book

Loss Data Analysis

Name: Loss Data Analysis
ISBN: 9783110516135

by Henryk Gzyl, Silvia Mayoral, Erika Gomes-Gonçalves

February 2018

Intermediate to advanced

210 pages

6h 34m

English

De Gruyter

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Cover
Title Page
Copyright
Preface
Contents
1 Introduction
1.1 The basic loss aggregation problem1.2 Description of the contents
2 Frequency models
2.1 A short list of examples2.1.1 The Poisson distribution2.1.2 Poisson mixtures2.1.3 The negative binomial distribution2.1.4 Binomial models2.2 Unified version2.3 Examples2.3.1 Determining the parameter of a Poisson distribution2.3.2 Binomial distribution
3 Individual severity models
3.1 A short catalog of distributions3.1.1 Exponential distribution3.1.2 The simple Pareto distribution3.1.3 Gamma distribution3.1.4 The lognormal distribution3.1.5 The beta distribution3.1.6 A mixture of distributions3.2 Numerical examples3.2.1 Data from a density
4 Some detailed examples
4.1 Claim distribution and operational risk losses4.2 Simple model of credit risk4.3 Shock and damage models4.4 Barrier crossing times4.5 Applications in reliability theory
5 Some traditional approaches to the aggregation problem
5.1 General remarks5.1.1 General issues5.2 Analytical techniques: The moment generating function5.2.1 Simple examples5.3 Approximate methods5.3.1 The case of computable convolutions5.3.2 Simple approximations to the total loss distribution5.3.3 Edgeworth approximation

5.4 Numerical techniques
5.4.1 Calculations starting from empirical or simulated data5.4.2 Recurrence relations5.4.3 Further numerical issues5.5 Numerical examples
5.6 Concluding remarks
6 Laplace transforms and fractional moment problems
6.1 Mathematical properties of the Laplace transform and its inverse6.2 Inversion of the Laplace transform6.3 Laplace transform and fractional moments6.4 Unique determination of a density from its fractional moments6.5 The Laplace transform of compound random variables6.5.1 The use of generating functions and Laplace transforms6.5.2 Examples6.6 Numerical determination of the Laplace transform6.6.1 Recovering a density on [0, 1] from its integer moments6.6.2 Computation of (6.5) by Fourier summation
7 The standard maximum entropy method
7.1 The generalized moment problem7.2 The entropy function7.3 The maximum entropy method7.4 Two concrete models7.4.1 Case 1: The fractional moment problem7.4.2 Case 2: Generic case7.5 Numerical examples7.5.1 Density reconstruction from empirical data7.5.2 Reconstruction of densities at several levels of data aggregation7.5.3 A comparison of methods
8 Extensions of the method of maximum entropy
8.1 Generalized moment problem with errors in the data8.1.1 The bounded error case8.1.2 The unbounded error case8.1.3 The fractional moment problem with bounded measurement error8.2 Generalized moment problem with data in ranges8.2.1 Fractional moment problem with data in ranges8.3 Generalized moment problem with errors in the data and data in ranges8.4 Numerical examples8.4.1 Reconstruction from data in intervals8.4.2 Reconstruction with errors in the data
9 Superresolution in maxentropic Laplace transform inversion
9.1 Introductory remarks9.2 Properties of the maxentropic solution9.3 The superresolution phenomenon9.4 Numerical example
10 Sample data dependence
10.1 Preliminaries10.1.1 The maximum entropy inversion technique10.1.2 The dependence of λ on μ10.2 Variability of the reconstructions10.2.1 Variability of expected values10.3 Numerical examples10.3.1 The sample generation process10.3.2 The ‘true’ maxentropic density10.3.3 The sample dependence of the maxentropic densities10.3.4 Computation of the regulatory capital
11 Disentangling frequencies and decompounding losses
11.1 Disentangling the frequencies11.1.1 Example: Mixtures of Poisson distributions11.1.2 A mixture of negative binomials11.1.3 A more elaborate case11.2 Decompounding the losses11.2.1 Simple case: A mixture of two populations11.2.2 Case 2: Several loss frequencies with different individual loss distributions
12 Computations using the maxentropic density
12.1 Preliminary computations12.1.1 Calculating quantiles of compound variables12.1.2 Calculating expected losses given that they are large12.1.3 Computing the quantiles and tail expectations12.2 Computation of the VaR and TVaR risk measures12.2.1 Simple case: A comparison test12.2.2 VAR and TVaR of aggregate losses12.3 Computation of risk premia
13 Review of statistical procedures
13.1 Parameter estimation techniques13.1.1 Maximum likelihood estimation13.1.2 Method of moments13.2 Clustering methods13.2.1 K-means13.2.2 EM algorithm13.2.3 EM algorithm for linear and nonlinear patterns13.2.4 Exploratory projection pursuit techniques13.3 Approaches to select the number of clusters13.3.1 Elbow method13.3.2 Information criteria approach13.3.3 Negentropy13.4 Validation methods for density estimations13.4.1 Goodness of fit test for discrete variables13.4.2 Goodness of fit test for continuous variables13.4.3 A note about goodness of fit tests13.4.4 Visual comparisons13.4.5 Error measurement13.5 Beware of overfitting 13.6 Copulas13.6.1 Examples of copulas13.6.2 Simulation with copulas
Index
Bibliography

Overview

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

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