Package 'SSReliabilityClaytonMWD'

All Istanbul Dams Data

Description

Daily data for 10 dams in Istanbul, Türkiye. The dataset consists of daily occupancy rates of Istanbul’s dams, retrieved in March 2026 from https://data.ibb.gov.tr/en. The data span the period from late October 2000 to mid-February 2024.

Usage

AllDams

Format

A data frame with 8520 rows and 13 variables.

Source

Istanbul Metropolitan Municipality Open Data Portal https://data.ibb.gov.tr/en. Licensed under CC BY 4.0.

---

Two-dimensional Clayton Copula

Description

Computes the joint cumulative distribution function (CDF) and probability density function (PDF) of the two-dimensional Clayton copula.

Usage

Clayton_Copula(u, v, theta)

Clayton_Copula_pdf(u, v, theta)

Arguments

u	Numeric vector of values in $[0,1]$. First marginal (uniform).
v	Numeric vector of values in $[0,1]$. Second marginal (uniform).
theta	Positive numeric scalar. Dependence parameter $\theta > 0$.

Details

The joint distribution function of the two-dimensional Clayton copula is

$C(u,v;\theta) = \left(u^{-\theta} + v^{-\theta} - 1\right)^{-1/\theta},$

where $\theta > 0$.

The corresponding joint density is given by

$c(u,v;\theta) = (\theta + 1) u^{-(\theta + 1)} v^{-(\theta + 1)} \left(u^{-\theta} + v^{-\theta} - 1\right)^{-\left(1/\theta + 2\right)}.$

Value

• Clayton_Copula: Numeric vector of CDF values.

• Clayton_Copula_pdf: Numeric vector of PDF values.

References

Nelsen, R. B. (2006). An Introduction to Copulas. Springer.

Examples

u <- c(0.2, 0.5, 0.8)
v <- c(0.3, 0.6, 0.9)

Clayton_Copula(u, v, theta = 2)
Clayton_Copula_pdf(u, v, theta = 2)

---

Estimation of SSR Model Parameters with MWD Marginals via Clayton Copula

Description

Fits a dependent stress–strength reliability (SSR) model in which both strength and stress follow the Modified Weibull Distribution (MWD), and dependence is modeled using a Clayton copula.

The function estimates marginal parameters $(a_1, b_1, \lambda_1)$ for strength $X$, $(a_2, b_2, \lambda_2)$ for stress $(Y$, and the copula dependence parameter $\theta$.

Estimation is performed using Maximum Likelihood Estimation (MLE), Least Squares Estimation (LSE), Weighted Least Squares Estimation (WLSE), and Maximum Product of Spacings (MPS).

Usage

fit.SSR.ClaytonMWD(
  data,
  ACI = FALSE,
  bootstrap = FALSE,
  B = NULL,
  seed = NULL,
  one.step = TRUE,
  alpha = 0.05,
  verbose = FALSE
)

Arguments

data	A list containing two numeric vectors: X (strength) and Y (stress).
ACI	Logical. If TRUE, asymptotic $95\%$ confidence intervals based on MLE are computed.
bootstrap	Logical. If TRUE, parametric bootstrap confidence intervals are computed.
B	Integer. Number of bootstrap replications.
seed	Integer. Random seed for reproducibility.
one.step	Logical. If TRUE,one-step LSE and WLSE estimators are used for $\theta$.
alpha	Numeric. Significance level for confidence intervals (e.g., 0.05 for a $95\%$ confidence interval).
verbose	Logical; if TRUE, progress and intermediate results from the optimization procedure are printed. Default is FALSE.

Details

Fit SSR Model with Modified Weibull Marginals via Clayton Copula

Returns point estimates and interval estimates of model parameters using MLE, LSE, WLSE, and MPS methods.

Further theoretical details are available in Kizilaslan (2026).

