Loss Data Analysis The Maximum Entropy Approach 2nd Extended Edition by Henryk Gzyl, Silvia Mayoral, Erika Gomes Gonçalves ISBN 9783111047386 3111047385

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ISBN 10: 3111047385
ISBN 13: 9783111047386
Author: Henryk Gzyl, Silvia Mayoral, Erika Gomes Gonçalves

Loss Data Analysis The Maximum Entropy Approach 2nd Extended Edition Table of contents:

1 Introduction

1.1 The basic loss aggregation problem

1.2 Description of the contents

2 Frequency models

2.1 A short list of examples

2.1.1 The Poisson distribution

2.1.2 Poisson mixtures

2.1.3 The negative binomial distribution

2.1.4 Binomial models

2.2 Unified version

2.3 Examples

2.3.1 Determining the parameter of a Poisson distribution

2.3.2 Binomial distribution

3 Individual severity models

3.1 A short catalog of distributions

3.1.1 Exponential distribution

3.1.2 The simple Pareto distribution

3.1.3 Gamma distribution

3.1.4 The lognormal distribution

3.1.5 The beta distribution

3.1.6 A mixture of distributions

3.2 Numerical examples

3.2.1 Data from a density

4 Some detailed examples

4.1 Claim distribution and operational risk losses

4.2 Simple model of credit risk

4.3 Shock and damage models

4.4 Barrier crossing times

4.5 Applications in reliability theory

5 Some traditional approaches to the aggregation problem

5.1 General remarks

5.1.1 General issues

5.2 Analytical techniques: The moment generating function

5.2.1 Simple examples

5.3 Approximate methods

5.3.1 The case of computable convolutions

5.3.2 Simple approximations to the total loss distribution

5.3.3 Edgeworth approximation

5.4 Numerical techniques

5.4.1 Calculations starting from empirical or simulated data

5.4.2 Recurrence relations

5.4.3 Further numerical issues

5.5 Numerical examples

5.6 Concluding remarks

6 Laplace transforms and fractional moment problems

6.1 Mathematical properties of the Laplace transform and its inverse

6.2 Inversion of the Laplace transform

6.3 Laplace transform and fractional moments

6.4 Unique determination of a density from its fractional moments

6.5 The Laplace transform of compound random variables

6.5.1 The use of generating functions and Laplace transforms

6.5.2 Examples

6.6 Numerical determination of the Laplace transform

6.6.1 Recovering a density on [0,1] from its integer moments

6.6.2 Computation of (6.5) by Fourier summation

7 The standard maximum entropy method

7.1 The generalized moment problem

7.2 The entropy function

7.3 The maximum entropy method

7.4 Two concrete models

7.4.1 Case 1: The fractional moment problem

7.4.2 Case 2: Generic case

7.5 Numerical examples

7.5.1 Density reconstruction from empirical data

7.5.2 Reconstruction of densities at several levels of data aggregation

7.5.3 A comparison of methods

8 Extensions of the method of maximum entropy

8.1 Generalized moment problem with errors in the data

8.1.1 The bounded error case

8.1.2 The unbounded error case

8.1.3 The fractional moment problem with bounded measurement error

8.2 Generalized moment problem with data in ranges

8.2.1 Fractional moment problem with data in ranges

8.3 Generalized moment problem with errors in the data and data in ranges

8.4 Numerical examples

8.4.1 Reconstruction from data in intervals

8.4.2 Reconstruction with errors in the data

9 Superresolution in maxentropic Laplace transform inversion

9.1 Introductory remarks

9.2 Properties of the maxentropic solution

9.3 The superresolution phenomenon

9.4 Numerical example

10 Sample data dependence

10.1 Preliminaries

10.1.1 The maximum entropy inversion technique

10.1.2 The dependence of λ on μ

10.2 Variability of the reconstructions

10.2.1 Variability of expected values

10.3 Numerical examples

10.3.1 The sample generation process

10.3.2 The ‘true’ maxentropic density

10.3.3 The sample dependence of the maxentropic densities

10.3.4 Computation of the regulatory capital

11 Disentangling frequencies and decompounding losses

11.1 Disentangling the frequencies

11.1.1 Example: Mixtures of Poisson distributions

11.1.2 A mixture of negative binomials

11.1.3 A more elaborate case

11.2 Decompounding the losses

11.2.1 Simple case: A mixture of two populations

11.2.2 Case 2: Several loss frequencies with different individual loss distributions

12 Computations using the maxentropic density

12.1 Preliminary computations

12.1.1 Calculating quantiles of compound variables

12.1.2 Calculating expected losses given that they are large

12.1.3 Computing the quantiles and tail expectations

12.2 Computation of the VaR and TVaR risk measures

12.2.1 Simple case: A comparison test

12.2.2 VAR and TVaR of aggregate losses

12.3 Computation of risk premia

13 A solution to the capital allocation problem

13.1 Introduction and preliminaries

Problem 1: How to assign risk capital within specified bounds?

Problem 2: Determination of the risk measure from the market prices of risk

13.2 Numerical results

13.2.1 The capital allocation

13.2.2 Problem 2: Determining the distortion function from given risk prices

13.3 Appendices

13.3.1 Application of MEM to determine the capital allocation

13.3.2 Distorted risk measure

13.3.3 Application of MEM to determine the distortion measure

13.3.4 A numerical instability issue

14 Review of statistical procedures

14.1 Parameter estimation techniques

14.1.1 Maximum likelihood estimation

14.1.2 Method of moments

14.2 Clustering methods

14.2.1 K-means

14.2.2 EM algorithm

14.2.3 EM algorithm for linear and nonlinear patterns

14.2.4 Exploratory projection pursuit techniques

14.3 Approaches to select the number of clusters

14.3.1 Elbow method

14.3.2 Information criteria approach

14.3.3 Negentropy

14.4 Validation methods for density estimations

14.4.1 Goodness of fit test for discrete variables

14.4.2 Goodness of fit test for continuous variables

14.4.3 A note about goodness of fit tests

14.4.4 Visual comparisons

14.4.5 Error measurement

14.5 Beware of overfitting

14.6 Copulas

Definition

Properties

Sklar’s theorem

14.6.1 Examples of copulas

14.6.2 Simulation with copulas

Bibliography

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Tags: Henryk Gzyl, Silvia Mayoral, Erika Gomes Gonçalves, Data Analysis, Maximum Entropy