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Fold-up derivatives of set-valued functions and the change-set problem: A Survey

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Abstract

We give a survey on fold-up derivatives, a notion which was introduced by Khmaladze (J Math Anal Appl 334:1055–1072, 2007) and extended by Khmaladze and Weil (J Math Anal Appl 413:291–310, 2014) to describe infinitesimal changes in a set-valued function. We summarize the geometric background and discuss in detail applications in statistics, in particular to the change-set problem of spatial statistics, and show how the notion of fold-up derivatives leads to the theory of testing statistical hypotheses about the change-set. We formulate Poisson limit theorems for the log-likelihood ratio in two versions of this problem and present also the route to a central limit theorem.

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Acknowledgements

The authors thank two anonymous referees for their useful remarks on a previous version of this work.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Victoria University of Wellington, P.O.Box 600, Wellington, New Zealand

    Estate Khmaladze

  2. Institute of Stochastics, Department of Mathematics, Karlsruhe Institute of Technology, 76128, Karlsruhe, Germany

    Wolfgang Weil

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  1. Estate Khmaladze
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  2. Wolfgang Weil
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Correspondence to Estate Khmaladze.

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Khmaladze, E., Weil, W. Fold-up derivatives of set-valued functions and the change-set problem: A Survey. Ann Inst Stat Math 70, 1–38 (2018). https://doi.org/10.1007/s10463-017-0628-7

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  • DOI: https://doi.org/10.1007/s10463-017-0628-7

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