Estimation of the copula density function

You are asked to revisit the i.i.d assumption regarding the likelihood function estimation . Copulas are able to model multivariate dependencies and allow you to construct a joint distribution by separating the dependence structure from the marginal distributions.

  1. Sklar’s Theorem states that any multivariate distribution ( F(X_1, X_2, , X_d) ) can be expressed using copulas: [ F(X_1, X_2, …, X_d) = C(F_1(X_1), F_2(X_2), …, F_d(X_d)) ] where:
    • ( C ) is a copula function capturing dependency.
    • ( F_i(X_i) ) are the marginal CDFs.
  2. Likelihood Function with Copulas
    The joint probability density function (PDF) is: [ f(X_1, …, X_d) = c(F_1(X_1), …, F_d(X_d)) _{i=1}^{d} f_i(X_i) ] where:
    • ( c(u_1, …, u_d) = ) is the copula density.
    • ( f_i(X_i) ) are the marginal densities.
  3. Log-Likelihood Function Given data ( (X_1^j, …, X_d^j) ) for ( j=1, …, n ), the log-likelihood function is: [ L(, ) = _{j=1}^{n} ]
    • ( ) are copula parameters.
    • ( _i ) are marginal distribution parameters.

Steps to Estimate the Likelihood

  1. Choose Marginal Distributions: Fit parametric or nonparametric distributions to each variable.
  2. Transform Data to Uniform Margins: Use the Probability Integral Transform (apply ( F_i(X_i) ) to each marginal).
  3. Select a Copula Model: Common choices include Gaussian, t-Copula, Clayton, Gumbel, or Frank.
  4. Estimate Copula Parameters: Maximum likelihood estimation (MLE) or pseudo-MLE (based on empirical margins).
  5. Compute the Log-Likelihood: Use the copula density function and marginal PDFs.

Use Cases

  • Anomaly Detection: Estimating likelihood to find low-probability events.
  • Risk Modeling: Financial portfolio risk, VaR.
  • Dependence Modeling: Climate data, insurance, etc.

Would you like a Python implementation?