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TitleLoss Data Analytics
TagsWeight Loss
LanguageEnglish
File Size2.3 MB
Total Pages168
Table of Contents
                            Preface
Introduction to Loss Data Analytics
	Relevance of Analytics
		What is Analytics?
		Short-term Insurance
		Insurance Processes
	Insurance Company Operations
		Initiating Insurance
		Renewing Insurance
		Claims and Product Management
		Loss Reserving
	Case Study: Wisconsin Property Fund
		Fund Claims Variables
		Fund Rating Variables
		Fund Operations
	Further Resources and Contributors
Frequency Distributions
	How Frequency Augments Severity Information
	Basic Frequency Distributions
		Foundations
		Probability Generating Function
		Important Frequency Distributions
	The (a, b, 0) Class
		The (a, b, 0) Class - Example
	Estimating Frequency Distributions
	Other Frequency Distributions
		Zero Truncation or Modification
	Mixture Distributions
		Mixtures of Finite Populations
		Mixtures of Infinitely Many Populations
	Goodness of Fit
	Exercises
	Technical Supplement: Iterated Expectations
Modeling Loss Severity
	Basic Distributional Quantities
		Moments
		Quantiles
		The Moment Generating Function
		Probability Generating Function
	Continuous Distributions for Modeling Loss Severity
		The Gamma Distribution
		The Pareto Distribution
		The Weibull Distribution
		The Generalized Beta Distribution of the Second Kind
	Methods of Creating New Distributions
		Functions of Random Variables and their Distributions
		Multiplication by a Constant
		Raising to a Power
		Exponentiation
		Finite Mixtures
		Continuous Mixtures
	Coverage Modifications
		Policy Deductibles
		Policy Limits
		Coinsurance
		Reinsurance
	Maximum Likelihood Estimation
		Maximum Likelihood Estimators for Complete Data
		Maximum Likelihood Estimators for Grouped Data
		Maximum Likelihood Estimators for Censored Data
		Maximum Likelihood Estimators for Truncated Data
	Further Resources and Contributors
	Exercises
Model Selection, Validation, and Inference
	Nonparametric Inference
		Nonparametric Estimation
		Tools for Model Selection
		Starting Values
	Model Validation
		Iterative Model Selection
		Summarizing Model Selection
		Out of Sample Validation
		Gini Statistic
	Modified Data
		Parametric Estimation using Modified Data
		Nonparametric Estimation using Modified Data
	Bayesian Inference
		Bayesian Model
		Decision Analysis
		Posterior Distribution
	Exercises
Simulation
	Generating Independent Uniform Observations
	Inverse Transform
	How Many Simulated Values?
Portfolio Management including Reinsurance
	Overview:
	Tails of Distributions
	Measures of Risk
	Reinsurance
		Proportional Reinsurance
		Surplus Share Proportional Treaty
		Excess of Loss Reinsurance
		Relations with Personal Insurance
		Layers of Coverage
Dependence Modeling
	Introduction
	Variable Types
		Qualitative Variables
		Quantitative Variables
		Multivariate Variables
	Classic Measures of Scalar Associations
		Association Measures for Quantitative Variables
		Rank Based Measures
		Nominal Variables
	Introduction to Copula
	Application Using Copulas
		Data Description
		Marginal Models
		Probability Integral Transformation
		Joint Modeling with Copula Function
	Types of Copulas
		Elliptical Copulas
		Archimedian Copulas
		Properties of Copulas
	Why is Dependence Modeling Important?
Technical Supplement: Statistical Inference
	Overview of Statistical Inference
	Estimation and Prediction
	Maximum Likelihood Theory
		Likelihood Function
		Information Criteria
                        
Document Text Contents
Page 84

84 CHAPTER 4. MODEL SELECTION, VALIDATION, AND INFERENCE

As the above example suggests, there is flexibility with the method of moments. For example, we could
have matched the second and third moments instead of the first and second, yielding different estimators.
Furthermore, there is no guarantee that a solution will exist for each problem. You will also find that
matching moments is possible for a few problems where the data are censored or truncated, but in general,
this is a more difficult scenario. Finally, for distributions where the moments do not exist or are infinite,
method of moments is not available. As an alternative for the infinite moment situation, one can use the
percentile matching technique.

Percentile Matching

Under percentile matching, we approximate the quantiles or percentiles of the parametric distribution using
the empirical (nonparametric) quantiles or percentiles described in Section 4.1.1.

Show Example

Example – Property Fund. For the 2010 property fund, we illustrate matching on quantiles. In particular,
the Pareto distribution is intuitively pleasing because of the closed-form solution for the quantiles. Recall
that the distribution function for the Pareto distribution is

F (x) = 1 −
(

θ

x + θ


Easy algebra shows that we can express the quantile as

F −1(q) = θ
(

(1 − q)−1/α − 1
)

for a fraction q, 0 < q < 1.

The 25th percentile (the first quartile) turns out to be 0.78853 and the 95th percentile is 50.98293 (both in
thousands of dollars). With two equations

0.78853 = θ
(

1 − (1 − .25)−1/α
)

and 50.98293 = θ
(

1 − (1 − .75)−1/α
)

and two unknowns, the solution is

α̂ = 0.9412076 and θ̂ = 2.205617.

We remark here that a numerical routine is required for these solutions as no analytic solution is available.
Furthermore, recall that the maximum likelihood estimates are α̂MLE = 0.9990936 and θ̂MLE = 2.2821147,
so the percentile matching provides a better approximation for the Pareto distribution than the method of
moments.

Exercise – Exam C Question 1. You are given:

(i) Losses follow a loglogistic distribution with cumulative distribution function:

F (x) =
(x/θ)γ

1 + (x/θ)γ

(ii) The sample of losses is:

10 35 80 86 90 120 158 180 200 210 1500

Calculate the estimate of θ by percentile matching, using the 40th and 80th empirically smoothed percentile
estimates.

Page 167

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