In Part Ⅰ the concept of the general regular variation of n-th order is proposed and its construction is discussed. The uniqueness of the standard expression and the higher order regularity of the auxiliary functions...In Part Ⅰ the concept of the general regular variation of n-th order is proposed and its construction is discussed. The uniqueness of the standard expression and the higher order regularity of the auxiliary functions are proved.展开更多
In this part II the fundamental inequality of the third order general regular variation is proved and the second order Edgeworth expansion of the distribution of the extreme values is discussed.
Logarithmic general error distribution is an extension of lognormal distribution. In this paper, with optimal norming constants the higher-order expansion of distribution of partial maximum of logarithmic general erro...Logarithmic general error distribution is an extension of lognormal distribution. In this paper, with optimal norming constants the higher-order expansion of distribution of partial maximum of logarithmic general error distribution is derived.展开更多
Cyber losses in terms of number of records breached under cyber incidents commonly feature a significant portion of zeros, specific characteristics of mid-range losses and large losses, which make it hard to model the...Cyber losses in terms of number of records breached under cyber incidents commonly feature a significant portion of zeros, specific characteristics of mid-range losses and large losses, which make it hard to model the whole range of the losses using a standard loss distribution. We tackle this modeling problem by proposing a three-component spliced regression model that can simultaneously model zeros, moderate and large losses and consider heterogeneous effects in mixture components. To apply our proposed model to Privacy Right Clearinghouse (PRC) data breach chronology, we segment geographical groups using unsupervised cluster analysis, and utilize a covariate-dependent probability to model zero losses, finite mixture distributions for moderate body and an extreme value distribution for large losses capturing the heavy-tailed nature of the loss data. Parameters and coefficients are estimated using the Expectation-Maximization (EM) algorithm. Combining with our frequency model (generalized linear mixed model) for data breaches, aggregate loss distributions are investigated and applications on cyber insurance pricing and risk management are discussed.展开更多
文摘In Part Ⅰ the concept of the general regular variation of n-th order is proposed and its construction is discussed. The uniqueness of the standard expression and the higher order regularity of the auxiliary functions are proved.
基金This work supported by the National Natural Science Foundation of China （Grand No. 10071003）
文摘In this part II the fundamental inequality of the third order general regular variation is proved and the second order Edgeworth expansion of the distribution of the extreme values is discussed.
基金Supported by the National Natural Science Foundation of China(11171275)the Natural Science Foundation Project of CQ(cstc2012jj A00029)the Doctoral Grant of University of Shanghai for Science and Technology(BSQD201608)
文摘Logarithmic general error distribution is an extension of lognormal distribution. In this paper, with optimal norming constants the higher-order expansion of distribution of partial maximum of logarithmic general error distribution is derived.
文摘Cyber losses in terms of number of records breached under cyber incidents commonly feature a significant portion of zeros, specific characteristics of mid-range losses and large losses, which make it hard to model the whole range of the losses using a standard loss distribution. We tackle this modeling problem by proposing a three-component spliced regression model that can simultaneously model zeros, moderate and large losses and consider heterogeneous effects in mixture components. To apply our proposed model to Privacy Right Clearinghouse (PRC) data breach chronology, we segment geographical groups using unsupervised cluster analysis, and utilize a covariate-dependent probability to model zero losses, finite mixture distributions for moderate body and an extreme value distribution for large losses capturing the heavy-tailed nature of the loss data. Parameters and coefficients are estimated using the Expectation-Maximization (EM) algorithm. Combining with our frequency model (generalized linear mixed model) for data breaches, aggregate loss distributions are investigated and applications on cyber insurance pricing and risk management are discussed.
