks Cole, 1988).