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Parameter Estimation for Dependent Risks: Comparing MLE and IFM in Bivariate Copula Models

This study evaluates different methods for estimating parameters in copula-based dependent risk models. It assesses the impact of marginal distributions, copulas, and sample size while discussing practical implications. The research covers copula theory, parameter estimation methods, experimental assumptions, and results analysis. The study compares maximum likelihood estimation (MLE) and inference function for margins (IFM) techniques, highlighting the accuracy and efficiency of IFM over MLE.

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Parameter Estimation for Dependent Risks: Comparing MLE and IFM in Bivariate Copula Models

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  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

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