Statistics of Extremes Book

Chapter 1	Aims and Tools	1
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" Distributions	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 2	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	55
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 3	Exact Distribution of Extremes	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.	Quantiles of Extremes	79
3.1.4.	Characteristic Extremes	82
3.1.5.	The Extremal 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 of Extremes	110
3.2.8.	The Extremal Quotient	111
Chapter 4	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'Hopital'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.	Extremes of the Exponential Type	126
4.2.0.	Problems	126
4.2.1.	The Logistic Distribution	126
4.2.2.	Normal Extremes, 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 of 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 5	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 Distributions	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 6	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	229
6.2.5.	Maximum Likelihood Estimations	231
6.2.6.	Approximate Solutions	232
6.2.7.	Asymptotic Variance of a Forecast	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.	Applications to Naval Engineering	251
6.3.9.	An Application to Geology	254
Chapter 7	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 Derivations	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 Two Parameters	293
7.3.3.	Analytical Examples	298
7.3.4.	Droughts	299
7.3.5.	Fatigue Failures	302
Chapter 8	The Range	306
8.1.	Asymptotic Distributions of Range and Midrange	306
8.1.0.	Problems	306
8.1.1.	The Range of Minima	307
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