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Authors: Shahzad, Mirza Naveed
Keywords: Natural Sciences
Probabilities & applied mathematics
Issue Date: 2016
Publisher: Quaid-i-Azam University Islamabad, Pakistan
Abstract: The purpose of extreme value frequency analysis is to analyze past records of extremes to estimate future occurrence probabilities, nature, intensity and frequency. It is only possible if most suitable probability distribution is employed with proper estimation method. Many probability distributions and parameter estimation methods have been proposed in last couple of decade, but the quest of best fit has always been of concern. In the continuity of this dimension, the fundamental aim of this dissertation is to model the extreme events by proper probability distributions using the most suitable method of estimation. This objective is achieved by reviewing and employing the concept of L- and TL-moments and quadratic rank transmutation map. The L- and TL-moments of some specific distributions are derived, and parameter estimation is approached through the method of L- and TL-moments. In this study three transmuted and two double-bounded transmuted distributions are developed and proposed with their properties and applications. Moreover, the generalized relationships are also established to obtain the properties of the transmuted distributions using their parent distribution. In the first part of the dissertation, it is observed that the Singh Maddala, Dagum, and generalized Power function distribution are suitable candidates for extreme value frequency analysis, as these densities are heavy-tailed in their range. In literature, the theory of L- and TL-moments is considered best and extensively used for such analysis. Therefore, the L- and TL-moments are derived, and the parameters of these densities are estimated by employing the method of L- and TL-moments. These estimation methods are compared with the method of maximum likelihood estimation and method of moments using some real extreme events data sets. Simulation studies have also been carried out for the same purpose. In these studies, superiority of the method of L- and TL-moments has been justified. In the second part of the dissertation, three heavy-tailed, flexible and versatile distributions are introduced using the quadratic rank transmutation map to model the extreme value data. The proposed distributions are the transmuted Singh Maddala, transmuted Dagum and transmuted New distribution. The mathematical properties viiiand reliability behaviors are derived for each of the proposed transmuted distribution. The densities of order statistics, generalized TL-moments, and its special cases are also studied. Parameters are estimated using the method of maximum likelihood estimation. The appropriateness of the transmuted distributions for modeling extreme value data is illustrated using some real data sets. The empirical results indicated that the proposed transmuted distributions perform better as compared to the parent distributions. In literature, continuous double-bounded data is fairly popular. However, it is quite unrealistic to analyze such kind of data using normal theory models. This type of data is also targeted, and two new double-bounded distributions have been introduced, in the third part of the dissertation. These developed distributions termed as transmuted Kumaraswamy and transmuted Power function distribution. The most common mathematical properties are derived, and it has been observed that the hazard rate function have either increasing or bathtub shaped for these distributions. The method of maximum likelihood estimation is employed for the parameter estimation and the construction of the confidence intervals. The application and potential of these distributions are investigated using real data sets. Comparatively, proposed double bounded transmuted distributions performed better than their parent distributions in real applications. Finally, it has already been proved that transmuted distributions are better than their parent distributions. But directly dealing with the transmuted density is complicated and exhaustive especially for order statistics analysis. To make it simple, the relationships between transmuted and parent distributions are established for the single and product moments of order statistics. In addition, the generalized TL- moments of the transmuted distribution and its special cases are derived using single moments of the parent distribution. The established relationships are used for parameter estimation, and a simulation study is also carried out to investigate the behavior of the estimators. Moreover, the transmuted and parent distributions relationships are illustrated through two well-known distributions and two real data sets. Furthermore, it can be claimed on the base of established results; now it is quite convenient to find the moments of order statistics, parameter estimates and especially generalized TL-moments for transmuted distributions.
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