CONTENTS
Chapter One : AIMS AND TOOLS
1.0. Aims..................................................................................................1
1.0.1. Conditions...............................................................................1
1.0.2. History....................................................................................2
1.0.3. The Flood Problem...............................................................4
1.0.4. Methodology...........................................................................5
1.0.5. Arrangement of Contents.....................................................5
1.1.General Tools ..................................................................................7
1.1.1. Linear Transformations.........................................................7
1.1.2. Other Transformations..........................................................7
1.1.3. Symmetry...............................................................................8
1.1.4. Measures of Dispersion........................................................9
1.1.5. Moments................................................................................10
1.1.6. Generating Function..............................................................11
1.1.7. Convolution............................................................................13
1.1.8. The Gamma Function...........................................................15
1.1.9. The Logarithmic Normal Distribution.................................16
1.2. Specific Tools.................................................................................20
1.2.0. Problems................................................................................20
1.2.1. The Intensity Function.........................................................20
1.2.2. The Distribution of Repeated Occurrences........................21
1.2.3. Analysis of Return Periods..................................................26
1.2.4. "Observed" Distribution........................................................28
1.2.5. Construction of Probability Papers.....................................28
1.2.6. The Plotting Problem...........................................................29
1.2.7. Conditions for Plotting Positions........................................32
1.2.8. Fitting Straight Lines on Probability Papers.....................34
1.2.9. Application to the Normal Distribution..............................38
Chapter Two : ORDER STATISTICS AND THEIR EXCEEDANCES....42
2.1. Order Statistics..............................................................................42
2.1.0. Problems...............................................................................42
2.1.1. Distributions.........................................................................42
2.1.2. Averages..............................................................................43
2.1.3. Distribution of Frequencies...............................................46
2.1.4. Asymptotic Distribution of mth Central Values..............47
2.1.5. The Order Statistic with Minimum Variance...................51
2.1.6. Control Band......................................................................52
2.1.7. Joint Distribution of Order Statistics...............................53
2.1.8. Distribution of Distances..................................................53
2.2. The Distribution of Exceedances..............................................57
2.2.0. Introduction........................................................................57
2.2.1. Distribution of the Number of Exceedances.................58
2.2.2. Moments............................................................................61
2.2.3. The Median.......................................................................63
2.2.4. The Probability of Exceedances as Tolerance Limit...64
2.2.5. Extrapolation from Small Samples..................................67
2.2.6. Normal and Rare Exceedances.......................................69
2.2.7. Frequent Exceedances.....................................................72
2.2.8. Summary............................................................................73
Chapter Three : EXACT DISTRIBUTION OF EXTERMES.............75
3.1. Averages of Extremes................................................................75
3.1.0. Problems.............................................................................75
3.1.1. Exact Distributions............................................................75
3.1.2. Return Periods of Largest and Large Values................78
3.1.3. Quantities of Extremes.....................................................79
3.1.4. Characteristic Extremes....................................................82
3.1.5. The Extermal Intensity Function......................................84
3.1.6. The Mode............................................................................85
3.1.7. Moments..............................................................................87
3.1.8. The Maximum of the Mean Largest Value.....................89
3.2. Extremal Statistics......................................................................94
3.2.0. Problems.............................................................................94
3.2.1. Absolute Extreme Values.................................................94
3.2.2. Exact Distribution of Range..............................................97
3.2.3. The Mean Range...............................................................100
3.2.4. The Range as Tolerance Limit........................................103
3.2.5. The Maximum of the Mean Range..................................106
3.2.6. Exact Distribution of the Midrange.................................108
3.2.7. Asymptotic Independence..................................................110
3.2.8. The Extremal Quotient......................................................111
Chapter Four : ANALYTICAL STUDY OF EXTREMES................113
4.1. The Exponential Type..............................................................113
4.1.0. Problems..........................................................................113
4.1.1. Largest Value for the Exponential Distribution...........113
4.1.2. Order Statistics for the Exponential Distribution........116
4.1.3. L''HÔpital''s Rule...............................................................118
4.1.4. Definition of the Exponential Type...............................120
4.1.5. The Three Classes..........................................................122
4.1.6. The Logarithmic Trend...................................................123
4.1.7. The Characteristic Product.............................................125
4.2. Extreme of the Exponential Type...........................................126
4.2.0. Problems...........................................................................126
4.2.1. The Logistic Distribution................................................126
4.2.2. Normal Extreme, Numerical Values...............................129
4.2.3. Analysis of Normal Extremes........................................136
4.2.4. Normal Extreme Deviates...............................................140