Value

A list containing:

all.results	Point estimates of all model parameters.
theta.Ktau	Kendall's tau estimate corresponding to $\theta$.
seed	Random seed used in the analysis.
data	Input dataset used in the analysis.
ACI.parameters	If ACI = TRUE, asymptotic $95\%$ confidence intervals for model parameters.
boot.mle	If bootstrap = TRUE, bootstrap confidence intervals based on MLE.
boot.lse	If bootstrap = TRUE, bootstrap confidence intervals based on LSE.
boot.wlse	If bootstrap = TRUE, bootstrap confidence intervals based on WLSE.
boot.mps	If bootstrap = TRUE, bootstrap confidence intervals based on MPS.
boot.samples	A list containing bootstrap samples for all parameters across all methods.

References

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

Examples

data <- list(X = TerkosDam, Y = OmerliDam)
fit.SSR <- fit.SSR.ClaytonMWD(data,  ACI = TRUE, bootstrap = TRUE, B = 5,
                             seed = 2026, one.step = TRUE, alpha = 0.05)
print(fit.SSR)

---

Estimation of the Clayton Copula Dependence Parameter

Description

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

Usage

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.

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).

Value

A list containing:

estimate	Estimate of the Clayton copula parameter, $\theta$.
opt.fit	Full optimization result.

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.

---

Estimation of Parameters for the Modified Weibull Distribution

Description

Estimates the parameters of the Modified Weibull Distribution (MWD) using classical estimation methods.

Usage

fitMWD(
  data,
  est.method,
  opt.method,
  starts,
  lower = NULL,
  upper = NULL,
  verbose = FALSE,
  ...
)

Arguments

data	Numeric vector of observations.
est.method	Character string specifying the estimation method. Options include "MLE", "LSE", "WLSE", and "MPS".
opt.method	Character string specifying the optimization method used in optim, such as "Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", or "Brent".
starts	Numeric vector of initial values for the parameters
lower	Numeric vector of lower bounds for parameters in constrained optimization. Ignored if NULL.
upper	Numeric vector of upper bounds for parameters in constrained optimization.
verbose	Logical. If TRUE, prints optimization progress.
...	Additional arguments passed to optim.

Details

Fit the Modified Weibull Distribution (MWD)

The Modified Weibull Distribution (Lai et al., 2003) has cumulative distribution function (CDF) and probability density function (PDF):

$F(x) = 1 - \exp\left(-a x^b \exp(\lambda x)\right),$

$f(x) = a (b + \lambda x) x^{b - 1} \exp(\lambda x) \exp\left(-a x^b \exp(\lambda x)\right),$

where $x > 0$, $a > 0$ is a scale parameter, $b \ge 0$ is a shape parameter, and $\lambda \ge 0$ is a flexibility parameter controlling the growth rate of the hazard function.

The parameters are estimated using the following methods:

• Maximum Likelihood Estimation (MLE): Maximizes the log-likelihood under the MWD 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): A modification of LSE that assigns weights to the squared differences. 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 CDF.

Further details are provided in Kizilaslan (2026).

Value

A list containing:

estimates	Named numeric vector of estimated parameters $(a, b, \lambda)$.
measures	Numeric vector of model selection criteria (log-likelihood, AIC, BIC).
initials	Initial values used in the optimization.
opt.fit	Full output from optim.

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.

Lai, C. D., Xie, M., and Murthy, D. N. P. (2003). A modified Weibull distribution. IEEE Transactions on Reliability, 52(1), 33–37.

