Estimation of the Clayton Copula Dependence Parameter

Estimates the dependence parameter \(\theta\) of the Clayton copula based on observed data from a stress–strength model.

fitClayton(
  x,
  y,
  est.method,
  opt.method,
  start,
  estimates,
  lower = NULL,
  upper = NULL,
  verbose = FALSE,
  ...
)

Arguments

x

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

y

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

est.method

Character string specifying the estimation method used. Options include "MLE", "LSE", "WLSE", and "MPS".

opt.method

Character string specifying the optimization method used in optim. Common options include "Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", and "Brent".

start

Numeric scalar. Initial value for \(\theta\).

estimates

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

lower

Numeric vector. Lower bounds for parameters in constrained optimization. Only used if supported by opt.method.

upper

Numeric vector. Upper bounds for parameters in constrained optimization. Only used if supported by opt.method.

verbose

Logical; if TRUE, progress and intermediate results from the optimization procedure are printed. Default is FALSE.

...

Additional arguments passed to optim.

Value

A list containing:

estimate

Estimate of the Clayton copula parameter, \(\theta\).

opt.fit

Full optimization result.

Details

Estimate the Clayton Copula Parameter

The Clayton copula is defined as $$ C(u,v;\theta) = \left(u^{-\theta} + v^{-\theta} - 1\right)^{-1/\theta}, $$ where \(\theta > 0\).

The parameter is estimated using the following methods:

• Maximum Likelihood Estimation (MLE): Maximizes the joint log-likelihood under the assumed model.

• Least Squares Estimation (LSE): Minimizes squared differences between empirical and theoretical CDFs. The empirical CDF uses Benard's approximation: \(F(x_{(i)}) = (i - 0.3)/(n + 0.4)\), for \(i = 1, \dots, n\).

• Weighted Least Squares Estimation (WLSE): Uses weights \(w_i = \frac{(n+1)^2(n+2)}{i(n-i+1)}\), for \(i = 1, \dots, n\).

• Maximum Product of Spacings (MPS): Maximizes the product of spacings of the fitted distribution function, providing a robust alternative to MLE.

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 preprint. Available at arXiv:2604.12130.