Loss Data Analysis

Loss Data Analysis

by Henryk Gzyl, Silvia Mayoral, Erika Gomes-Gonçalves
Released February 2018
Publisher(s): De Gruyter
ISBN: 9783110516135

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Book description

This volume deals with two complementary topics. On one hand the book deals with the problem of determining the the probability distribution of a positive compound random variable, a problem which appears in the banking and insurance industries, in many areas of operational research and in reliability problems in the engineering sciences.

On the other hand, the methodology proposed to solve such problems, which is based on an application of the maximum entropy method to invert the Laplace transform of the distributions, can be applied to many other problems.

The book contains applications to a large variety of problems, including the problem of dependence of the sample data used to estimate empirically the Laplace transform of the random variable.

Contents
Introduction
Frequency models
Individual severity models
Some detailed examples
Some traditional approaches to the aggregation problem
Laplace transforms and fractional moment problems
The standard maximum entropy method
Extensions of the method of maximum entropy
Superresolution in maxentropic Laplace transform inversion
Sample data dependence
Disentangling frequencies and decompounding losses
Computations using the maxentropic density
Review of statistical procedures

Table of contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Preface
  5. Contents
  6. 1 Introduction
    1. 1.1 The basic loss aggregation problem
    2. 1.2 Description of the contents
  7. 2 Frequency models
    1. 2.1 A short list of examples
      1. 2.1.1 The Poisson distribution
      2. 2.1.2 Poisson mixtures
      3. 2.1.3 The negative binomial distribution
      4. 2.1.4 Binomial models
    2. 2.2 Unified version
    3. 2.3 Examples
      1. 2.3.1 Determining the parameter of a Poisson distribution
      2. 2.3.2 Binomial distribution
  8. 3 Individual severity models
    1. 3.1 A short catalog of distributions
      1. 3.1.1 Exponential distribution
      2. 3.1.2 The simple Pareto distribution
      3. 3.1.3 Gamma distribution
      4. 3.1.4 The lognormal distribution
      5. 3.1.5 The beta distribution
      6. 3.1.6 A mixture of distributions
    2. 3.2 Numerical examples
      1. 3.2.1 Data from a density
  9. 4 Some detailed examples
    1. 4.1 Claim distribution and operational risk losses
    2. 4.2 Simple model of credit risk
    3. 4.3 Shock and damage models
    4. 4.4 Barrier crossing times
    5. 4.5 Applications in reliability theory
  10. 5 Some traditional approaches to the aggregation problem
    1. 5.1 General remarks
      1. 5.1.1 General issues
    2. 5.2 Analytical techniques: The moment generating function
      1. 5.2.1 Simple examples
    3. 5.3 Approximate methods
      1. 5.3.1 The case of computable convolutions
      2. 5.3.2 Simple approximations to the total loss distribution
      3. 5.3.3 Edgeworth approximation
    4. 5.4 Numerical techniques
      1. 5.4.1 Calculations starting from empirical or simulated data
      2. 5.4.2 Recurrence relations
      3. 5.4.3 Further numerical issues
    5. 5.5 Numerical examples
    6. 5.6 Concluding remarks
  11. 6 Laplace transforms and fractional moment problems
    1. 6.1 Mathematical properties of the Laplace transform and its inverse
    2. 6.2 Inversion of the Laplace transform
    3. 6.3 Laplace transform and fractional moments
    4. 6.4 Unique determination of a density from its fractional moments
    5. 6.5 The Laplace transform of compound random variables
      1. 6.5.1 The use of generating functions and Laplace transforms
      2. 6.5.2 Examples
    6. 6.6 Numerical determination of the Laplace transform
      1. 6.6.1 Recovering a density on [0, 1] from its integer moments
      2. 6.6.2 Computation of (6.5) by Fourier summation