Examples

# generate data from MWD(a, b, lambda)
n <- 100
a <- 0.75; b <- 1.25; lambda <- 0.60
set.seed(123)
dat <- rMweibull(n, a, b, lambda)
init <- runif(3)

# Fit MWD to dat.
fit.mle <- fitMWD(data = dat, est.method = "mle", opt.method = "L-BFGS-B", starts = init,
                  lower = c(1e-05,1e-05,1e-05), upper = c(Inf,Inf,Inf), hessian = FALSE )
fit.mle$estimates

fit.lse <- fitMWD(data = dat, est.method = "lse", opt.method = "L-BFGS-B", starts = init,
                  lower = c(1e-05,1e-05,1e-05), upper = c(Inf,Inf,Inf), hessian = FALSE )
fit.lse$estimates

fit.wlse <- fitMWD(data = dat, est.method = "wlse", opt.method = "L-BFGS-B", starts = init,
                   lower = c(1e-05,1e-05,1e-05), upper = c(Inf,Inf,Inf), hessian = FALSE )
fit.wlse$estimates

fit.mps <- fitMWD(data = dat, est.method = "mps", opt.method = "L-BFGS-B", starts = init,
                  lower = c(1e-05,1e-05,1e-05), upper = c(Inf,Inf,Inf), hessian = FALSE )
fit.mps$estimates

---

Estimating parameters of the two-parameter Weibull distribution

Description

Estimates the parameters of the two-parameter Weibull distribution using classical methods.

Usage

fitWD(
  data,
  est.method,
  opt.method,
  starts,
  lower = NULL,
  upper = NULL,
  verbose = FALSE,
  ...
)

Arguments

data	Numeric vector of observations.
est.method	Character string specifying the estimation method. Options include "MLE", "LSE", "WLSE", and "MPS".
opt.method	Character string specifying the optimization method used in optim, such as "Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", or "Brent".
starts	Numeric vector of initial values for the parameters
lower	Numeric vector of lower bounds for parameters in constrained optimization. Ignored if NULL.
upper	Numeric vector of upper bounds for parameters in constrained optimization.
verbose	Logical. If TRUE, prints optimization progress.
...	Additional arguments passed to optim.

Details

Classical estimations fo the parameters of the two-parameter Weibull distribution

The two-parameter Weibull Distribution has cumulative distribution function (CDF) and probability density function (PDF):

$F(x) = 1 - \exp(-a x^b),$

$f(x) = a b x^{b - 1} \exp(-a x^b ),$

where $x > 0$, $a > 0$ is the scale parameter and $b > 0$ is the shape parameter.

The parameters are estimated using the following methods:

• Maximum Likelihood Estimation (MLE): Maximizes the log-likelihood under the MWD 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): A modification of LSE that assigns weights to the squared differences. 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 CDF.

Value

A list containing:

estimates	Named numeric vector of estimated parameters $(a, b)$.
measures	Numeric vector of model selection criteria (log-likelihood, AIC, BIC).
initials	Initial values used in the optimization.
opt.fit	Full output from optim.

Examples

# generate data from WD(a, b)
n <- 50
a <- 0.75; b <- 1.25; lambda <- 0 # reduces two-parameter Weibull distribution
set.seed(123)
X <- rMweibull(n, a, b, lambda)
init <- runif(2)

# fit model
fit.mle <- fitWD(data = X, est.method = "mle", opt.method = "L-BFGS-B", starts = init,
                 lower = c(1e-05,1e-05), upper = c(Inf,Inf), hessian = FALSE )
fit.mle$estimates

fit.lse <- fitWD(data = X, est.method = "lse", opt.method = "L-BFGS-B", starts = init,
                 lower = c(1e-05,1e-05), upper = c(Inf,Inf), hessian = FALSE )
fit.lse$estimates

fit.wlse <- fitWD(data = X, est.method = "wlse", opt.method = "L-BFGS-B", starts = init,
                  lower = c(1e-05,1e-05), upper = c(Inf,Inf), hessian = FALSE )
fit.wlse$estimates

fit.mps <- fitWD(data = X, est.method = "mps", opt.method = "L-BFGS-B", starts = init,
                 lower = c(1e-05,1e-05), upper = c(Inf,Inf), hessian = FALSE )
fit.mps$estimates

---

One-Step Least Squares Estimation of the Clayton Copula Parameter

Description

Computes a one-step least squares estimator (LSE) 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.

Usage

LSE_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.

Details

One-Step LSE 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 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).

