Research On The Extended Composite Model And Its Application

Distribution theory is one important part of statistical theory.With the deepening of research and the promotion of modern computer technology,more and more statistical distributions with flexible forms have been proposed in different disciplines for application.And until now,research results about statistical distributions have been continued to bring out.The composite distribution proposed by Cooray&Ananda(2005)is one important development of statistical distribution theory in recent years.This theory has been improved and extended continuously because of its simple idea and flexible forms.This paper proposes a new kind of composite distribution,which can be left skewed,symmetrical and right skewed based on different parameters.This makes it possible for applications in a wider scope and in more fields.This paper mainly discusses the expansion and application of composite distribution model.In the introduction section,the research background and significance were discussed.Literature review of the income distribution,city size and the insurance claims fitting were also presented.Research design and possible break-through of this dissertation were outlined.This dissertation can be divided into five chapters:The first chapter discusses conventional composite distribution theory;The extent of composite distributions are discussed in detail in the second chapter;The third chapter discusses the parameter estimation of the extended composite distribution;The fourth chapter assesses the test of fitness of the extended composite distribution;And in the fifth chapter,the extended composite distribution model is applied to the study of income,county population size and insurance losses.Finally,conclusion and prospects is addressed.The main contents of each chapter are:Chapter 1 introduces the composite models,which can be divided into two-restricted mixing weights and unrestricted mixing weights.Restricted mixing weights composite models include composite Lognormal-Pareto model proposed by Cooray&Ananda(2005),and composite Lognormal-Weibull model proposed by Ciumara(2006)?Cooray(2009).There are two kinds of composite models with mixing weights.One owed to Scollnik(2007),Scollnik(2007)firstly designed the first and the second composite Lognormal-Pareto model,other such models include composite Weibull-Pareto model and composite Weibull-Pareto II model(Scollnik&Chen,2012).Another kind of composite model is Nadarajah&Bakar model(Nadarajah&Bakar,2014),which includes the Lognormal-Pareto II model and the Lognormal-Burr model(Nadarajah&Bakar,2014).But actually these two kinds of composite models with mixing weights are the same.Chapter 2 introduces the extended composite models,which also can be divided into two-restricted mixing weights and unrestricted mixing weights.Different with the original composite models,this new composite models choose the Reverse Pareto distribution as the first part,Lognormal distribution and Weibull distribution as the second part respectively.Thus four kinds of new composite models were proposed:Type I Reverse Pareto-Lognormal composite distribution(RPLC-I),Type I Reverse Pareto-Weibull composite distribution(RPWC-I),Type II Reverse Pareto-Lognormal composite distribution(RPLC-?)and Type ? Reverse Pareto-Weibull composite distribution(RPWC-?).Then a comprehensive treatment of the mathematical properties of the new distribution including expressions for the rth moment,skewness and kurtosis is provided.Also it showed that these four distributions can be left skewed,symmetrical and right skewed based on different parameters.Chapter 3 accesses the parameter estimation of the four extended composite models,RPLC-I,RPWC-?,RPLC-II and RPWC-?.First,four kinds of estimation methods,the method of maximum likelihood(MLE),the method of moments(MM),the method of nonlinear least squares(NLS)and the method of Bayes(Bayes),are proposed.Also the expected Fisher information matrix and the observed Fisher information matrix are obtained.Then the Monte Carlo simulation was carried to show the performance of different estimation methods,the results can be as a guideline for practical use.Chapter 4 focuses on the model validity of the extended composite models.The probability plot,the correlation coefficient,and the goodness-of-fit tests are provided.The first two methods are based on some transformation of the cumulative distributions.The goodness-of-fit tests,Kolmogorov-Smimov test statistics(K-S),the Cramer-von Mises test statistics(C-vM),and the Anderson-Darling test statistics(A-D),are based on the 'distance' between the EDF constructed from the data and the cdf of the fitted models.Tables of critical values are presented for the correlation coefficients method and three kinds of goodness-of-fit tests.Also the power comparisons were made to compare the performance to these test methods.Chapter 5 illustrates the use of the extended composite models.Three real data sets from the practical area are analyzed.First,the Chinese rural household income data are from China Health and Nutrition Survey(CHNS),it is found that RPLC-II distribution can properly fit this data than other distributions.Moreover,the Gini coefficients,theGeneralized Entropy index,the Theil index?the Atkinson index?the Bonferroni index and the Zenga index of Chinese rural household income are also computed based on RPLC-II distribution.Second,the Chinese county size data is from the national population census,it is also found that the RPLC-? distribution can fit it best among all the proposal distributions.Furthermore,the six inequality indexes are also computed.The third data is the amount paid to settle and close a claim from a large US property and casualty insurer for private passenger.It is also found that he RPLC-II distribution can fit it best among all the proposal distributions.On this basis,the Risk measurement indexes such as Value at Risk(VaR)and Tail Value at Risk(TVaR)are calculated.Finally,conclusions and prospects are given in the last section.Both the theoretical and practical findings are presented.Furthermore,some existing problems for current research and further improvement are also proposed.The main work and major findings of this dissertation are:1.In this dissertation,two kinds of the extended composite models,the Type I and Type ? extended composite models,are proposed.These models were developed based on the method suggested by Cooray&Ananda(2005)by specifying two of the model parameters in terms of other parameters of the models,thus,reducing the number of parameters to be estimated.Weighted Reverse Pareto distribution forming as the head of the model,four models,Type I Reverse Pareto-Lognormal composite distribution(RPLC-?),Type I Reverse Pareto-Weibull composite distribution(RPWC-I),Type ?Reverse Pareto-Lognormal composite distribution(RPLC-?)and Type II Reverse Pareto-Weibull composite distribution(RPWC-?)are proposed.And all the four models can be left skewed,symmetrical and right skewed based on different parameters.2.Details of the given models property,and inferential theory are derived.Four parameter estimation methods,the method of maximum likelihood(MLE),the method of moments(MM),the method of nonlinear least squares(NLS)and the method of Bayes(Bayes),are proposed.Simulation results show that in the case of RPLC-?,MM shows the best,then Bayes;in the case of RPWC-I,MLE shows the best,then MM or NLS2;in the case of RPLC-II,MLE shows the best,then MM or NLS1;and in the case of RPWC-?,MLE and MM show best,then NLS1 or NLS2.3.The extended composite models can be test by probability plotting,correlation coefficient,or goodness-of-fit tests.The power comparison of correlation coefficient tests results show that the power of RPLC-? and RPWC-? can reach over 0.6,and power of RPLC-I and RPWC-I can reach 1.0 when the sample size reach 1000.The power comparison of goodness-of-fit tests results show that the power of RPLC-I and RPWC-?can reach nearly 0.9,the power of RPLC-II can reach over 0.9,and the power of RPWC-I can reach 1.0.Overall,among the three goodness-of-fit tests,A-D test has the highest power,then C-vM test,and the K-S test has the lowest power.4.The extended composite models can be used for modelling income distribution,county size distribution,and insurance losses.Moreover,such inequality index as the Gini coefficient,the Generalized Entropy index,the Theil index,the Atkinson index?the Bonferroni index and the Zenga index,can be obtained based on the extended composite models,which can be used to analyze the inequality.In addition,based on the extended composite distribution,it is also possible to calculate the risk measures such as Value-at-Risk(VaR)and Tail Value at Risk(TVaR),so as to facilitate the timely control of risks by enterprises.