One-Step Weighted Least Squares Estimation of the Clayton Copula Parameter

Computes a one-step weighted least squares estimator (WLSE) of the Clayton copula dependence parameter \(\theta\). The estimator is obtained via a second-order Taylor expansion of the Clayton copula \(C_{\theta}(u, v)\) around an initial value \(\theta_0\), typically the Kendall's tau-based moment estimate.

WLSE_clayton_onestep(par, x, y, estimates)

Arguments

par

Numeric scalar. Initial estimate of \(\theta\), typically obtained from Kendall's tau.

x

Numeric vector. Observations of the strength variable \(X\).

y

Numeric vector. Observations of the stress variable \(Y\).

estimates

A named list of marginal parameter estimates: \((a_1, b_1, \lambda_1)\) for strength and \((a_2, b_2, \lambda_2)\) for stress.

Value

Numeric scalar. One-step WLSE estimate of the dependence parameter \(\theta\).

Details

One-Step WLSE Estimator for the Clayton Copula Parameter

The one-step estimator is constructed by substituting a second-order Taylor expansion of the Clayton copula \(C_{\theta}(u, v)\) into the weighted least squares estimating equation, evaluated at \(\theta_0\), and solving analytically for \(\theta\).This avoids iterative numerical optimisation and yields a closed-form estimation of \(\theta\).

Further theoretical details are provided in Kizilaslan (2026).

References

Kizilaslan, F. (2026). Reliability estimation in dependent stress–strength model with Clayton copula and modified Weibull margins. arXiv:2604.12130

See also

LSE_clayton_onestep for the LSE version, theta_Ktau_estimate for the Kendall's Tau-based estimate.