Value

Numeric scalar. One-step LSE estimate of the dependence parameter $\theta$.

References

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

See Also

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

---

The Modified Weibull Distribution (MWD)

Description

Density, distribution function, quantile function, random generation, and hazard function for the Modified Weibull distribution (MWD) introduced by Lai et al. (2003).

Usage

dMweibull(x, a, b, lambda, log = FALSE)

pMweibull(q, a, b, lambda, lower.tail = TRUE, log = FALSE)

qMweibull(p, a, b, lambda, lower.tail = TRUE)

rMweibull(n, a, b, lambda)

hMweibull(x, a, b, lambda, log = FALSE)

Arguments

x	Numeric vector of observations.
a	Positive scale parameter ($a > 0$).
b	Non-negative shape parameter ($b \ge 0$).
lambda	Non-negative parameter ($\lambda \ge 0$) controlling the growth rate of the hazard function.
log	Logical. If TRUE, returns log-density, log-distribution, or log-hazard values where applicable.
q	Numeric vector of quantiles.
lower.tail	Logical. If FALSE, returns $1 - F(x)$ and computes quantiles for $1 - p$. #' The Modified Weibull distribution with parameters $a$, $b$, and $\lambda$ has cumulative distribution function (CDF), probability density function (PDF), and hazard function given by
p	Numeric vector of probabilities in $[0,1]$.
n	Integer; number of observations to be generated.

Details

Modified Weibull Distribution

The Modified Weibull distribution with parameters $a$, $b$ and $\lambda$ has cumulative distribution function (CDF), probability density function (PDF), and hazard function given by

$F(x) = 1 - \exp\left(-a x^b \exp(\lambda x)\right),$

$f(x) = a (b + \lambda x) x^{b - 1} \exp(\lambda x) \exp\left(-a x^b \exp(\lambda x)\right),$

$h(x) = a (b + \lambda x) x^{b - 1} \exp(\lambda x),$

where $x > 0$, $a > 0$ is the scale parameter, $b \ge 0$ is a shape parameter, and $\lambda \ge 0$ is an acceleration or flexibility parameter that controls how quickly the hazard grows over time.

Special cases:

• If $\lambda = 0$, the MWD reduces to the Weibull distribution $F(x) = 1 - \exp(-a x^b)$.

• If $b = 0$, the MWD reduces to a type I extreme-value (log-gamma) distribution $F(x) = 1 - \exp(-a \exp(\lambda x))$.

Value

• dMweibull: Density values.

• pMweibull: Distribution function values.

• qMweibull: Quantiles.

• rMweibull: Random deviates.

• hMweibull: Hazard function values.

References

Lai, C. D., Xie, M., and Murthy, D. N. P. (2003). A modified Weibull distribution. IEEE Transactions on Reliability, 52(1), 33–37.

Examples

n <- 25
a <- 0.75; b <- 1.25; lambda <- 0.60
set.seed(123)
x <- rlnorm(n)
ff <- dMweibull(x, a, b, lambda)
FF <- pMweibull(x, a, b, lambda)
qq <- qMweibull(runif(n), a, b, lambda)
dat <- rMweibull(n, a, b, lambda)
hf <- hMweibull(x, a, b, lambda )

---

Omerli Dam Data

Description

Omerli Dam is the largest dam supplying Istanbul, Türkiye, and is located on the Anatolian side. The dataset consists of daily occupancy rates of Istanbul’s dams, retrieved in March 2026 from Istanbul Metropolitan Municipality datasets website https://data.ibb.gov.tr/en.

Usage

OmerliDam

Format

A numeric vector of length 95, representing monthly average occupancy rates.

Details

The data span the period from late October 2000 to mid-February 2024. Monthly average occupancy rates are computed based on the daily data for the period September-December of each year, resulting in a total of 95 observations.

Source

Istanbul Metropolitan Municipality Open Data Portal https://data.ibb.gov.tr/en. Licensed under CC BY 4.0.

