Finite-Size Scaling In Extreme Statistics

With the rapid development of science and technology, the theory and method of Extremevalue statistics analysis are continuously developed and improved. Extreme value statistics (EVS)has attracted many interests in engineering, finance, and environmental sciences where extremeevents may have disastrous consequences. Unfortunately, the application of EVS is hampered bythe cost of acquiring quality statistics, EVS is derived from the extremes of subsets of a data set,requiring abundant data for reasonable statistics. Data analysis is further complicated while theEVS limit distributions may be known, the convergence with increasing sample size becameslow. Clearly, a detailed finite-size (FS) analysis providing the convergence rate and shapecorrections to the limit distribution is much needed.Chapter 2 introduces the basic theory of the EVS limit distribution, it obtains thedistribution functions of depending on order statistics, meanwhile three kinds of EVS limitdistribution and the generalized extreme value distribution are obtained．As far as Gumbeldistribution, its attractive field was found. And then it introduces threshold models of thegeneralized Pareto distribution. At the same time the means of parameter estimation has beengiven．The most important method is Likelihood estimation.Chapter 3 systematically discusses a renormalization group method. And the deviation fromthe limit distributions in extreme value statistics arising due to the finite size of data sets isstudied. It shows that the universality classes are subdivided according to the exponent of thefinite-size convergence. With temperature data, the approach provides an intuitive and accessiblesummary of the mathematical results for the leading FS correction, including the explicit formsof the shape corrections (scaling functions). For example, we take extreme temperatures aboutNanjing station from 1960 to 2005, the gap is analyzed between the shape corrections withsimulations.Chapter 4 emphases on the weighted distribution and the goodness of fit by Kolmogorovtest. Taking normal distribution as example, we fit a set of data with weighted distribution whichis better than with single normal distribution. Weighted distribution often fits well in the centralpart, but it is not the case in the tail．In this paper, weighted distribution can reflect the tail bytaking extreme micro herm date form Nanjing station. The gap which has been analyzed amongall the above distribution shrink.