Aims

The section starts by introducing one of the simplest methods of parametric estimation - the method of moments - but note that other methods (maximum likelihood methods and least squares methods) will be introduced in later sections.

For discrete observations representing counts, the correct choice of parametric model is often determined by the context. However, for continuous uni-modal data, the correct choice of model can be more difficult. There are often a number of feasible models whose distributions appear at first glance to similar while differing in essential details. In this second half of section we introduce probability plots (plots of the quantiles of the data against the quantiles of the fitted distribution) way of assessing the fit of the data to the chosen parametric family.

Objectives

The following objectives will help you to assess how well you have mastered the relevant material. By the end of this section you should be able to:

• Understand how parametric families can be used to provide flexible models for the distribution of population random variables.

• Recall the formulae for the probability mass function and the probability density function for the standard parametric families of discrete and continuous distributions (Binomial, Geometric, Poisson, Exponential, Gamma, Normal, Uniform) and be able to relate the parameters of each distribution to population quantities such as the population mean and the population variance.

• Write down equation(s) defining the method of moments estimators for parametric families defined in terms of one or more unknown parameters.

• Solve the equations for the methods of moments estimators in simple one and two parameter cases, and hence compute appropriate methods of moments estimates from a given data set.

• Use R to numerically calculate the quantiles of simple standard distributions (Uniform, Exponential, Gamma, Normal).

• Construct simple probability plots of the quantiles of a data set against the quantiles of a given or fitted distribution (by hand or in R), and use the plot to assess the fit of the data to the specified distribution.

Suggested Reading

Handouts and Problem Sheets

Questions - set this week

PROBLEM SHEET 2 -- Questions 1, 3, 5

Interesting links