Pioneering Works on Extreme Value Theory: In Honor of Masaaki Sibuya (JSS Research Series in Statistics)

This book presents the state of the art in extreme value theory, with a collection of articles related to a seminal paper on the bivariate extreme value distribution written by Professor Masaaki Sibuya in 1960, demonstrating various developments of the original idea over the last half-century. Written by active researchers, the unique combination of articles allows readers to gain a sense of the excellence of the field, ranging from theory to practice, and the tradition of theoretical developments motivated by practically important issues such as tsunamis and financial crises. The contributions discuss a range of topics, including the parameter estimation of the generalized beta distribution, resampling with the empirical beta copula, and regression analysis on imbalanced binary data, as well as the semiparametric estimation of the upper bound of extrema, the long-term analysis of extreme precipitation over Japanese river basins, and various rules of thumb in hydrology. Preface 6 Contents 9 1 Parameter Estimation of Generalized Beta Distributions and Its Application to a Historical Tsunami Magnitude Dataset 10 1.1 Introduction 10 1.2 Estimation Methods 13 1.2.1 NagaBalaKama Transform for Estimating Shape Parameters 13 1.2.2 Hall and Wang's Empirical Prior for Estimating Location-Scale Parameters 15 1.2.3 Kachiashvili's Iteration for Joint Estimation 16 1.2.4 A New Estimator of the Generalized Beta Distributions 16 1.3 Evaluation and Application to a Historical Tsunami Dataset 18 1.3.1 Evaluation of the New Estimator 18 1.3.2 Fitting the Generalized Beta Distribution to a Historical Tsunami Dataset 29 References 34 2 Resampling Procedures with Empirical Beta Copulas 36 2.1 Introduction 37 2.2 Review on Bootstrapping Empirical Copula Processes 38 2.2.1 Straightforward Bootstrap 39 2.2.2 Multiplier Bootstrap with Estimated Partial Derivatives 40 2.3 Resampling with Empirical Beta Copulas 41 2.3.1 Standard Bootstrap for the Empirical Beta Copula 42 2.3.2 Bootstrap by Drawing Samples from the Empirical Beta Copula 43 2.3.3 Approximating the Sampling Distributions of Rank Statistics by Resampling from the Empirical Beta Copula 44 2.4 Simulation Studies 45 2.4.1 Covariance of the Limiting Process 45 2.4.2 Confidence Intervals for Rank Correlation Coefficients 45 2.4.3 Confidence Intervals for a Copula Parameter 50 2.4.4 Testing the Symmetry of a Copula 50 2.5 Concluding Remarks 54 References 61 3 Regression Analysis for Imbalanced Binary Data: Multi-dimensional Case 63 3.1 Introduction 63 3.2 Quasi-linear Logistic Regression Model and Its Imbalance Limit 65 3.2.1 The Quasi-linear Logistic Regression Model 65 3.2.2 Imbalance Limit 66 3.3 Extension of the Model and Its Copula Representation 67 3.3.1 Detectable Model 67 3.3.2 Copula Representation 68 3.4 The Imbalance Limit of Detectable Models 71 3.5 Examples of Equivariant Predictors 75 3.6 Conclusion 77 References 78 4 An Analysis of Extremes: Semiparametric Efficiency in Regression 79 4.1 Introduction 79 4.2 Modeling Extremes 81 4.3 Adaptive and Efficient Estimation 81 4.3.1 Parametric Models and Efficient Estimators 82 4.3.2 Semiparametric Models and Adaptive and Efficient Estimators 84 4.4 Semiparametric Boundary Parameter Regression with a Known Boundary Structure 84 4.5 Semiparametric Boundary Parameter Regression with an Unknown Boundary Structure 86 4.6 Real-Data Analysis 93 4.6.1 Example 1: Survival Time of Lung Cancer Patients (Two-Sample Models) 93 4.6.2 Example 2: Survival Time of Lung Cancer Patients (calorie) 93 4.7 Conclusion 97 References 98 5 Comparison of AMS and POT Analysis with Long Historical Precipitation and Future Change Analysis Using ``d4PDF'' 100 5.1 Introduction 101 5.2 Data Used 102 5.2.1 Historical Precipitation Observations 102 5.2.2 Large Ensemble of Climate Simulations 103 5.3 Frequency Analysis 104 5.3.1 Block Size of Block Maxima 105 5.3.2 Threshold Selection 105 5.3.3 AMS and POT Analysis of Long Historical Precipitation 108 5.3.4 Considerations Regarding AMS and POT Analyses 113 5.4 Analysis of Climate Simulations 114 5.5 Discussion 117 5.6 Conclusion 118 References 118 6 History and Perspectives of Hydrologic Frequency Analysis in Japan 120 6.1 Introduction 120 6.2 Hydrologic Frequency Analysis Method 121 6.3 Various Probability Distribution Functions 123 6.4 Goodness of Fit 124 6.4.1 SLSC 125 6.4.2 COR, MLL, and AIC 126 6.4.3 Comparison of Goodness-of-Fit Criteria 127 6.5 Resampling Methods for Bias Correction and Stability Analysis 130 6.6 Future Perspectives 131 6.6.1 AMS or PDS 131 6.6.2 Nonparametric Analysis for Large Samples 133 6.6.3 Probability Distribution Functions with Lower and Upper Bounds 137 6.7 Conclusions 138 References 139