《Statistics of Extremes》

Chapter One:AIMS AND TOOLS1

1．0．Aims1

1．0．1．Conditions1

1．0．2．History2

1．0．3．The Flood Problem4

1．0．4．Methodology5

1．0．5．Arrangement of Contents5

1．1．General Tools7

1．1．1．Linear Transformations7

1．1．2．Other Transformations7

1．1．3．Symmetry8

1．1．4．Measures of Dispersion9

1．1．5．Moments10

1．1．6．Generating Function11

1．1．7．Convolution13

1．1．8．The Gamma Function15

1．1．9．The Logarithmic Normal Distribution16

1．2．Specific Tools20

1．2．0．Problems20

1．2．1．The Intensity Function20

1．2．2．The Distribution of Repeated Occurrences21

1．2．3．Analysis of Return Periods26

1．2．4．＂Observed＂Distributions28

1．2．5．Construction of Probability Papers28

1．2．6．The Plotting Problem29

1．2．7．Conditions for Plotting Positions32

1．2．8．Fitting Straight Lines on Probability Papers34

1．2．9．Application to the Normal Distribution38

Chapter Two:ORDER STATISTICS AND THEIR EXCEEDANCES42

2．1．Order Statistics42

2．1．0．Problems42

2．1．1．Distributions42

2．1．2．Averages43

2．1．3．Distribution of Frequencies46

2．1．4．Asymptotic Distribution of mth Central Values47

2．1．5．The Order Statistic with Minimum Variance51

2．1．6．Control Band52

2．1．7．Joint Distribution of Order Statistics53

2．1．8．Distribution of Distances55

2．2．The Distribution of Exceedances57

2．2．0．Introduction57

2．2．1．Distribution of the Number of Exceedances58

2．2．2．Moments61

2．2．3．The Median63

2．2．4．The Probability of Exceedances as Tolerance Limit64

2．2．5．Extrapolation from Small Samples67

2．2．6．Normal and Rare Exceedances69

2．2．7．Frequent Exceedances72

2．2．8．Summary73

Chapter Three:EXACT DISTRIBUTION OF EXTREMES75

3．1．Averages of Extremes75

3．1．0．Problems75

3．1．1．Exact Distributions75

3．1．2．Return Periods of Largest and Large Values78

3．1．3．Quantiles of Extremes79

3．1．4．Characteristic Extremes82

3．1．5．The Extremal Intensity Function84

3．1．6．The Mode85

3．1．7．Moments87

3．1．8．The Maximum of the Mean Largest Value89

3．2．Extremal Statistics94

3．2．0．Problems94

3．2．1．Absolute Extreme Values94

3．2．2．Exact Distribution of Range97

3．2．3．The Mean Range100

3．2．4．The Range as Tolerance Limit103

3．2．5．The Maximum of the Mean Range106

3．2．6．Exact Distribution of the Midrange108

3．2．7．Asymptotic Independence of Extremes110

3．2．8．The Extremal Quotient111

Chapter Four:ANALYTICAL STUDY OF EXTREMES113

4．1．The Exponential Type113

4．1．0．Problems113

4．1．1．Largest Value for the Exponential Distribution113

4．1．2．Order Statistics for the Exponential Distribution116

4．1．3．L＇Hopital＇s Rule118

4．1．4．Definition of the Exponential Type120

4．1．5．The Three Classes122

4．1．6．The Logarithmic Trend123

4．1．7．The Characteristic Product125

4．2．Extremes of the Exponential Type126

4．2．0．Problems126

4．2．1．The LOgistic Distribution126

4．2．2．Normal Extremes,Numerical Values129

4．2．3．Analysis of Normal Extremes136

4．2．4．Normal Extreme Deviates140

4．2．5．Gamma Distribution143

4．2．6．Logarithmic Normal Distribution146

4．2．7．The Normal Distribution as a Distribution of Extremes147

4．3．The Cauchy Type149

4．3．0．Problems149

4．3．1．The Exponential Type and the Existence of Moments149

4．3．2．Pareto＇s Distribution151

4．3．3．Definition of the Pareto and the Cauchy Types152

4．3．4．Extremal Properties153

4．3．5．Other Distributions without Moments154

4．3．6．Summary155

Chapter Five:THE FIRST ASYMPTOTIC DISTRIBUTION156

5．1．The Three Asymptotes156

5．1．0．Introduction156

5．1．1．Preliminary Derivation156

5．1．2．The Stability Postulate157

