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

  • Estate Khmaladze
  • Wolfgang Weil
Invited Review Article


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.


Infinitesimal image analysis Generalized functions Fold-up derivatives Local Steiner formula Local point process Set-valued mapping Derivative set Normal cylinder Change-set problem 



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


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Copyright information

© The Institute of Statistical Mathematics, Tokyo 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsVictoria University of WellingtonWellingtonNew Zealand
  2. 2.Institute of Stochastics, Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany

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