---

Parametric Bootstrap Confidence Intervals

Description

Computes parametric bootstrap confidence intervals for unknown model parameters and reliability $R$, based on maximum likelihood estimation (MLE), least squares estimation (LSE), weighted least squares estimation (WLSE), and maximum product of spacing estimation (MPS).

Usage

parametric_bootstrap(
  est.method,
  opt.method,
  boot.estimates,
  n,
  B = 1000,
  seed = NULL,
  one.step = TRUE,
  alpha = 0.05
)

Arguments

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".
boot.estimates	A named list of initial parameter estimates. The elements $(a_1, b_1, \lambda_1)$ correspond to the strength variable, $(a_2, b_2, \lambda_2)$ correspond to the stress variable, and $\theta$ is the Clayton copula dependence parameter.
n	Integer. Sample size.
B	Integer. Number of bootstrap replications.
seed	Integer. Random seed for reproducibility.
one.step	Logical. If TRUE, one-step LSE and WLSE estimators are used for $\theta$.
alpha	Numeric. Significance level for confidence intervals (e.g., 0.05 for a $95\%$ confidence interval).

Details

This function implements a parametric bootstrap percentile method to construct confidence intervals for unknown parameters and reliability $R$ under different estimation methods (MLE, LSE, WLSE, and MPS).

Further theoretical details are provided in Kizilaslan (2026).

Value

A list containing:

parameters.quantiles	A numeric matrix with lower and upper $100(1-\alpha)\%$ bootstrap percentile confidence limits.
boot.results	A matrix of bootstrap estimates for all parameters over $B$ replications.

References

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

---

Print Method for SSR Fit Objects

Description

Prints results of an object of class SSRfit produced by fit.SSR.ClaytonMWD. The output includes parameter estimates and confidence interval estimates obtained using MLE, LSE, WLSE, and MPS, for the marginal parameters, the Clayton copula parameter $\theta$, and the reliability measure $R$.

Usage

## S3 method for class 'SSRfit'
print(x, ...)

Arguments

x	An object of class SSRfit returned by fit.SSR.ClaytonMWD.
...	Additional arguments passed to the print method. For example, digits controls the number of decimal places used in printed output.

Details

Print SSR Fit Results (Clayton Copula with MWD Marginals)

This method organizes and displays results in a structured format, separating point estimates and interval estimates for all model components.

Value

Invisibly returns the input object x of class "SSRfit". The function is called for its side effects, namely printing formatted summaries of parameter estimates, dependence parameter estimates, and associated confidence intervals to the console.

Examples

data = list(X = TerkosDam, Y = OmerliDam)
fit.SSR = fit.SSR.ClaytonMWD(data, ACI = TRUE, bootstrap = TRUE, B = 5,
                             seed = 2026, one.step = TRUE, alpha = 0.05)
print(fit.SSR)
print(fit.SSR, 3)

---

Stress–Strength Reliability for MWD under Clayton Copula

Description

Computes the stress–strength reliability (SSR) $R = P(X > Y)$, where $X \sim \text{MWD}(a_1, b_1, \lambda_1)$ (strength) and $Y \sim \text{MWD}(a_2, b_2, \lambda_2)$ (stress), with dependence modeled using a Clayton copula.

Usage

Reliability_Clayton_MWD(a1, b1, lambda1, a2, b2, lambda2, theta)

Arguments

a1, b1, lambda1	Parameters of the strength variable $X$, with $a_1 > 0$, $b_1 \ge 0$, and $\lambda_1 \ge 0$.
a2, b2, lambda2	Parameters of the stress variable $Y$, with $a_2 > 0$, $b_2 \ge 0$, and $\lambda_2 \ge 0$.
theta	Clayton copula dependence parameter, $\theta > 0$.

Details

Stress–Strength Reliability under Clayton Copula with MWD Marginals

The stress–strength reliability is defined as $R = P(X > Y)$, which can be expressed using the joint distribution induced by the Clayton copula.

