Parameter Estimation for Dependent Risks: Experiments with Bivariate Copula Models

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 FX(x1,…xd) = Pr(X1 x1,…, Xd xd) • Marginal distributions denoted by FX1,…,FXd where FXi(xi) = Pr (Xi  xd)

5. Copulas • Joint distributions can be written as FX(x1, …, xd) = Pr(X1 x1,…, Xd xd) = Pr(F1(X1)  F1(x1),…, Fd(Xd)  Fd(xd)) = Pr(U1 F1(x1),…, Ud Fd(xd)) where each Ui is uniform (0, 1).

6. Copulas • Sklar’s Theorem – any continuous multivariate distribution has a unique copula given by FX(x1, …, xd) = C(F1(x1), … ,Fd(xd)) • For discrete distributions the copula exists but may not be unique.

7. Copulas • We will consider bivariate cumulative distribution F(x,y) = C(F1(x), F2(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

12. Parameter Estimation – IFM

13. Experimental Assumptions • Experiments “True distribution” • All cases assume Kendall’s tau = 0.5 • Gumbel copula with parameter = 2 and Lognormal marginals • Gumbel copula with parameter = 2 and Gamma marginals • Frank copula with parameter = 5.75 and Lognormal marginals • 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[X1] = E[X2] = 1 • Std. Dev[X1] = Std. Dev[X2] = 1 • Case 2: • E[X1] = E[X2] = 1 • Std. Dev[X1] = Std. Dev[X2] = 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

21. 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