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52669 results in Statistics and Probability

Page 265 of 2634

  • 1 - A Panorama of Lebesgue Integration

  • René L. Schilling, Technische Universität, Dresden, Franziska Kühn, Technische Universität, Dresden
  • Book:
    Counterexamples in Measure and Integration
    Published online:
    27 May 2021
    Print publication:
    17 June 2021, pp 1-35

  • Summary

    This chapter contains a brief non-technical overview on Lebesgue’s integration theory with respect to an abstract measure. Further topics include key theorems from the theory, such as convergence theorems (dominated convergence, monotone convergence, Fatou’s lemma), product measures (Fubini’s and Tonelli’s theorem) and abstract differentiation (Radon--Nikodým theorem).

  • Conditional mean risk sharing in the individual model with graphical dependencies

  • Michel Denuit, Christian Y. Robert
  • Journal:
    Annals of Actuarial Science / Volume 16 / Issue 1 / March 2022
    Published online by Cambridge University Press:
    17 June 2021, pp. 183-209

  • Conditional mean risk sharing appears to be effective to distribute total losses amongst participants within an insurance pool. This paper develops analytical results for this allocation rule in the individual risk model with dependence induced by the respective position within a graph. Precisely, losses are modelled by zero-augmented random variables whose joint occurrence distribution and individual claim amount distributions are based on network structures and can be characterised by graphical models. The Ising model is adopted for occurrences and loss amounts obey decomposable graphical models that are specific to each participant. Two graphical structures are thus used: the first one to describe the contagion amongst member units within the insurance pool and the second one to model the spread of losses inside each participating unit. The proposed individual risk model is typically useful for modelling operational risks, catastrophic risks or cybersecurity risks.

Page 265 of 2634