Statistics 1

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2. Parametric models, Method of moments estimation & Assessment of fit

Aims

This section introduces the idea of modelling the distribution of a variable in a population in terms of a family of parametric distributions, where the parameters of the distribution relate to specific quantities of interest in the population, such as the population mean or variance. If our model is appropriate for the data, then we can make inferences about the population from which data was obtained simply by estimating the parameters for the distribution.

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.

Handouts and Problem Sheets

Handout for Section 2 | Problem sheet 3 | Solutions sheet 3

I have just noticed that unfortunately I mistakenly printed and distributed an old version of the Section 2 handout, and I suppose this may cause some confusion. The only substantial error is that the handout uses o_i for the order statistics, while I meant to use x_(i) (and did so in the overhead slides and on the board). The online version of the handout has now been corrected (3.23pm, 17 February 2011).

Copyright notice

All material in these pages is copyright of the University unless explicitly stated otherwise. It is provided exclusively for educational purposes at the University and is to be downloaded or copied for your private study only, and not for distribution to anyone else.

Please also note that material from previous years' delivery of this unit is not necessarily a reliable indicator of what will be covered or examined this year.

Questions - set this week

PROBLEM SHEET 3 -- Questions 1, 4, 5

Interesting links

The report of the National Equality Panel,
which I mentioned in Lecture 3, along with background data and the Government response.

R demos - the functions I used in lectures 4 and 5 to compare histograms and density functions.

Data and Story Library
To get you in the mood for the start of the Statistics part of the unit, the Data and Story Library is an online library of datafiles and stories that illustrate the use of basic statistics methods.

Engineering Statistics Handbook
The Explore secion of the Engineering Statistics Handbook has some nice sections on Exploratory Data Analysis. It is one of a number of sites that explain different aspects of probability plots and quantile plots, but there are not that many with nice applets.

The Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an collection of interactive illustrations of concepts in science, technology, mathematics, finance, etc. There are some particularly nice demonstrations on statistics topics that you might like to browse.

Vestac
Under its Basics section this site has various applets relevant to the material covered in the last two weeks, including one or two simple applets visualising histograms and boxplots for samples from Normal and Binomial distributions, and a simple applet demonstrating Normal probability plots.

Note that I have no control over the content or availability of these external web pages. The links may be slow to load, or may sometimes fail altogether - please email me to report if a link goes down. Similarly applets may be slow to load or run, but beware that you may experience problems if you try to exit them before they have finished loading.

Professor Peter Green, School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK.
Email link Telephone: +44 (0)117 928 7967; Fax: +44 (0)117 928 7999