4.2.5. Gamma Distribution..........................................................143
4.2.6. Logarithmic Normal Distribution......................................146
4.2.7. The Normal Distribution as a Distribution Extremes....147
4.3. The Cauchy Type.......................................................................149
4.3.0. Problems.............................................................................149
4.3.1. The Exponential Type and the Existence of Moments..149
4.3.2. Pareto''s Distribution..........................................................151
4.3.3. Definition of the Pareto and the Cauchy Types............152
4.3.4. Extremal Properties...........................................................153
4.3.5. Other Distributions without moments...............................154
4.3.6. Summary..............................................................................155
Chapter Five : THE FIRST ASYMPTOTIC DISTRIBUTION............156
5.1. The Three Asymptotes................................................................156
5.1.0. Introduction..........................................................................156
5.1.1. Preliminary Derivation........................................................156
5.1.2. The Stability Postulate.......................................................157
5.1.3. Outline of Other Derivations..............................................162
5.1.4. Interdependence..................................................................164
5.2. The Double Exponential Distribution..........................................166
5.2.0. Introduction..........................................................................166
5.2.1. Derivations...........................................................................166
5.2.2. The Methods of Cramer and Von Mises...........................170
5.2.3. Mode and Median.................................................................172
5.2.4. Generating Functions............................................................173
5.2.5. Standard and Mean Deviations............................................174
5.2.6. Probability Paper and Return Period..................................176
5.2.7. Comparison with Other Distribution....................................179
5.2.8. Barricelli''s Generalization.....................................................184
5.3. Extreme Order Statistics...............................................................187
5.3.0. Problems.................................................................................187
5.3.1. Distribution of the mth Extreme..........................................187
5.3.2. Probabilities of the mth Extreme........................................189
5.3.3. Generating Functions............................................................192
5.3.4. Cramer''s Distribution of mth Extremes..............................194
5.3.5. Extreme Distances................................................................197
5.3.6. The Largest Absolute Value and the Two Sample Problem .....198
Chapter Six : USES OF THE FIRST ASYMPTOTE...........................201
6.1. Order Statistics from the Double Exponential Distribution.......201
6.1.0. Problems................................................................................201
6.1.1. Maxima of Largest Values..................................................201
6.1.2. Minima of Largest Values...................................................203
6.1.3. Consecutive Modes..............................................................206
6.1.4. Consecutive Means and Variances.....................................208
6.1.5. Standard Errors....................................................................212
6.1.6. Extension of the Control Band...........................................216
6.1.7. The Control Curve of Dick and Darwin............................218
6.2. Estimation of Parameters.............................................................219
6.2.0. Problems...............................................................................219
6.2.1. Exponential and Normal Extremes....................................219
6.2.2. Use of Order Statistics.......................................................223
6.2.3. Estimates for Probability Paper.........................................226
6.2.4. Sufficient Estimation Functions, by B .F. Kimball............229
6.2.5. Maximum Likelihood Estimations, by B. F. Kimball ........231
6.2.6. Approximate Solutions..........................................................232
6.2.7. Asymptotic Variance of a Forecast, by B. F. Kimball......234
6.3. Numerical Examples........................................................................236
6.3.0. Problems.................................................................................236
6.3.1. Floods.....................................................................................236
6.3.2. The Design Flood....................................................................238
6.3.3. Meteorological Examples........................................................241
6.3.4. Application to Aeronautics......................................................245
6.3.5. Oldest Ages..............................................................................246
6.3.6. Breaking Strength...................................................................248
6.3.7. Breakdown Voltage ................................................................249
6.3.8. Application to Naval Engineering...........................................251
6.3.9. An Application to Geology......................................................254
Chapter Seven : THE SECOND AND THIRD ASYMPTOTES................255
7.1. The Second Asymptote......................................................................255
7.1.0. Problems....................................................................................255
7.1.1. Frechet''s Derivation.................................................................255
7.1.2. The Cauchy Type.....................................................................259
7.1.3. Averages and Moments ..........................................................264
7.1.4. Estimation of the Parameters..................................................266
7.1.5. The Increase of the Extremes.................................................269
7.1.6. Generalization.............................................................................270
7.1.7. Applications................................................................................271
7.1.8. Summary.....................................................................................272
7.2. The Third Asymptote..........................................................................272
7.2.0. Introduction.................................................................................272
7.2.1. The Von Mises Derivation........................................................273
7.2.2. Other Derivation.........................................................................276
7.2.3. Averages and Moments of Smallest Values............................280