  12. 7 The standard maximum entropy method
    1. 7.1 The generalized moment problem
    2. 7.2 The entropy function
    3. 7.3 The maximum entropy method
    4. 7.4 Two concrete models
      1. 7.4.1 Case 1: The fractional moment problem
      2. 7.4.2 Case 2: Generic case
    5. 7.5 Numerical examples
      1. 7.5.1 Density reconstruction from empirical data
      2. 7.5.2 Reconstruction of densities at several levels of data aggregation
      3. 7.5.3 A comparison of methods
  13. 8 Extensions of the method of maximum entropy
    1. 8.1 Generalized moment problem with errors in the data
      1. 8.1.1 The bounded error case
      2. 8.1.2 The unbounded error case
      3. 8.1.3 The fractional moment problem with bounded measurement error
    2. 8.2 Generalized moment problem with data in ranges
      1. 8.2.1 Fractional moment problem with data in ranges
    3. 8.3 Generalized moment problem with errors in the data and data in ranges
    4. 8.4 Numerical examples
      1. 8.4.1 Reconstruction from data in intervals
      2. 8.4.2 Reconstruction with errors in the data
  14. 9 Superresolution in maxentropic Laplace transform inversion
    1. 9.1 Introductory remarks
    2. 9.2 Properties of the maxentropic solution
    3. 9.3 The superresolution phenomenon
    4. 9.4 Numerical example
  15. 10 Sample data dependence
    1. 10.1 Preliminaries
      1. 10.1.1 The maximum entropy inversion technique
      2. 10.1.2 The dependence of λ on μ
    2. 10.2 Variability of the reconstructions
      1. 10.2.1 Variability of expected values
    3. 10.3 Numerical examples
      1. 10.3.1 The sample generation process
      2. 10.3.2 The ‘true’ maxentropic density
      3. 10.3.3 The sample dependence of the maxentropic densities
      4. 10.3.4 Computation of the regulatory capital
  16. 11 Disentangling frequencies and decompounding losses
    1. 11.1 Disentangling the frequencies
      1. 11.1.1 Example: Mixtures of Poisson distributions
      2. 11.1.2 A mixture of negative binomials
      3. 11.1.3 A more elaborate case
    2. 11.2 Decompounding the losses
      1. 11.2.1 Simple case: A mixture of two populations
      2. 11.2.2 Case 2: Several loss frequencies with different individual loss distributions
  17. 12 Computations using the maxentropic density
    1. 12.1 Preliminary computations
      1. 12.1.1 Calculating quantiles of compound variables
      2. 12.1.2 Calculating expected losses given that they are large
      3. 12.1.3 Computing the quantiles and tail expectations
    2. 12.2 Computation of the VaR and TVaR risk measures
      1. 12.2.1 Simple case: A comparison test
      2. 12.2.2 VAR and TVaR of aggregate losses
    3. 12.3 Computation of risk premia
  18. 13 Review of statistical procedures
    1. 13.1 Parameter estimation techniques
      1. 13.1.1 Maximum likelihood estimation
      2. 13.1.2 Method of moments
    2. 13.2 Clustering methods
      1. 13.2.1 K-means
      2. 13.2.2 EM algorithm
      3. 13.2.3 EM algorithm for linear and nonlinear patterns
      4. 13.2.4 Exploratory projection pursuit techniques
    3. 13.3 Approaches to select the number of clusters
      1. 13.3.1 Elbow method
      2. 13.3.2 Information criteria approach
      3. 13.3.3 Negentropy
    4. 13.4 Validation methods for density estimations
      1. 13.4.1 Goodness of fit test for discrete variables
      2. 13.4.2 Goodness of fit test for continuous variables
      3. 13.4.3 A note about goodness of fit tests
      4. 13.4.4 Visual comparisons
      5. 13.4.5 Error measurement
    5. 13.5 Beware of overfitting
    6. 13.6 Copulas
      1. 13.6.1 Examples of copulas
      2. 13.6.2 Simulation with copulas
  19. Index
  20. Bibliography

Product information

  • Title: Loss Data Analysis
  • Author(s): Henryk Gzyl, Silvia Mayoral, Erika Gomes-Gonçalves
  • Release date: February 2018
  • Publisher(s): De Gruyter
  • ISBN: 9783110516135