This book is a blend of rigorous probability theories and interesting statistical methodologies. It contains two parts. In Part I we propose two new asymmetric kernels called Birnbaum-Saunders and Lognormal kernels. These two kernels can be applied to the estimation of the probability density functions of ultra-high frequency financial data. In Part II we study in detail the domain of attraction (DOA) approach for the estimation of extreme value index (EVI). We apply both the parametric approach and the DOA approach to estimate EVI and the extreme quantile. We conclude that the DOA approach is more flexible than the parametric approach, whereas in the DOA approach there is no single estimator dominates all the other estimators.

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