7.2.4. Special Cases..............................................................................285
7.2.5. The Increase of the Extremes..................................................287
7.2.6. The 15 Relations Among the 3 Asymptotes............................288
7.3. Applications of the Third Asymptote.................................................289
7.3.0.Problems.......................................................................................289
7.3.1. Estimation of the Three Parameters........................................289
7.3.2. Estimation of T재 Parameters...................................................293
7.3.3. Analytical Examples....................................................................298
7.3.4. Droughts.......................................................................................299
7.3.5. Fatigue Failures..........................................................................302
Chapter Eight : THE RANGE......................................................................306
8.1. Asymptotic Distribution of Range and Midrange...............................306
8.1.0. Problems.........................................................................................306
8.1.1. The Range of Minima....................................................................306
8.1.2. Generating Function of the Range...............................................308
8.1.3. The Reduced Range......................................................................309
8.1.4. Asymptotic Distribution of the Midrange....................................311
8.1.5. A Bivariate Transformation..........................................................312
8.1.6. Asymptotic Distribution of the Range.........................................316
8.1.7. Boundary Conditions.....................................................................320
8.1.8. Extreme Ranges.............................................................................321
8.1.9. Summary.........................................................................................324
8.2. Extremal Quotient and Geometric Range.............................................324
8.2.0. Problems........................................................................................324
8.2.1. Definitions......................................................................................325
8.2.2. The Extremal Quotient.................................................................326
8.2.3. The Geometric Range..................................................................327
8.3. Applications..............................................................................................331
8.3.0. Problems.........................................................................................331
8.3.1. The Midrange.................................................................................331
8.3.2. The Parameters in the Distribution of Range............................333
8.3.3. Normal Ranges...............................................................................336
8.3.4. Estimation of Initial Standard Deviation......................................338
8.3.5. Climatological Examples................................................................340
Summary...........................................................................................................345
Bibliography......................................................................................................349
Index..................................................................................................................373
Chapter One : AIMS AND TOOLS
1.0. Aims..................................................................................................1
1.0.1. Conditions...............................................................................1
1.0.2. History....................................................................................2
1.0.3. The Flood Problem...............................................................4
1.0.4. Methodology...........................................................................5
1.0.5. Arrangement of Contents.....................................................5
1.1.General Tools ..................................................................................7
1.1.1. Linear Transformations.........................................................7
1.1.2. Other Transformations..........................................................7
1.1.3. Symmetry...............................................................................8
1.1.4. Measures of Dispersion........................................................9
1.1.5. Moments................................................................................10
1.1.6. Generating Function..............................................................11
1.1.7. Convolution............................................................................13
1.1.8. The Gamma Function...........................................................15
1.1.9. The Logarithmic Normal Distribution.................................16
1.2. Specific Tools.................................................................................20
1.2.0. Problems................................................................................20
1.2.1. The Intensity Function.........................................................20
1.2.2. The Distribution of Repeated Occurrences........................21
1.2.3. Analysis of Return Periods..................................................26
1.2.4. "Observed" Distribution........................................................28
1.2.5. Construction of Probability Papers.....................................28
1.2.6. The Plotting Problem...........................................................29
1.2.7. Conditions for Plotting Positions........................................32
1.2.8. Fitting Straight Lines on Probability Papers.....................34
1.2.9. Application to the Normal Distribution..............................38
Chapter Two : ORDER STATISTICS AND THEIR EXCEEDANCES....42
2.1. Order Statistics..............................................................................42
2.1.0. Problems...............................................................................42
2.1.1. Distributions.........................................................................42
2.1.2. Averages..............................................................................43
2.1.3. Distribution of Frequencies...............................................46
2.1.4. Asymptotic Distribution of mth Central Values..............47
2.1.5. The Order Statistic with Minimum Variance...................51
2.1.6. Control Band......................................................................52
2.1.7. Joint Distribution of Order Statistics...............................53
2.1.8. Distribution of Distances..................................................53
2.2. The Distribution of Exceedances..............................................57
2.2.0. Introduction........................................................................57
2.2.1. Distribution of the Number of Exceedances.................58
