- Understanding extreme value statistics with analysis using Visual Basic

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# Understanding extreme value statistics with analysis using Visual Basic

Author

Lee, Hwee Sin

Supervisor

Phang, Rosalind Lay Ping

Abstract

Extreme Value Statistics (EVS) has been widely used in many fields, such as biological, engineering, and environmental studies.

This Academic Exercise (AE) aims at understanding the development of EVS, right from the exact distribution of extremes and its limitations, to the Generalised Extreme Value (GEV) distribution. I have attempted to link up the whole development. giving as much details as and when necessary.

Chapter 2 discusses the exact distribution of extremes. I have demonstrated using examples that, theoretically, the exact distribution of extremes can be written down as functions of the initial distribution and of sample size n. The limitation of such is given, which provides us a motivation to look for distributions which are simpler and yet able to represent the distribution of extremes.

Chapter 3 discusses two approaches of deriving the asymptotic distributions, originally derived by Fisher and Tippett(l928), and Gumbel(1958). As the derivations are not straight forward, I have included further explanations. Three forms have been derived.

Chapter 4 rounds up the derivations of the 3 asymptotic forms to a single form, which is known as the GEV distribution. The sign of parameter k in the GEV distribution decides what type the distribution belongs to. 1 have shown how the GEV distribution is of the same form as each of the three asymptotic forms as derived in chapter 3. Estimation of the parameters in the GEV distribution is the next natural thing to discuss. The Method of Moments and the Maximum Likelihood Estimation are discussed in this AE. Graphical plots are usually preferred to complicated equations. Therefore, after understanding the theory, we move on to discuss its corresponding results, one of which, is the Gumbel Probability Paper. How the plot comes about and the uses of such plots are further explained in chapter 5. The three types of plots from the three types of distributions are related by a transformation.

After understanding the GEV distribution of EVS and graphical plots, one naturally hopes to find some tools that will simplify the job of plotting on the Gumbel probability paper, as well as doing the graphical plot. The final part of my AE is to produce such a user-friendly computer program using Visual Basic, :is described in chapter 6. The steps involved in using this program are described and illustrated using a known set of data. Although the program has its limitations, it is still a very useful tool to begin with.

This Academic Exercise (AE) aims at understanding the development of EVS, right from the exact distribution of extremes and its limitations, to the Generalised Extreme Value (GEV) distribution. I have attempted to link up the whole development. giving as much details as and when necessary.

Chapter 2 discusses the exact distribution of extremes. I have demonstrated using examples that, theoretically, the exact distribution of extremes can be written down as functions of the initial distribution and of sample size n. The limitation of such is given, which provides us a motivation to look for distributions which are simpler and yet able to represent the distribution of extremes.

Chapter 3 discusses two approaches of deriving the asymptotic distributions, originally derived by Fisher and Tippett(l928), and Gumbel(1958). As the derivations are not straight forward, I have included further explanations. Three forms have been derived.

Chapter 4 rounds up the derivations of the 3 asymptotic forms to a single form, which is known as the GEV distribution. The sign of parameter k in the GEV distribution decides what type the distribution belongs to. 1 have shown how the GEV distribution is of the same form as each of the three asymptotic forms as derived in chapter 3. Estimation of the parameters in the GEV distribution is the next natural thing to discuss. The Method of Moments and the Maximum Likelihood Estimation are discussed in this AE. Graphical plots are usually preferred to complicated equations. Therefore, after understanding the theory, we move on to discuss its corresponding results, one of which, is the Gumbel Probability Paper. How the plot comes about and the uses of such plots are further explained in chapter 5. The three types of plots from the three types of distributions are related by a transformation.

After understanding the GEV distribution of EVS and graphical plots, one naturally hopes to find some tools that will simplify the job of plotting on the Gumbel probability paper, as well as doing the graphical plot. The final part of my AE is to produce such a user-friendly computer program using Visual Basic, :is described in chapter 6. The steps involved in using this program are described and illustrated using a known set of data. Although the program has its limitations, it is still a very useful tool to begin with.

Date Issued

1998

Call Number

QA273.6 Lee

Date Submitted

1998