5．1．3．Outline of Other Derivations162

5．1．4．Interdependence164

5．2．The Double Exponential Distribution166

5．2．0．Introduction166

5．2．1．Derivations166

5．2．2．The Methods of Cramér and Von Mises170

5．2．3．Mode and Median172

5．2．4．Generating Functions173

5．2．5．Standard and Mean Deviations174

5．2．6．Probability Paper and Return Period176

5．2．7．Comparison with Other Distributions179

5．2．8．Barricelli＇s Generalization184

5．3．Extreme Order Statistics187

5．3．0．Problems187

5．3．1．Distribution of the mth Extreme187

5．3．2．Probabilities of the mth Extreme189

5．3．3．Generating Functions192

5．3．4．Cramér＇s Distribution of mth Extremes194

5．3．5．Extreme Distances197

5．3．6．The Largest Absolute Value and the Two Sample Problem198

Chapter Six:USES OF THE FIRST ASYMPTOTE201

6．1．Order Statistics from the Double Exponential Distribution201

6．1．0．Problems201

6．1．1．Maxima of Largest Values201

6．1．2．Minima of Largest Values203

6．1．3．Consecutive Modes206

6．1．4．Consecutive Means and Variances208

6．1．5．Standard Errors212

6．1．6．Extension of the Control Band216

6．1．7．The Control Curve of Dick and Darwin218

6．2．Estimation of Parameters219

6．2．0．Problems219

6．2．1．Exponential and Normal Extremes219

6．2．2．Use of Order Statistics223

6．2．3．Estimates for Probability Paper226

6．2．4．Sufficient Estimation Functions＆by B．F．Kimball229

6．2．5．Maximum Likelihood Estimations＆by B．F．Kimball231

6．2．6．Approximate Solutions232

6．2．7．Asymptotic Variance of a Forecast＆by B．F．Kimball234

6．3．Numerical Examples236

6．3．0．Problems236

6．3．1．Floods236

6．3．2．The Design Flood238

6．3．3．Meteorological Examples241

6．3．4．Application to Aeronautics245

6．3．5．Oldest Ages246

6．3．6．Breaking Strength248

6．3．7．Breakdown Voltage249

6．3．8．Applications to Naval Engineering251

6．3．9．An Application to Geology254

Chapter Seven:THE SECOND AND THIRD ASYMPTOTES255

7．1．The Second Asymptote255

7．1．0．Problems255

7．1．1．Fréchet＇s Derivation255

7．1．2．The Cauchy Type259

7．1．3．Averages and Moments264

7．1．4．Estimation of the Parameters266

7．1．5．The Increase of the Extremes269

7．1．6．Generalization270

7．1．7．Applications271

7．1．8．Summary272

7．2．The Third Asymptote272

7．2．0．Introduction272

7．2．1．The Von Mises Derivation273

7．2．2．Other Derivations276

7．2．3．Averages and Moments of Smallest Values280

7．2．4．Special Cases285

7．2．5．The Increase of the Extremes287

7．2．6．The 15 Relations Among the 3 Asymptotes288

7．3．Applications of the Third Asymptote289

7．3．0．Problems289

7．3．1．Estimation of the Three Parameters289

7．3．2．Estimation of Two Parameters293

7．3．3．Analytical Examples298

7．3．4．Droughts299

7．3．5．Fatigue Failures302

Chapter Eight:THE RANGE306

8．1．Asymptotic Distributions of Range and Midrange306

8．1．0．Problems306

8．1．1．The Range of Minima307

8．1．2．Generating Function of the Range308

8．1．3．The Reduced Range309

8．1．4．Asymptotic Distribution of the Midrange311

8．1．5．A Bivariate Transformation312

8．1．6．Asymptotic Distribution of the Range316

8．1．7．Boundary Conditions320

8．1．8．Extreme Ranges321

8．1．9．Summary324

8．2．Extremal Quotient and Geometric Range324

8．2．0．Problems324

8．2．1．Definitions325

8．2．2．The Extremal Quotient326

8．2．3．The Geometric Range327

8．3．Applications331

8．3．0．Problems331

8．3．1．The Midrange331

8．3．2．The Parameters in the Distribution of Range333

8．3．3．Normal Ranges336

8．3．4．Estimation of Initial Standard Deviation338

8．3．5．Climatological Examples340

Summary345

Bibliography349

Index373