Joe Copula Sampling, Density, Conditional Probabilities, and Kendall’s Tau

Summary

This technical reference implements the bivariate Joe copula, a dependence model with a parameter theta in the range from 1 upward. It provides formulas for the copula cumulative distribution, density, and conditional probability, along with a sampling method. Sampling starts with independent uniform draws and numerically inverts a conditional distribution to produce dependent pairs; a threshold keeps extreme percentile values away from zero.

The class also estimates theta from a supplied Kendall’s tau by numerically integrating a tau relationship and solving for the parameter with a root finder. These tools can support dependence modeling in quantitative finance, including joint-risk analysis, though the document does not provide a trading strategy, market data, or empirical results. Its practical use depends on valid parameter inputs and numerical behavior near boundaries; the implementation’s finite root search and percentile threshold are modeling choices that can affect estimates and generated samples.

Key ideas

• The Joe copula represents bivariate dependence with a parameter theta of at least 1.
• Its cumulative distribution, density, and conditional probability are provided analytically.
• Dependent samples are generated by transforming uniform draws through numerical conditional inversion.
• The implementation estimates theta by matching Kendall’s tau through numerical integration and root finding.
• A finite threshold and search interval constrain numerical behavior near boundary values.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.