UNIVERSITI PUTRA MALAYSIA

NAHDIYA ZAINAL ABIDIN

IPM 2013 4

GOODNESS-OF-FIT TESTS FOR EXTREME VALUE DISTRIBUTIONS

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GOODNESS-OF-FIT TESTS FOR EXTREME VALUEDISTRIBUTIONS

By

NAHDIYA ZAINAL ABIDIN

Thesis Submitted to the School of Graduate Studies, Universiti PutraMalaysia, in Fulfilment of the Requirements for the Degree of Master

of Science

July 2013

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Abstract of thesis presented to the Senate of Universiti Putra Malaysia infulfilment of the requirement for the degree of Master of Science

GOODNESS-OF-FIT TESTS FOR EXTREME VALUEDISTRIBUTIONS

By

NAHDIYA ZAINAL ABIDIN

July 2013

Chair: Mohd Bakri Adam, PhD

Faculty: Institute for Mathematical Research

This study concentrates on the Goodness-of-fit (GoF) test for the extreme value

distributions. The distributions involved are Generalized Extreme Value (GEV)

Type-I, Type-II and Type-III distributions. In this study, the types of GoF tests

involved are the graphical plots as well as the statistical tests. In the graphical

plot, the existing QQ plot is built based on the quantiles of the hypothetical and

empirical distributions. However, the QQ plot suffers from the deviation at the

tail of the distribution which particularly occurs very often for the case of heavy

tailed distributions. In order to reduce the deviation, the conditional quantiles is

recommended. The conditional quantiles plots the end points of the hypothetical

and empirical distributions closer to each other. In addition, the alternative plot

suggested is hybrid plot. Unlike the QQ plot which plots the original values of the

quantiles, the hybrid plot illustrates the quantiles deviation between the hypothet-

ical and empirical values. Moreover, the hybrid plot lets several statistical models

to be plotted into a single graph. These plots are done in a graph because the

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degree of fit for different statistical models can be visually compared. This is be-

cause the horizontal axis is restricted between 0 to 1 for any statistical distribution.

The parameters of GEV Type-I, Type-II and Type-III are estimated by maxi-

mum likelihood estimation (MLE). The statistical tests involved in the GoF test

are Anderson-Darling (AD), Cramer-von Mises (CVM), Zhang Anderson-Darling

(ZAD), Zhang Cramer-von Mises (ZCVM) and Shimokawa (Ln) tests. To deter-

mine the most powerful statistical test, the critical values of these statistical tests

are generated first. Then, the reliability of the critical values are validated by the

power study. If the rejection rate of the critical value is close to the respective sig-

nificance level, that particular critical value is reliable. In addition, it is of interest

to make use of the critical values developed by other researchers. These critical

values are done for GEV distribution. These critical values were generated from

AD, ZAD and Ahmad tests. For this study, they are labelled as AD-GEV, ZAD-

GEV and Ahmad-GEV respectively. These critical values are tested for reliability

as well. The power of the statistical tests are examined by the power study as

well. Next, to evaluate the power, the alternative distributions are fitted to the

extreme value distribution model. Based on the alternative distributions, the most

powerful test should be able to produce the highest rejection rate.

The results for graphical plot show that conditional quantiles plot is better than

the traditional quantiles plot to illustrate the agreement between two identical dis-

tributions as well as the discrepancy between two different distributions. Besides,

for the statistical tests, the results state that the AD test is the most powerful test

for GEV Type-I. For GEV Type-II, the most powerful test are devided according

to the cluster of sample size n. The AD test can generally be used for cluster n=15

to 17, while the ZAD test is powerful for the cluster n=18 to 49. The cluster of

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n=50 to 100 has AD-GEV test as the powerful test. Besides, for GEV Type-III,

the ZAD test is generally powerful for cluster n= 18 to 100, but for cluster, n=15

to 17, the ZCVM test is more powerful.

In the application part, two types of data were used. The first type is the data that

was collected from extreme value distribution while the second type is the data

that is normally distributed. The extreme value distribution models are fitted

to both types of data. The data that is distributed according to the extreme

values distribution is used to verify the agreement between the extreme value

distribution and the extreme value distribution model. On the other hand, data

that is normally distributed is employed to verify that extreme value distribution

model does not fit the non extreme value distribution. The result signifies that

the findings in graphical and statistical method of GoF are applicable.

