Although advanced statistical models have been proposed to fit complex data better,the advances of science and technology have generated more complex data,e.g.,Big Data,in which existing probability theory and statist...Although advanced statistical models have been proposed to fit complex data better,the advances of science and technology have generated more complex data,e.g.,Big Data,in which existing probability theory and statistical models find their limitations.This work establishes probability foundations for studying extreme values of data generated from a mixture process with the mixture pattern depending on the sample length and data generating sources.In particular,we show that the limit distribution,termed as the accelerated max-stable distribution,of the maxima of maxima of sequences of random variables with the above mixture pattern is a product of three types of extreme value distributions.As a result,our theoretical results are more general than the classical extreme value theory and can be applicable to research problems related to Big Data.Examples are provided to give intuitions of the new distribution family.We also establish mixing conditions for a sequence of random variables to have the limit distributions.The results for the associated independent sequence and the maxima over arbitrary intervals are also developed.We use simulations to demonstrate the advantages of our newly established maxima of maxima extreme value theory.展开更多
There continues to be unfading interest in developing parametric max-stable processes for modelling tail dependencies and clustered extremes in time series data.However,this comes with some difficulties mainly due to ...There continues to be unfading interest in developing parametric max-stable processes for modelling tail dependencies and clustered extremes in time series data.However,this comes with some difficulties mainly due to the lack of models that fit data directly without transforming the data and the barriers in estimating a significant number of parameters in the existing models.In thiswork,we study the use of the sparsemaxima ofmovingmaxima(M3)process.After introducing random effects and hidden Fréchet type shocks into the process,we get an extended maxlinear model.The extended model then enables us to model cases of tail dependence or independence depending on parameter values.We present some unique properties including mirroring the dependence structure in real data,dealing with the undesirable signature patterns found in most parametricmax-stable processes,and being directly applicable to real data.ABayesian inference approach is developed for the proposed model,and it is applied to simulated and real data.展开更多
文摘This discussion reviews the paper by Zhengjun Zhang in the context of broader research on multivariate extreme value theory and max-stable processes.
基金partially supported by NSF-DMS-1505367 and Wisconsin Alumni Research Foundation#MS N215758partially supported by National Science Foundation NSF-DMS-1505367 and NSFDMS-2012298.
文摘Although advanced statistical models have been proposed to fit complex data better,the advances of science and technology have generated more complex data,e.g.,Big Data,in which existing probability theory and statistical models find their limitations.This work establishes probability foundations for studying extreme values of data generated from a mixture process with the mixture pattern depending on the sample length and data generating sources.In particular,we show that the limit distribution,termed as the accelerated max-stable distribution,of the maxima of maxima of sequences of random variables with the above mixture pattern is a product of three types of extreme value distributions.As a result,our theoretical results are more general than the classical extreme value theory and can be applicable to research problems related to Big Data.Examples are provided to give intuitions of the new distribution family.We also establish mixing conditions for a sequence of random variables to have the limit distributions.The results for the associated independent sequence and the maxima over arbitrary intervals are also developed.We use simulations to demonstrate the advantages of our newly established maxima of maxima extreme value theory.
文摘There continues to be unfading interest in developing parametric max-stable processes for modelling tail dependencies and clustered extremes in time series data.However,this comes with some difficulties mainly due to the lack of models that fit data directly without transforming the data and the barriers in estimating a significant number of parameters in the existing models.In thiswork,we study the use of the sparsemaxima ofmovingmaxima(M3)process.After introducing random effects and hidden Fréchet type shocks into the process,we get an extended maxlinear model.The extended model then enables us to model cases of tail dependence or independence depending on parameter values.We present some unique properties including mirroring the dependence structure in real data,dealing with the undesirable signature patterns found in most parametricmax-stable processes,and being directly applicable to real data.ABayesian inference approach is developed for the proposed model,and it is applied to simulated and real data.
