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Parameter Estimation for Dependent Parameter Estimation for Dependent Risks: Experiments with Bivariate Risks: Experiments with Bivariate
Copula ModelsCopula ModelsAuthors:
Florence Wu
Michael Sherris
Date: 11 November 2005
Aims of Research: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.
CoverageCoverage
A (very) brief review of copulas. Outline methods of parameter estimation (MLE,
IFM). Outline experimental assumptions. Report and discuss results and implications.
CopulasCopulas
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)
CopulasCopulas
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).
CopulasCopulas
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.
CopulasCopulas
We will consider bivariate cumulative distribution F(x,y) = C(F1(x), F2(y)) with density given by
CopulasCopulas
We will use Gumbel and Frank copulas (often used in insurance risk modelling)
Gumbel copula is:
Frank copula is :
Parameter EstimationParameter Estimation
Parameter Estimation - MLEParameter Estimation - MLE
Parameter Estimation – IFMParameter Estimation – IFM
Parameter Estimation – IFMParameter Estimation – IFM
Experimental AssumptionsExperimental Assumptions
Experiments “True distribution” All cases assume Kendall’s tau = 0.51. Gumbel copula with parameter = 2 and
Lognormal marginals2. Gumbel copula with parameter = 2 and Gamma
marginals3. Frank copula with parameter = 5.75 and
Lognormal marginals4. Frank copula with parameter = 5.75 and Gamma marginals
Experimental AssumptionsExperimental 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
Experiment Results – Experiment Results – Goodness of Fit ComparisonGoodness of Fit Comparison
Case 1 (50 Samples):
Experiment Results – Goodness of Fit Experiment Results – Goodness of Fit ComparisonComparison
Case 1 (5000 Samples):
Experiment Results – Goodness of Fit Experiment Results – Goodness of Fit ComparisonComparison
Case 2 (50 Samples):
Experiment Results – Goodness of Fit Experiment Results – Goodness of Fit ComparisonComparison
Case 2 (5000 Samples):
Experiment Results – Parameter Experiment Results – Parameter Estimated Standard Errors (Case 2)Estimated Standard Errors (Case 2)
Experiment Results – Run timeExperiment Results – Run time
Experiment Results – Run timeExperiment Results – Run time
ConclusionsConclusions
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