1. Parameter Estimation for Dependent Risks: Experiments with Bivariate Copula Models Authors: Florence Wu Michael Sherris Date: 11 November 2005

2. Aims of Research: Assess, under varying assumptions, the performance of different methods for estimation of parameters, full MLE, and IFM, for copula base dependent risk models. Assess the impact of marginal distribution, copula and sample size on parameter estimation for commonly used marginal distributions (log-normal and gamma) and copulas (Frank and Gumbel). Report and discuss Implications for practical applications.

3. Coverage A (very) brief review of copulas. Outline methods of parameter estimation (MLE, IFM). Outline experimental assumptions. Report and discuss results and implications.

4. Copulas Portfolio of d risks each with continuous strictly increasing distribution functions with joint probability distribution F X (x 1,…x d ) = Pr(X 1  x 1,…, X d  x d ) Marginal distributions denoted by F X1,…, F Xd where F Xi (x i ) = Pr (X i  x d )

5. Copulas Joint distributions can be written as F X (x 1, …, x d ) = Pr(X 1  x 1,…, X d  x d ) = Pr(F 1 (X 1 )  F 1 (x 1 ),…, F d (X d )  F d (x d )) = Pr(U 1  F 1 (x 1 ),…, U d  F d (x d )) where each U i is uniform (0, 1).

6. Copulas Sklar’s Theorem – any continuous multivariate distribution has a unique copula given by F X (x 1, …, x d ) = C(F 1 (x 1 ), …,F d (x d )) For discrete distributions the copula exists but may not be unique.

7. Copulas We will consider bivariate cumulative distribution F(x,y) = C(F 1 (x), F 2 (y)) with density given by

8. Copulas We will use Gumbel and Frank copulas (often used in insurance risk modelling) Gumbel copula is: Frank copula is :

9. Parameter Estimation

10. Parameter Estimation - MLE

11. Parameter Estimation – IFM

13. Experimental Assumptions Experiments “True distribution” All cases assume Kendall’s tau = 0.5 1. Gumbel copula with parameter = 2 and Lognormal marginals 2. Gumbel copula with parameter = 2 and Gamma marginals 3. Frank copula with parameter = 5.75 and Lognormal marginals 4. Frank copula with parameter = 5.75 and Gamma marginals

14. Experimental Assumptions Case Assumptions – all marginals with same mean and variance: – Case 1 (Base): E[X 1 ] = E[X 2 ] = 1 Std. Dev[X 1 ] = Std. Dev[X 2 ] = 1 – Case 2: E[X 1 ] = E[X 2 ] = 1 Std. Dev[X 1 ] = Std. Dev[X 2 ] = 0.4 Generate small and large sample sizes and use Nelder-Mead to estimate parameters

15. Experiment Results – Goodness of Fit Comparison Case 1 (50 Samples):

16. Experiment Results – Goodness of Fit Comparison Case 1 (5000 Samples):

17. Experiment Results – Goodness of Fit Comparison Case 2 (50 Samples):

18. Experiment Results – Goodness of Fit Comparison Case 2 (5000 Samples):

19. Experiment Results – Parameter Estimated Standard Errors (Case 2)

20. Experiment Results – Run time

22. Conclusions IFM versus full MLE: – IFM surprisingly accurate estimates especially for the dependence parameter and for the lognormal marginals Goodness of Fit: – Clearly improves with sample size, satisfactory in all cases for small sample sizes Run time: – Surprisingly MLE, with one numerical fit, takes the longest time to run compared to IFM with separate numerical fitting of marginals and dependence parameters IFM performs very well compared to full MLE