2.2.2. Moments............................................................................61
2.2.3. The Median.......................................................................63
2.2.4. The Probability of Exceedances as Tolerance Limit...64
2.2.5. Extrapolation from Small Samples..................................67
2.2.6. Normal and Rare Exceedances.......................................69
2.2.7. Frequent Exceedances.....................................................72
2.2.8. Summary............................................................................73
Chapter Three : EXACT DISTRIBUTION OF EXTERMES.............75
3.1. Averages of Extremes................................................................75
3.1.0. Problems.............................................................................75
3.1.1. Exact Distributions............................................................75
3.1.2. Return Periods of Largest and Large Values................78
3.1.3. Quantities of Extremes.....................................................79
3.1.4. Characteristic Extremes....................................................82
3.1.5. The Extermal Intensity Function......................................84
3.1.6. The Mode............................................................................85
3.1.7. Moments..............................................................................87
3.1.8. The Maximum of the Mean Largest Value.....................89
3.2. Extremal Statistics......................................................................94
3.2.0. Problems.............................................................................94
3.2.1. Absolute Extreme Values.................................................94
3.2.2. Exact Distribution of Range..............................................97
3.2.3. The Mean Range...............................................................100
3.2.4. The Range as Tolerance Limit........................................103
3.2.5. The Maximum of the Mean Range..................................106
3.2.6. Exact Distribution of the Midrange.................................108
3.2.7. Asymptotic Independence..................................................110
3.2.8. The Extremal Quotient......................................................111
Chapter Four : ANALYTICAL STUDY OF EXTREMES................113
4.1. The Exponential Type..............................................................113
4.1.0. Problems..........................................................................113
4.1.1. Largest Value for the Exponential Distribution...........113
4.1.2. Order Statistics for the Exponential Distribution........116
4.1.3. L''HÔpital''s Rule...............................................................118
4.1.4. Definition of the Exponential Type...............................120
4.1.5. The Three Classes..........................................................122
4.1.6. The Logarithmic Trend...................................................123
4.1.7. The Characteristic Product.............................................125
4.2. Extreme of the Exponential Type...........................................126
4.2.0. Problems...........................................................................126
4.2.1. The Logistic Distribution................................................126
4.2.2. Normal Extreme, Numerical Values...............................129
4.2.3. Analysis of Normal Extremes........................................136
4.2.4. Normal Extreme Deviates...............................................140
4.2.5. Gamma Distribution..........................................................143
4.2.6. Logarithmic Normal Distribution......................................146
4.2.7. The Normal Distribution as a Distribution Extremes....147
4.3. The Cauchy Type.......................................................................149
4.3.0. Problems.............................................................................149
4.3.1. The Exponential Type and the Existence of Moments..149
4.3.2. Pareto''s Distribution..........................................................151
4.3.3. Definition of the Pareto and the Cauchy Types............152
4.3.4. Extremal Properties...........................................................153
4.3.5. Other Distributions without moments...............................154
4.3.6. Summary..............................................................................155
Chapter Five : THE FIRST ASYMPTOTIC DISTRIBUTION............156
5.1. The Three Asymptotes................................................................156
5.1.0. Introduction..........................................................................156
5.1.1. Preliminary Derivation........................................................156
5.1.2. The Stability Postulate.......................................................157
5.1.3. Outline of Other Derivations..............................................162
5.1.4. Interdependence..................................................................164
5.2. The Double Exponential Distribution..........................................166
5.2.0. Introduction..........................................................................166
5.2.1. Derivations...........................................................................166
5.2.2. The Methods of Cramer and Von Mises...........................170
5.2.3. Mode and Median.................................................................172
5.2.4. Generating Functions............................................................173
5.2.5. Standard and Mean Deviations............................................174
5.2.6. Probability Paper and Return Period..................................176
5.2.7. Comparison with Other Distribution....................................179
5.2.8. Barricelli''s Generalization.....................................................184
5.3. Extreme Order Statistics...............................................................187
5.3.0. Problems.................................................................................187
5.3.1. Distribution of the mth Extreme..........................................187
5.3.2. Probabilities of the mth Extreme........................................189
5.3.3. Generating Functions............................................................192
5.3.4. Cramer''s Distribution of mth Extremes..............................194
5.3.5. Extreme Distances................................................................197
5.3.6. The Largest Absolute Value and the Two Sample Problem .....198
Chapter Six : USES OF THE FIRST ASYMPTOTE...........................201
6.1. Order Statistics from the Double Exponential Distribution.......201
6.1.0. Problems................................................................................201
6.1.1. Maxima of Largest Values..................................................201
6.1.2. Minima of Largest Values...................................................203
6.1.3. Consecutive Modes..............................................................206
