New extreme value theory for maxima of maxima
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 foundatio...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Taylor & Francis Group
2021-07-01
|
| Series: | Statistical Theory and Related Fields |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1080/24754269.2020.1846115 |
| _version_ | 1797677053276848128 |
|---|---|
| author | Wenzhi Cao Zhengjun Zhang |
| author_facet | Wenzhi Cao Zhengjun Zhang |
| author_sort | Wenzhi Cao |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T22:39:24Z |
| format | Article |
| id | doaj.art-fbbcf80f89c64a778fa441cddab61538 |
| institution | Directory Open Access Journal |
| issn | 2475-4269 2475-4277 |
| language | English |
| last_indexed | 2024-03-11T22:39:24Z |
| publishDate | 2021-07-01 |
| publisher | Taylor & Francis Group |
| record_format | Article |
| series | Statistical Theory and Related Fields |
| spelling | doaj.art-fbbcf80f89c64a778fa441cddab615382023-09-22T09:19:46ZengTaylor & Francis GroupStatistical Theory and Related Fields2475-42692475-42772021-07-015323225210.1080/24754269.2020.18461151846115New extreme value theory for maxima of maximaWenzhi Cao0Zhengjun Zhang1University of Wisconsin-MadisonUniversity of Wisconsin-MadisonAlthough 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.http://dx.doi.org/10.1080/24754269.2020.1846115maximum domain of attractionmax-stable distributioncompeting-maximum domain of attractionsaccelerated max-stable distributionaccelerated extreme value distribution |
| spellingShingle | Wenzhi Cao Zhengjun Zhang New extreme value theory for maxima of maxima Statistical Theory and Related Fields maximum domain of attraction max-stable distribution competing-maximum domain of attractions accelerated max-stable distribution accelerated extreme value distribution |
| title | New extreme value theory for maxima of maxima |
| title_full | New extreme value theory for maxima of maxima |
| title_fullStr | New extreme value theory for maxima of maxima |
| title_full_unstemmed | New extreme value theory for maxima of maxima |
| title_short | New extreme value theory for maxima of maxima |
| title_sort | new extreme value theory for maxima of maxima |
| topic | maximum domain of attraction max-stable distribution competing-maximum domain of attractions accelerated max-stable distribution accelerated extreme value distribution |
| url | http://dx.doi.org/10.1080/24754269.2020.1846115 |
| work_keys_str_mv | AT wenzhicao newextremevaluetheoryformaximaofmaxima AT zhengjunzhang newextremevaluetheoryformaximaofmaxima |