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Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia sebagaimemenuhi keperluan untuk ijazah Master Sains

UJIAN KEBAGUSAN KESESUAIAN BAGI TABURAN NILAIEKSTREM

Oleh

NAHDIYA ZAINAL ABIDIN

Julai 2013

Pengerusi: Mohd Bakri Adam, PhD

Fakulti: Institut Penyelidikan Matematik

Penyelidikan ini memfokuskan pada ujian kebagusan kesesuaian (GoF) bagi tabu-

ran nilai ekstrem. Taburan yang terlibat adalah taburan Nilai Ekstrem Teritlak

(GEV) Jenis-I, Jenis-II dan Jenis-III. Dalam kajian ini, jenis GoF yang terlibat

ialah plot grafik dan ujian statistik. Dalam plot grafik, plot QQ yang sedia ada dib-

ina berdasarkan kuantil dari taburan hipotesis dan empirikal. Walaubagaimana-

pun, plot QQ mengalami sisihan pada bahagian ekor taburan dimana ianya kerap

berlaku terutama bagi kes taburan berekor panjang. Bagi mengurangkan sisi-

han, kuantil bersyarat dicadangkan. Kuantil bersyarat memplot poin-poin akhir

hipotesis dan empirical lebih hampir di antara satu sama lain. Di samping itu,

plot alternatif yang dicadangkan ialah plot hibrid. Tidak seperti plot QQ yang

memplotkan nilai asal kuantil, plot hibrid memaparkan sisihan kuantil di antara

nilai hipotesis dan empirikal. Tambahan lagi, plot hibrid membenarkan beber-

apa model statistik untuk diplotkan di dalam satu graf. Plot ini dibuat di dalam

satu graf kerana darjah kesesuaian bagi model statistik yang berbeza dapat dilihat

secara perbandingan.Ini kerana paksi mendatar dihadkan diantara 0 dan 1 bagi

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mana-mana taburan statistik.

Parameter GEV Jenis-I, Jenis-II dan Jenis-III dianggarkan oleh Kemungkinan

Anggaran Maksimum (MLE). Ujian statistik yang terlibat dalam ujian GoF ialah

ujian Anderson-Darling (AD), Cramer-von Mises (CVM), Zhang Anderson-Darling

(ZAD), Zhang Cramer-von Mises (ZCVM) and Shimokawa (Ln). Untuk menda-

patkan ujian statistik yang paling berkuasa, nilai kritikal bagi ujian statistik ini

dibina terlebih dahulu. Kemudian, kebolehpercayaan nilai kritikal disahkan oleh

kajian kuasa. Sekiranya kadar penolakan bagi nilai kritikal adalah hampir dengan

aras keertian masing-masing, nilai kritikal tersebut adalah dipercayai. Di samping

itu, adalah penting untuk menggunakan nilai kritikal yang dibina oleh penyelidik-

penyelidik lain. Nilai-nilai kritikal tersebut dibuat untuk taburan GEV. Nilai-nilai

kritikal tersebut dijana daripada ujian AD, ZAD dan Ahmad. Untuk kajian ini,

mereka masing-masing dilabelkan sebagai AD-GEV, ZAD-GEV dan Ahmad-GEV.

Nilai kritikal ini juga diuji kebolehpercayaannya. Kuasa ujian statistik diperiksa

oleh kajian kuasa juga. Kemudian,untuk menilai kuasa, taburan alternatif di-

padankan dengan model nilai ekstrem. Berdasarkan taburan alternatif, ujian yang

paling berkuasa sepatutnya mampu menghasilkan kadar penolakan yang tertinggi.

Keputusan bagi plot grafik menunjukkan plot kuantil bersyarat adalah lagi bagus

berbanding plot kuantil tradisional untuk menggambarkan kesefahaman di an-

tara dua taburan yang sama serta percanggahan di antara dua taburan berbeza.

