Goodness-of-fit tests for extreme value distributions

This study concentrates on the Goodness-of-fit (GoF) test for the extreme value distributions. The distributions involved are Generalized Extreme Value (GEV) Type-I, Type-II and Type-III distributions. In this study, the types of GoF tests involved are the graphical plots as well as the statistical tests. In the graphical plot, the existing QQ plot is built based on the quantiles of the hypothetical and empirical distributions. However, the QQ plot suffers from the deviation at the tail of the distribution which particularly occurs very often for the case of heavy tailed distributions. In order to reduce the deviation, the conditional quantiles is recommended. The conditional quantiles plots the end points of the hypothetical and empirical distributions closer to each other. In addition, the alternative plot suggested is hybrid plot. Unlike the QQ plot which plots the original values of the quantiles, the hybrid plot illustrates the quantiles deviation between the hypothetical and empirical values. Moreover, the hybrid plot lets several statistical models to be plotted into a single graph. These plots are done in a graph because the degree of fit for different statistical models can be visually compared. This is because the horizontal axis is restricted between 0 to 1 for any statistical distribution. The parameters of GEV Type-I, Type-II and Type-III are estimated by maximum likelihood estimation (MLE). The statistical tests involved in the GoF test are Anderson-Darling (AD), Cramer-von Mises (CVM), Zhang Anderson-Darling (ZAD), Zhang Cramer-von Mises (ZCVM) and Shimokawa (Ln) tests. To determine the most powerful statistical test, the critical values of these statistical tests are generated first. Then, the reliability of the critical values are validated by the power study. If the rejection rate of the critical value is close to the respective significance level, that particular critical value is reliable. In addition, it is of interest to make use of the critical values developed by other researchers. These critical values are done for GEV distribution. These critical values were generated from AD, ZAD and Ahmad tests. For this study, they are labelled as AD-GEV, ZADGEV and Ahmad-GEV respectively. These critical values are tested for reliability as well. The power of the statistical tests are examined by the power study as well. Next, to evaluate the power, the alternative distributions are fitted to the extreme value distribution model. Based on the alternative distributions, the most powerful test should be able to produce the highest rejection rate. The results for graphical plot show that conditional quantiles plot is better than the traditional quantiles plot to illustrate the agreement between two identical distributions as well as the discrepancy between two different distributions. Besides,for the statistical tests, the results state that the AD test is the most powerful test for GEV Type-I. For GEV Type-II, the most powerful test are devided according to the cluster of sample size n. The AD test can generally be used for cluster n=15 to 17, while the ZAD test is powerful for the cluster n=18 to 49. The cluster of n=50 to 100 has AD-GEV test as the powerful test. Besides, for GEV Type-III,the ZAD test is generally powerful for cluster n= 18 to 100, but for cluster, n=15 to 17, the ZCVM test is more powerful. In the application part, two types of data were used. The first type is the data that was collected from extreme value distribution while the second type is the data that is normally distributed. The extreme value distribution models are fitted to both types of data. The data that is distributed according to the extreme values distribution is used to verify the agreement between the extreme value distribution and the extreme value distribution model. On the other hand, data that is normally distributed is employed to verify that extreme value distribution model does not fit the non extreme value distribution. The result signifies that the findings in graphical and statistical method of GoF are applicable.

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