# 臺灣博碩士論文加值系統

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 以Edgeworth展開式修正來自一般化柏拉圖分配其L-moments統計量的漸近分配
 VaR is an important risk management tool in finance and economics, which focuses on tail estimation, especially when financial loss follows a heavy-tailed distribution. It is recently found that Generalized Pareto Distributions (GPDs) from Extreme Value Theory (EVT) perform better than certain existing models, e.g., the lognormal distribution, in describing excessive losses. L-moments, expectations of certain linear combinations of order statistics, can describe a probability distribution well like the conventional moments and are more robust to outliers, because their power measurements are also in cumulative distribution functions. In this paper, we obtain certain properties of the unbiased estimators of the L-moments, namely the UMVU property. We then obtain Edgeworth expansions for the sampling distributions of L-moments when sampling from a generalized Pareto distribution. These Edgeworth approximations are expected to give better approximations than the CLT’s normal approximation, because these Edgeworth approximations add more information about loss severity to describing the distribution through some adjustments of skewness and kurtosis. We will also show how the scale and shape parameters of a GPD may affect the fitting performance of the derived Edgeworth approximation of L-moments, by some numerical analyses. Finally we will illustrate the method using real data from an insurance company.
 1.Introduction ------------------------------------12.Extreme Value Theory ----------------------------4 2.1The Generalized Extreme Value Distributions ---4 2.2 The Generalized Pareto Distribution ----------73.L-moments and Probability-weighted Moments Estimation -------------------------------------10 3.1 L-moments -----------------------------------10 3.2 Probability-weighted Moments and Their Unbiased Estimators -------------------------11 3.3 Asymptotic Distributions of Sample PWMs -----134.Estimations and Edgeworth Expansions of Sampling Distributions of L-moments ---------------------14 4.1 L-moments and UMVUEs-------------------------14 4.2 Asymptotic Sampling Distributions of Sample L-moments -----------------------------------18 4.3 Edgeworth Approximations of the Sampling Distributions of L-moments ------------------195.Numerical Illustration -------------------------276.Data Analysis ----------------------------------357.Conclusions ------------------------------------39References ---------------------------------------40Appendix A ---------------------------------------43Appendix B ---------------------------------------45Appendix C ---------------------------------------50
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