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Calibration Method

This method judges the estimate of α for a 1-parameter Archimedean copula using Kendall’s τ. It employs a consistent estimator for τ and compares the estimated Kendall distribution with the empirical one. Steps involve estimating τ, constructing distributions, and measuring closeness graphically and numerically.

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Calibration Method

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  1. Calibration Method Bob Fountain May 17, 2005

  2. Problem Judge the estimate of the parameter α for a 1-parameter Archimedean copula. The method will use the relationship between α and Kendall’s τ.

  3. Sample tau A consistent estimator of Kendall’s τ for two series is

  4. Kendall variable The Kendall random variable is defined as where F is the true (unknown) joint distribution function of the two random variables.

  5. Kendall distribution The Kendall distribution is the cdf of the Kendall random variable. Genest and Rivest show that the Kendall distribution is related to the generator of an Archimedean copula by

  6. Step 1 Construct an estimate of the Kendall distribution by substituting a previous estimate of α. Call this distribution

  7. Step 2 Define Then the empirical Kendall distribution is

  8. Step 3 Compare the Kendall distribution based on the estimated α with the empirical Kendall distribution. This can be done both graphically and using any numerical measure of closeness of two distribution functions.

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