Higher-order Expansions On Distributions Of Extremes

Extreme random events, such as. the 9.11 terrorist attacks in 2001, the Indian Ocean tsunami in 2004, the Hurricane Kartrina in 2005, the Wenchuan earthquake in 2008, the Haiti’s earthquake in 2010. the Japan earthquake and the Fukushima nuclear leak in 2011. the Malaysia’s civil aviation disaster in 2014. and the global financial crisis and the economic bubbles, were catastrophic tragedies to human beings and the environments. Those arc rare events with extremely small probabilities, new probability and statistics tools beyond the classical central limit theorems are needed to characterize those extreme random phenomena.Extreme value theory may provide an efficient way to achieve this goal. The study of extreme random phenomena was from the seminal work of Fisher and Tippctt (1928) by investigating the types of limiting distributions of extreme order statistics. Nowadays, standard and excellent monographs in extreme value theory are Lcadbetter et al. (1983). Galambos (1987). Embrechts et al. (1997), de Haan and Ferreira (2006). Resnick (2007), among others. With rich theoretical support, extreme value theory has been widely used in a variety of areas like financial econometrics, insurance, engineering and environmental sciences, where the analysis of asymptotics play an important role providing the behavior of distributional tail.Asymptotic behavior can give many uncovered performance and relation-ships of risk measures, ruin probabilities, and distributional tails of random sum-mation by their first-order approximations. The first-order approximation may be the first step to understand of the tail behaviors of risks, ruin probabilities, subordinated distributions. The second step is the second-order approximations which may provide more precise asymptotic information, and determine the con-vergence rates of the first-order approximations. The study of higher-order ap-proximations in extreme value theory has received more and more attention in the recent literature.Empirical studies show that most real data sets with asymmetries and heavy-tailedness can be modeled well by skew distributions and heavy-tailed distribu-tions. Asymptotic properties of those distributions such as the distributional tail behavior, the limiting distribution of extreme and its associated higher-order dis-tributional expansions may provide precisely information for further study. This thesis is devoted to the investigation of higher-order asymptotics and convergence rates of extremes of random sequences following skew distributions, exponential tail distributions and bivariate Gaussian distributions.Part Ⅰ focuses on the asymptotics of two kinds of skew distributions, i.e., the skew-normal and logarithmic skew-normal distribution. Skew distribution families are flexible parametric families with additional parameters allowing to regulate skewness and tails. It is especially important to fit real data sets such as from environment, insurance and finance. For univariate random sequences following respectively the skew-normal and the logarithmic skew-normal distri-bution, probabilistic properties such as the Mills’ type inequalities, Mills’type ratios, and the asymptotics including the higher-order expansions of the normal-ized order statistics and the moments of the normalized maxima are studied. The pointwise convergence rates of the distributions and the moments of the normalized maxima to their limits are also considered.Part Ⅱ considers the second-order asymptotics of convolution of distributions with exponential tail. Asymptotic behavior of convolution of sum of random variables with heavy tail are widely studied in many fields such as queueing theory, branching processes, renewal theory, finance and insurance. Cline (1986) derived the tail asymptotics of convolution of distributions with exponential tail. which are constructed by regularly varying functions. Based on the work of Cline (1986), Part Ⅱ constructs exponential tail distributions by using second-order regularly varying functions (2RV). Using properties of 2RV. the second-order tail asymptotics of convolution of those distributions are established, respectively.Part Ⅲ, the last part of this thesis, is interested in the asymptotics of ex-tremes of Husler-Reiss model and its extension. The Husler-Reiss model is formed by an independent bivariate Gaussian random triangular arrays. The limiting dis-tribution of maxima was studied by the seminal paper of Husler and Reiss (1989) provided that the so-called Husler-Reiss condition holds. and the limit distribu-tion is the max-stable Hiisler-Reiss distribution. The limit distribution showed that the components in maxima can be asymptotically dependent. By the refined Hiisler-Reiss conditions, Hashorva et al. (2014) established higher-order distribu- tional expansions of maxima. For Husler-Reiss model and its extension, Part Ⅲ totally finishes the following work:Establishing the uniform convergence rates of distributions of the normalized maxima to the max-stable Husler-Reiss distri-bution under the second-order Husler-Reiss condition given by Hashorva et al. (2014):Extending the Husler-Reiss model to the non-identically distributed case, where each vector of the nth row follows from a bivariate Gaussian distribution with correlation coefficient being a monotone, continuous function of i/n. For this extended model, the first and second-order asymptotics of distributions of extremes are established, respectively. Estimation and asymptotic distributions of parameters in the extended Hiisler-Reiss model are considered. Some simula-tion study and empirical data analysis are provided to support the main results: Deriving the limiting distributions of the normalized maxima and the normalized minima of Husler-Reiss model, which shows that the maxima and the minima are asymptotic independent. Furthermore, under the refined Husler-Reiss condition, the second-order expansions of the joint distributions of the normalized maxima and the normalized minima are established.The main contributions of the thesis are as follows.1. The higher-order expansions of distributions and moments of maxima from SN(λ) and LSN(λ) samples are established.2. Using properties of 2RV functions, the second-order tail asymptotics of con-volution of distributions with exponential tails are derived.3. For Husler-Reiss model, the uniform convergence rates of maxima, the higher-order expansions of joint distributions of maxima and minima are presented. For the extended Husler-Reiss model with non-identically distribution, the first and second-order asymptotics of distributions of normalized maxima are established. Furthermore, the parameters of correlation coefficient are esti-mated through maximum likelihood method and the asymptotic normality of those estimators are derived, which can be employed to test the Husler-Reiss condition.