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# ACJJ

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## Caroline Santos

on 8 April 2013

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#### Transcript of ACJJ

- Consider the excess losses over a certain threshold u:

- Limiting distribution for the distribution of the excesses (Balkema & Pickands, 1975) Data Description Empirical
Density Function Modeling the Bulk Empirical Summary Statistics ACTU 459 Project Presented by:
Alessandra Fraquelli
Caroline Hilario Santos
Jiaying Xu
Jacques Zang Empirical Cumulative Distribution Function Summary Statistics Tail Distribution Bulk Distribution Generalized Pareto Distribution Why GPD? Choosing the
Threshold Empirical Mean Excess Function - Linear GPD Mean Excess
Function: GPD Parameter
Estimation - Generalized Pareto Distribution Function: - Two values of u: 5000 & 6500 Method of Moments Percentile Matching MOM vs. Percentile
Matching - GPD log-likelihood function MOM estimates are chosen Confidence Intervals for
GPD MOM Parameters - Simulations used to construct confidence intervals - Parameter estimates fall within respective confidence intervals at level α = 0.05 Outliers - Outliers affect more than
- Choose to include outliers in model Graphical Comparison GPD PP
Plots GPD QQ
Plots u = 6500 u = 5000 u = 5000 u = 6500 Log-Logistic and Weibull Distributions Empirical Graphs Possible Bulk Distributions With more advanced optimization tools, confidence intervals for MLEs could be obtained by the following method:

1) Simulate 100,000 random Log-Logistic/Weibull variables with MLE estimated parameters.
2) Since one parameter cannot be expressed in terms of another, optimize the partial derivatives of the log-likelihood and solve the two equations for the two unknown parameters.
3) Repeat this 10,000 times.
4) Use the 10,000 simulations in order to estimate the mean and the variance of the unknown parameters.
5) Construct confidence intervals using the Normal Approximation. - Log-Normal
- Weibull
- Log-Logistic
- Log-Gamma

(all with large scale parameters and
shape parameters larger than 1) Graphical Comparison: Weibull Density Function PP Plots u = 5000 u = 6500 Graphical Comparison: Log-Logistic Weibull and Log-Logistic
Distributions Weibull Density Function Log-Logistic Density Function Weibull Maximum
Likelihood Estimates Score Functions: Log-Logistic Maximum
Likelihood Estimates Weibull Percentile Matching Estimates Log-Logistic Percentile Matching Estimates Confidence Intervals Score Functions: TESTS: Chi-Square
Goodness-of-Fit and Kolmogorov-Smirnov Bulk Tests Tail Tests u = 5000 Bulk Tests u = 6500 - P&C:
Car insurance
Home insurance Applications - Commonly used distributions inaccurate for both bulk and tail simultaneously
- Solution: using a mixture provides
Good fit for large claims (General Pareto)
Good fit for small and medium claims (Weibull) Conclusion Final Model Selection - Computationally intensive due to conditioning:
Parameters
Confidence intervals
- Requires resources and time QQ Plots Density Function PP Plots QQ Plots u = 5000 u = 6500 Limitations Questions
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