6.1.4. Consecutive Means and Variances.....................................208
6.1.5. Standard Errors....................................................................212
6.1.6. Extension of the Control Band...........................................216
6.1.7. The Control Curve of Dick and Darwin............................218
6.2. Estimation of Parameters.............................................................219
6.2.0. Problems...............................................................................219
6.2.1. Exponential and Normal Extremes....................................219
6.2.2. Use of Order Statistics.......................................................223
6.2.3. Estimates for Probability Paper.........................................226
6.2.4. Sufficient Estimation Functions, by B .F. Kimball............229
6.2.5. Maximum Likelihood Estimations, by B. F. Kimball ........231
6.2.6. Approximate Solutions..........................................................232
6.2.7. Asymptotic Variance of a Forecast, by B. F. Kimball......234
6.3. Numerical Examples........................................................................236
6.3.0. Problems.................................................................................236
6.3.1. Floods.....................................................................................236
6.3.2. The Design Flood....................................................................238
6.3.3. Meteorological Examples........................................................241
6.3.4. Application to Aeronautics......................................................245
6.3.5. Oldest Ages..............................................................................246
6.3.6. Breaking Strength...................................................................248
6.3.7. Breakdown Voltage ................................................................249
6.3.8. Application to Naval Engineering...........................................251
6.3.9. An Application to Geology......................................................254
Chapter Seven : THE SECOND AND THIRD ASYMPTOTES................255
7.1. The Second Asymptote......................................................................255
7.1.0. Problems....................................................................................255
7.1.1. Frechet''s Derivation.................................................................255
7.1.2. The Cauchy Type.....................................................................259
7.1.3. Averages and Moments ..........................................................264
7.1.4. Estimation of the Parameters..................................................266
7.1.5. The Increase of the Extremes.................................................269
7.1.6. Generalization.............................................................................270
7.1.7. Applications................................................................................271
7.1.8. Summary.....................................................................................272
7.2. The Third Asymptote..........................................................................272
7.2.0. Introduction.................................................................................272
7.2.1. The Von Mises Derivation........................................................273
7.2.2. Other Derivation.........................................................................276
7.2.3. Averages and Moments of Smallest Values............................280
7.2.4. Special Cases..............................................................................285
7.2.5. The Increase of the Extremes..................................................287
7.2.6. The 15 Relations Among the 3 Asymptotes............................288
7.3. Applications of the Third Asymptote.................................................289
7.3.0.Problems.......................................................................................289
7.3.1. Estimation of the Three Parameters........................................289
7.3.2. Estimation of T재 Parameters...................................................293
7.3.3. Analytical Examples....................................................................298
7.3.4. Droughts.......................................................................................299
7.3.5. Fatigue Failures..........................................................................302
Chapter Eight : THE RANGE......................................................................306
8.1. Asymptotic Distribution of Range and Midrange...............................306
8.1.0. Problems.........................................................................................306
8.1.1. The Range of Minima....................................................................306
8.1.2. Generating Function of the Range...............................................308
8.1.3. The Reduced Range......................................................................309
8.1.4. Asymptotic Distribution of the Midrange....................................311
8.1.5. A Bivariate Transformation..........................................................312
8.1.6. Asymptotic Distribution of the Range.........................................316
8.1.7. Boundary Conditions.....................................................................320
8.1.8. Extreme Ranges.............................................................................321
8.1.9. Summary.........................................................................................324
8.2. Extremal Quotient and Geometric Range.............................................324
8.2.0. Problems........................................................................................324
8.2.1. Definitions......................................................................................325
8.2.2. The Extremal Quotient.................................................................326
8.2.3. The Geometric Range..................................................................327
8.3. Applications..............................................................................................331
8.3.0. Problems.........................................................................................331
8.3.1. The Midrange.................................................................................331
8.3.2. The Parameters in the Distribution of Range............................333
8.3.3. Normal Ranges...............................................................................336
8.3.4. Estimation of Initial Standard Deviation......................................338
8.3.5. Climatological Examples................................................................340
Summary...........................................................................................................345
Bibliography......................................................................................................349
Index..................................................................................................................373