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4 results

  • 6 - Copulas

  • Mary R. Hardy, University of Waterloo, Ontario, David Saunders, University of Waterloo, Ontario
  • Book:
    Quantitative Enterprise Risk Management
    Published online:
    28 July 2022
    Print publication:
    05 May 2022, pp 163-201

  • Summary

    “In this chapter, we consider how individual, univariate distributions can be combined to create multivariate, joint distributions, using copula functions. This can be very valuable when a firm is looking at aggregating dependent risks from different business units. We present Sklar’s seminal theorem, which states that for continuous distributions, every joint distribution can be expressed with a unique copula, and every copula defines a valid joint distribution.

    We present some important copulas, both explicit and implicit, and discuss their features. We show how measures of rank dependency can be more informative than traditional correlation. In keeping with our interest in tail behaviour of loss distributions, we consider how different copulas exhibit different dependency in the tails of the marginal distributions.

    Finally, we discuss construction and estimation of copulas.”

Structural and System Reliability
  • Armen Der Kiureghian
  • Published online:
    13 January 2022
    Print publication:
    13 January 2022
  • Based on material taught at the University of California, Berkeley, this textbook offers a modern, rigorous and comprehensive treatment of the methods of structural and system reliability analysis. It covers the first- and second-order reliability methods for components and systems, simulation methods, time- and space-variant reliability, and Bayesian parameter estimation and reliability updating. It also presents more advanced, state-of-the-art topics such as finite-element reliability methods, stochastic structural dynamics, reliability-based optimal design, and Bayesian networks. A wealth of well-designed examples connect theory with practice, with simple examples demonstrating mathematical concepts and larger examples demonstrating their applications. End-of-chapter homework problems are included throughout. Including all necessary background material from probability theory, and accompanied online by a solutions manual and PowerPoint slides for instructors, this is the ideal text for senior undergraduate and graduate students taking courses on structural and system reliability in departments of civil, environmental and mechanical engineering.

  • Entropy inequalities for classes of probability distributions II. The multivariate case

  • Samuel Karlin, Yosef Rinott
  • Journal:
    Advances in Applied Probability / Volume 13 / Issue 2 / June 1981
    Published online by Cambridge University Press:
    01 July 2016, pp. 325-351
    Print publication:
    June 1981

  • In this paper we continue our investigation of entropy comparisons with emphasis on multivariate distributions. For multiparameter cases of the multinomial and negative multinomial distributions we consider various higher-order forms of multivariate convexity. For the multinormal, Wishart, and t-distributions we define a partial ordering on the set of covariance matrices and determine monotonicity of the entropy functional. We further indicate some entropy inequalities for different sampling schemes. Because of the complex nature of multivariate partial ordering relations several problems remain open.