Extreme Value Theory Of Generalized Modified Skew Slash Distribution And Its Related Distribution

In statistics,the norrmal distribution is undoubtedly one of the most widely used distributions,but it was found that many financial and meteorological data usually exhibited certain skew characteristics and did not strictly obey the normal distribution.In 1985,the skew normal distribution was proposed by Azzalini,which fitted the skew data more accurately and attracted wide attention from scholars.Since then,the skew slash,distribution has been proposed and gradually developed on this basis.Because the skew slash distribution has a heavier tail than the skew normal distribution,it is often used in the model fitting of heavy-tailed financial data.To describe the characteristics of the data more accurately and provide more models to choose from,this paper proposes a generalized modified skew slash distribution,and studies its tail dependence.In addition,the extreme value and tail dependence of the alpha skew normal distribution are theoretically deduced.Finally,combining the slash distribution and the Lomax distribution,the slash Lomax distribution,is proposed,and the linear regression model in which the error terms follow the slash Lomax distribution is studied.The main research results are as follows:(1)Based on the skew slash distribution and the gamma distribution,a new distribution is proposed,which is called the generalized modified skew slash distribution.After theoretical deduction,a series of basic properties such as probability density function,moment,expectation,and variance are obtained.The curves of the probability density function are used to illustrate that the generalized modified skew slash distribution has the good characteristics of flexible kurtosis and heavy tail.Subsequently,the generalized modified skew slash distribution is extended into the multi-dimensional,and the tail dependence of the distribution is studied under the two-dimensional case,and asymptotically dependent on the tail is derived.(2)Considering that extreme value theory has now been widely used in fields such as weather forecasting,architectural design,and market risk,as well as the wide application of skewed distribution families in practice.This paper conducts extreme value theoretical research on the alpha skew normal distribution proposed by Elal-Olivero in 2010.Through theoretical derivation,it is obtained that the extreme value distribution of the alpha skew normal distribution belongs to the Gumbel distribution,and the corresponding normalization constants in different situations are also given.In addition,the theoretical derivation of the tail dependence of the two-dimensional alpha skew normal distribution is carried out.(3)The slash distribution is further extended and supplemented.According to the stochastic representation of slash distribution,the slash Lomax distribution is proposed after combining the Beta distribution and Lomax distribution.The basic properties of slash Lomax distribution,such as probability density function,moments,and Reyi entropy are studied.The parameter estimations of this distribution through the ECM algorithm are discussed.In addition,the linear regression model when the error terms follow the slash Lomax distribution is studied.In the application,two sets of actual data are used to verify the validity and practicability of the slash Lomax distribution and its regression model.