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

Author

Listed:
  • Estate Khmaladze

    (Victoria University of Wellington)

  • Wolfgang Weil

    (Karlsruhe Institute of Technology)

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.

Suggested Citation

  • Estate Khmaladze & Wolfgang Weil, 2018. "Fold-up derivatives of set-valued functions and the change-set problem: A Survey," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(1), pages 1-38, February.
    • Handle: RePEc:spr:aistmt:v:70:y:2018:i:1:d:10.1007_s10463-017-0628-7
    DOI: 10.1007/s10463-017-0628-7
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    References listed on IDEAS

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    1. Einmahl, John H. J., 1997. "Poisson and Gaussian approximation of weighted local empirical processes," Stochastic Processes and their Applications, Elsevier, vol. 70(1), pages 31-58, October.
    2. Müller, Hans-Georg & Song, Kai-Sheng, 1996. "A set-indexed process in a two-region image," Stochastic Processes and their Applications, Elsevier, vol. 62(1), pages 87-101, March.
    3. Estate Khmaladze & Wolfgang Weil, 2008. "Local empirical processes near boundaries of convex bodies," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(4), pages 813-842, December.
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