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This publication focuses on advances made in the last twenty years in statistical analysis and hydrological frequency analysis and includes Matlab codes for Halphen fitting and generating.

Title: HALPHEN DISTRIBUTION FAMILY with Application in Hydrological Frequency Analysis

Authors:
Salaheddine El Adlouni and Bernard Bobée

Specifications: Soft Cover, 155 pp


ISBN-13: 9781887201902

Price: US $35
Cat No: HDF

From the TABLE OF CONTENTS

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DESCRIPTION

Studying the HALPHEN DISTRIBUTION FAMILY was suggested by Professor Jacques Bernier, a visiting scientist at the Chair in Statistical Hydrology. In this book the authors put together theoretical works from the advances that have been made in the last twenty years, such as:

- Statistical analysis, which has become an important tool in water resources management and planning, and;

- Hydrological frequency analysis, which is a particularly useful method to estimate the probability of occurrence of a given event by fitting a statistical distribution to a sample of independent and identically distributed data.

The desirable statistical properties of these three distributions (complete system and optimal properties of the estimators) motivated the researchers to thoroughly study this system. Chapters 2 to 4 comprehensively discuss each of the three Halphen distributions:

- Mathematical and statistical properties;
- Asymptotic behavior and model selection;
- Parameter estimation methods.

Chapter 5 is concerned with the simulation and generation of samples from Halphen distributions. Matlab codes for Halphen fitting and generating are included in Chapter 6.

This book will be a useful resource to scientists, engineers, and specialists involved in statistical hydrology and water resources management and planning.


TABLE OF CONTENTS [top]

FOREWORD
PREFACE
LIST OF TABLES
LIST OF FIGURES

CHAPTER 1 - INTRODUCTION TO FREQUENCY ANALYSIS USING HALPHEN DISTRIBUTIONS

CHAPTER 2 - HALPHEN DISTRIBUTION FAMILY: MATHEMATICAL AND STATISTICAL PROPERTIES

2.1 - HISTORICAL CONTEXT
2.2 - HALPHEN DISTRIBUTION TYPE A (HA)
2.2.1 - Introduction
2.2.2 - Moments and Moment Ratios
2.2.3 - Sufficient Statistics
2.2.4 - Related Distributions
2.3 - HALPHEN DISTRIBUTION TYPE B (HB)
2.3.1 - Introduction
2.3.2 - Moments and Moment Ratios
2.3.3 - Sufficient Statistics
2.3.4 - Related Distribution
2.4 - HALPHEN DISTRIBUTION TYPE INVERSE B (HIB)
2.4.1 - Introduction
2.4.2 - Moments and Moment Ratios
2.4.3 - Sufficient Statistics
2.4.4 - Related Distribution
2.5 - SUMMARY

CHAPTER 3 - ASYMPTOTIC BEHAVIOR OF THE HALPHEN DISTRIBUTIONS
3.1. - INTRODUCTION
3.2. - CLASSES OF HEAVY TAILED DISTRIBUTIONS AND EXTREME VALUE THEORY
3.3. - ASYMPTOTIC BEHAVIOR AND DECISION SUPPORT SYSTEM
3.3.1. - Asymptotic Behavior
3.3.2. - Asymptotic Behavior for the Halphen Distributions
3.3.3. - Decision Support System
3.4. - THE HALPHEN FAMILY AND THE GENERALIZED EXTREME VALUE (GEV) SYSTEM, A COMPARISON
3.4.1. - Introduction
3.4.2. - Theoretical Arguments
3.5. - TESTING HA AND HB AGAINST THE GAMMA DISTRIBUTION
3.5.1. - Hypothesis and Alternatives
3.5.2. - Moment Ratio Diagram: Test Elaboration
3.5.3. - Power Study
3.5.4. - Conclusion

 

CHAPTER 4 - PARAMETER ESTIMATION METHODS FOR THE HALPHEN DISTRIBUTIONS
4.1. - INTRODUCTION
4.2. - MAXIMUM LIKELIHOOD METHOD (ML)
4.2.1. - ML for the HA Distribution
4.2.2. - ML for the HB Distribution
4.2.3. - ML for the HIB Distribution
4.3. - THE METHOD OF MOMENT (MM)
4.3.1. - MM for HA Distribution
4.3.2. - MM for HB Distribution
4.3.3. - MM for HIB Distribution
4.4. - The ITERATIVE MIXED METHOD (IMM)
4.5. - The BAYESIAN APPROACH FOR HALPHEN DISTRIBUTIONS
4.6. - CONCLUSION

CHAPTER 5 - MONTE CARLO SIMULATIONS FOR HALPHEN DISTRIBUTIONS
5.1. - INTRODUCTION
5.2. - MONTE CARLO SIMULATION AND RANDOM VARIABLE GENERATION
5.3. - THE ACCEPTATION-REJECTION APPROACH FOR THE HALPHEN DISTRIBUTIONS
5.3.1. - The Acceptance-Rejection Method (A-R)
5.3.2. - Halphen Type A Distribution (HA)
5.3.3. - Halphen Type B Distribution (HB)
5.3.4. - Halphen Type IB Distribution (HIB)
5.4. - MARKOV CHAIN MONTE CARLO METHOD (MCMC)
5.5. - CONCLUSION

CHAPTER 6 - MATLAB CODES FOR HALPHEN FITTING AND GENERATING (HFG)
6.1. - INTRODUCTION
6.2. - DSS CODE
6.3. - GENERATION OF HALPHEN DISTRIBUTIONS
6.4. - ESTIMATION OF THE PARAMETERS OF THE HALPHEN DISTRIBUTIONS
6.5. - QUANTILE FUNCTIONS
6.6. - CASE STUDY
6.7. - CONCLUSION

APPENDIX A: BESSEL FUNCTIONS
A.1. - Recursive relations
A.2. - Asymptotic properties
A.3. - Dispersion function
APPENDIX B: EXPONENTIAL FACTORIAL FUNCTION
B.1. - Recurrence relations
B.2. - Asymptotic relations
B.3. - Dispersion function

APPENDIX C: DECISION SUPPORT SYSTEM
C.1. - Log-Log Plot
C.2. - The Mean Excess Function Diagram (MEF)
C.3. - Hill's Ratio Plot
C.4. - Jackson Statistic

APPENDIX D: BAYESIAN FRAMEWORK
References
Author Index
Subject Index
Abbreviations and Symbols (Citation Page)



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