Di samping itu, bagi ujian statistik, keputusan menyatakan bahawa ujian AD

adalah paling berkuasa untuk GEV Jenis-I . Bagi GEV Jenis-II, ujian yang paling

berkuasa dibahagikan mengikut kelompok saiz sampel n. Ujian AD secara umum-

nya digunakan untuk kelompok n=15 hingga 17, manakala ujian ZAD berkuasa

untuk kelompok n=18 hingga 49. Kelompok n=50 hingga 100 mempunyai AD-

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GEV sebagai berkuasa. Selain itu, untuk GEV Jenis-III, ujian ZAD secara umum-

nya berkuasa bagi kelompok n=18 hingga 100, tetapi bagi kelompok n=15 hingga

17, ujian ZCVM lebih berkuasa.

Dalam bahagian aplikasi, dua jenis data digunakan. Data yang pertama ialah data

yang dikumpul daripada taburan nilai ekstrem sementara data jenis kedua ialah

data yang bertaburan secara normal. Model taburan nilai ekstrem dipadankan

dengan kedua-dua data. Data yang bertaburan mengikut taburan nilai ekstrem

digunakan untuk mengesahkan kesefahaman diantara taburan nilai ekstrem dan

model taburan nilai ekstrem. Selain itu, data yang bertaburan secara normal di-

gunakan untuk mengesahkan bahawa model taburan nilai ekstrem tidak padan

dengan taburan yang bukan nilai ekstrem. Keputusan mengesahkan bahawa pen-

emuan dalam kaedah GoF grafik dan ujian statistik boleh diaplikasikan.

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ACKNOWLEDGEMENTS

First of all, I would like to thank my supervisor, Dr. Mohd Bakri Adam as well as

my co supervisor, Prof. Dr. Habshah Midi for supervising me in my study. Their

supervision and encouragement have helped me to accomplished my study. Their

guidance istruly appreciated.

I would like to express my gratitude to my parents Mr. Zainal Abidin Ali and

Mrs. Normala Ahmad who have been supporting me from the day I have decided

to enroll my master study. The appreciation also goes to my husband, Abdul

Muhaimin Abdullah and my parents in law, Prof. Dr. Abdullah Mohd Zin and

Assoc. Prof. Dr. Saadiyah Darus. All the works that I have done would be

nothing without the love and encouragement from my family.

Special thanks to my dear collegues; Mrs. Norhaslinda Ali, Ms. Nurain Sharul

Saimi and Mr. Nurazman Nordin who helped me to understand the statistical

extreme fields. Last but not least, my grateful thanks to all the staff at Institute for

Mathematical Research (INSPEM), UPM for helping me throughout the semesters.

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I certify that a Thesis Examination Committee has met on 24 July 2013 toconduct the final examination of Nahdiya Zainal Abidin her thesis entitled“Goodness-of-fit Tests for Extreme Value Distributions” in accordancewith the Universities and University Colleges Act 1971 and the Constitution ofthe Universiti Putra Malaysia [P.U.(A) 106] 15 March 1998. The Committee rec-ommends that the student be awarded the Master of Science.

Members of the Thesis Examination Committee were as follows:

Noor Akma bt Ibrahim, Ph.D.ProfessorFakulti of Science Universiti Putra Malaysia(Chairperson)

Isthrinayagy a/p S.Krishnarajah, Ph.D.Senior LecturerFakulti of Science Universiti Putra Malaysia(Internal Examiner)

Jayanthi a/p Arasan, Ph.D.Associate ProfessorFakulti of Science Universiti Putra Malaysia(Internal Examiner)

Ibrahim Mohamad, Ph.D.Associate ProfessorInstitut Sains MatematikUniversiti MalayaMalaysia(External Examiner)

NORITAH OMAR, PhDAssociate Professor and Deputy DeanSchool of Graduate StudiesUniversiti Putra Malaysia

Date:

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This thesis was submitted to the Senate of Universiti Putra Malaysia and has beenaccepted as fulfilment of the requirement for the degree of Master of Science. Themembers of the Supervisory Committee were as follows:

Mohd Bakri Adam, PhDSenior LecturerFaculty of ScienceUniversiti Putra Malaysia(Chairperson)

Habshah Midi, PhDProfessorFaculty of ScienceUniversiti Putra Malaysia(Member)

BUJANG BIN KIM HUAT, PhDProfessor and DeanSchool of Graduate StudiesUniversiti Putra Malaysia

Date:

