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