Review Articles

New extreme value theory for maxima of maxima

Wenzhi Cao ,

Department of Statistics, University of Wisconsin-Madison, Madison, WI, USA

Zhengjun Zhang

Department of Statistics, University of Wisconsin-Madison, Madison, WI, USA

Pages 232-252 | Received 08 May. 2020, Accepted 01 Nov. 2020, Published online: 20 Dec. 2020,
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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.

​​​​​​To cite this article: Wenzhi Cao & Zhengjun Zhang (2021) New extreme value theory for maxima of maxima, Statistical Theory and Related Fields, 5:3, 232-252, DOI:10.1080/24754269.2020.1846115
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