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DECLARATION

Declaration by graduate student

I hereby confirm that:� this thesis is my original work;� quotations, illustrations and citations have been duly referenced;� this thesis has not been submitted previously or concurrently for any other

degree at any other institutions;� intellectual property from the thesis and copyright of thesis are fully-owned

by Universiti Putra Malaysia, as according to the Universiti Putra Malaysia(Research) Rules 2012;

� written permission must be obtained from supervisor and the office of DeputyVice-Chancellor (Research and Innovation) before thesis is published (in theform of written, printed or in electronic form) including books, journals, mod-ules, proceedings, popular writings, seminar papers, manuscripts, posters,reports, lecture notes, learning modules or any other materials as stated inthe Universiti Putra Malaysia (Research) Rules 2012;

� there is no plagiarism or data falsification/fabrication in the thesis, andscholarly integrity is upheld as according to the Universiti Putra Malaysia(Graduate Studies) Rules 2003 (Revision 2012-2013) and the Universiti Pu-tra Malaysia (Research) Rules 2012. The thesis has undergone plagiarismdetection software.

Signature: Date:

Name and Matric No.:

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Declaration by Members of Supervisory Committee

This is to confirm that:� the research conducted and the writing of this thesis was under our supervi-

sion;� supervision responsibilities as stated in the Universiti Putra Malaysia (Grad-

uate Studies) Rules 2003 (Revision 2012-2013) are adhered to.

Signature: Signature:Name of Name ofChairman of Member ofSupervisory SupervisoryCommittee: Committee:

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TABLE OF CONTENTS

Page

ABSTRACT ii

ABSTRAK v

ACKNOWLEDGEMENTS viii

APPROVAL ix

DECLARATION xi

LIST OF TABLES xv

LIST OF FIGURES xvii

LIST OF ABBREVIATIONS xxiii

CHAPTER

1 INTRODUCTION 11.1 Extreme Value Theory 1

1.1.1 Extreme Value Distribution 21.1.2 Statistical Modeling 3

1.2 The Goodness-of-fit 81.3 Maximum Likelihood Estimation 101.4 Problem Statement 111.5 Objectives of the Research 121.6 Structure of the Thesis 12

2 LITERATURE REVIEW 142.1 Goodness-of-fit Plot 142.2 Statistical Goodness-of-fit Test 172.3 Parameter Estimation 22

3 GOODNESS-OF-FIT PLOT FOR EXTREME VALUE DISTRI-BUTION 233.1 Introduction 233.2 Methodology 24

3.2.1 Estimation of Parameters 253.2.2 Classical Plot for Extreme Value Distribution 273.2.3 Conditional Plot 293.2.4 Stabilized Plot 293.2.5 Hybrid Plot 303.2.6 Bias and Root Mean Square Error 303.2.7 Plot Between the Alternative Distributions and GEV Models 31

3.3 Results 313.3.1 GEV Type-I Distribution 31

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3.3.2 GEV Type-II Distribution 443.3.3 GEV Type-III Distribution 56

3.4 Discussion 633.5 Conclusion 66

4 STATISTICAL GOODNESS-OF-FIT TEST FOR EXTREME VALUEDISTRIBUTION 674.1 Introduction 674.2 Methodology 68

4.2.1 Statistics Values and Critical Values 684.2.2 Power Study of Goodness-of-fit 71

4.3 Results 724.3.1 Critical Values for GEV Type-I Distribution, Reliability and

Power of the Tests 734.3.2 Critical Values for GEV Type-II Distribution, Reliability

and Power of the Tests 774.3.3 Critical Values for GEV Type-III Distribution, Reliability

and Power of the Tests 844.4 Discussion 914.5 Conclusion 94

5 VERIFICATION OF THE ALTERNATIVE GOODNESS-OF-FITTEST FOR GENERALIZED EXTREME VALUE DISTRIBU-TION: AN APPLICATION 955.1 Introduction 955.2 Methodology 965.3 Results 985.4 Discussion 1035.5 Conclusion 106

6 OVERALL CONCLUSION AND FUTURE WORK 1076.1 Overall Conclusion 1076.2 Future Work 110

REFERENCES/BIBLIOGRAPHY 112

APPENDICES 117

BIODATA OF STUDENT 133

LIST OF PUBLICATIONS 134

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