Statistical Inference Of The Generalized Pareto Distribution

The generalized Pareto distribution (GPD) was first explicitly introduced byPickands (1975), since then was further studied by more and more researchers. Itwas one of distributions with location, shape and scale parameters. At present,it has been widely used in numerous fields, such as extreme value analysis, fit-ting insurance loss, climatic diagnosis, reliability study, financial risk managementfields and so on. As one of important branches of extreme value theory, the GPDbecomes new hot point in extreme value framework. In recent years, extremevalue theory is used to research not only maximum and minimum data, but alsothe data over a given threshold. The GPD just could fit that data excellently,so it has become mainstream in extreme value theory and obtains the universalapproval. It is of theoretical significance and practical value to study statisticalinference of the GPD.In this thesis, we mainly discuss with the statistical inference of the three-parameter GPD which includes parameter estimation, hypothesis test and appli-cation in hydrology, financial risk management and reliability fields.Firstly, the estimation of the three-parameter GPD is concerned. While tra-ditional methods, such as the maximum likelihood (ML), the methods of moments(MOM) and the probability weighted moments (PWM) have been extensively ap-plied, there are several problems and limitations when they are used. The MLmethod may be inapplicable whenever the algorithm used for estimating the pa-rameters fails to converge. The MOM and PWM estimators require restriction forthe shape parameter of the GPD. Most of those methods provide estimators forthe two-parameter GPD with full observation, but censored data often appearsin hydrological, meteorological and financial fields. Estimating methods with cen-sored data are diferent from full observation. Researchers, while concentrating on estimated methods with censored data, applied the procedures only to the gener-alized extreme value distribution. To solve these problems, alternative methodshave been paid much attention. We provide several estimated techniques, includ-ing methods of moments, modified methods of moments, probability weightedmoments, generalized partial probability weighted moments, maximum likelihoodestimation, least squares estimation, approximated generalized least squares esti-mation, L-and LH-moments, partial L-and partial LH-moments methods, alsosummarize advantages and disadvantages of those estimated methods. In addition,we compare the estimation precision among those estimated methods by MonteCarlo simulation.Secondly, the hypothesis test of the three-parameter GPD is concerned. Forthreshold selection, the too large threshold may result in large estimated variance,while the too small threshold may lead to biased estimation and the excesses overthe threshold which are not follow the GPD. In order to choose an acceptablethreshold, we give graphic and calculated methods. The graphic methods containempirical mean excess function plot and discrimination method of estimation sta-bility. However, sometimes could not find the threshold accurately by the graphicmethods in practice, so study calculated method of basing on exponential regres-sion model to select the threshold. For goodness of fit test, also provide graphicand calculated methods. Using the graphic method verdict sample whether comesfrom the GPD is to study the linear relation of points in Q-Q plot. The method issubjective to large extent and is lack of quantitative standard, so discuss the cal-culated methods including correlation coefcient R2test, χ2test, W2and A2testand give theoretical basis and test steps. In addition, we add tables of percentagepoints for W2and A2goodness-of-fit statistics with diferent shape parameter bythe PWM and partial L-moments methods. The graphic and calculated methodsshould be used together for the test. Finally, the application of the three-parameter GPD is concerned. The GPDis applied in hydrology and financial risk management fields to analyze annualmaximum rainfall depth and return of Shanghai composite index, and forecast themaximum rainfall depth and maximum investment loss. In addition, we comparethe estimation precision of diferent estimated methods and fitting performance ofdiferent distribution by Monte Carlo simulation. The GPD is also applied in relia-bility field. We supply reliability formulas for charging system once time charging.Furthermore, simulate real charging process of the system and compare the sim-ulated with calculated reliability of system once time charging. Consequently,simulation verifies accuracy of the formulas and provides theoretical basis.