In copula form, the reliability is computed as

$R = \int_{0}^{\infty} F_X(x)^{-(\theta + 1)} \left( F_X(x)^{-\theta} + G_Y(x)^{-\theta} - 1 \right)^{-\left(\frac{1}{\theta}+1\right)} f_X(x)\, dx,$

which can also be written as

$R = \int_{0}^{1} t^{-(\theta + 1)} \left( t^{-\theta} + G_Y(F_X^{-1}(t))^{-\theta} - 1 \right)^{-\left(\frac{1}{\theta}+1\right)} dt,$

where $F_X(x) = 1 - \exp(-a_1 x^{b_1} e^{\lambda_1 x})$ and $G_Y(y) = 1 - \exp(-a_2 y^{b_2} e^{\lambda_2 y})$.

Further theoretical details can be found in vignette("ssr-theory", package = "SSReliabilityClaytonMWD") and Kizilaslan (2026).

Value

A list identical to the output of stats::integrate:

value	Numerical value of the integral (reliability value).
abs.error	Estimated absolute error of the numerical integration.

References

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

Examples

a1 <- 0.75; b1 <- 1.5; lambda1 <- 0.6
a2 <- 1.2; b2 <- 0.5; lambda2 <- 0.9
theta <- 1 # 2, 3, 4, 5
R <- Reliability_Clayton_MWD(a1, b1, lambda1, a2, b2, lambda2, theta)
R$value
# approximated reliability R based on MC method
R_MC <- Reliability_Clayton_MWD_MC(a1, b1, lambda1, a2, b2, lambda2, theta, N = 50000)
R_MC$R

---

Monte Carlo Estimation of Stress–Strength Reliability (MWD + Clayton Copula)

Description

Computes an approximate value of the stress–strength reliability $R = P(X > Y)$ using Monte Carlo integration method under the modified Weibull model with dependence induced by a Clayton copula.

It calculates an approximate value of $R$ using the Monte Carlo integration method based on a random generated MWD sample from $MWD(a_1, b_1, \lambda_1)$.

Usage

Reliability_Clayton_MWD_MC(a1, b1, lambda1, a2, b2, lambda2, theta, N = 10000)

Arguments

a1, b1, lambda1	Parameters of the strength variable $X$, with $a_1 > 0$, $b_1 \ge 0$, and $\lambda_1 \ge 0$.
a2, b2, lambda2	Parameters of the stress variable $Y$, with $a_2 > 0$, $b_2 \ge 0$, and $\lambda_2 \ge 0$.
theta	Clayton copula dependence parameter, $\theta > 0$.
N	Integer. Number of Monte Carlo samples from $X \sim MWD(a_1,b_1,\lambda_1)$.

Details

Monte Carlo Estimation of Stress–Strength Reliability under Clayton Copula

The approximate stress–strength reliability $R$ is approximated via Monte Carlo integration:

$R \approx \frac{1}{N} \sum_{i=1}^{N} T(x_i; \boldsymbol{\Omega_1}, \boldsymbol{\Omega_2}, \theta) = \tilde{R},$

where $\boldsymbol{\Omega_1} = (a_1, b_1, \lambda_1)$ and $\boldsymbol{\Omega_2} = (a_2, b_2, \lambda_2)$ denote the marginal parameter vectors. The function

$T(x; \boldsymbol{\Omega_1}, \boldsymbol{\Omega_2}, \theta) = F_X(x)^{-(\theta + 1)} \left( F_X(x)^{-\theta} + G_Y(x)^{-\theta} - 1 \right)^{-\left(\frac{1}{\theta}+1\right)},$

where $x_1, \dots, x_N$ is a random sample generated from the Modified Weibull distribution $MWD(a_1, b_1, \lambda_1)$.

The accuracy of the approximation improves as sample size $N$ increases. In practice, values around $N = 10^5$ typically provide stable results.

Further details can be found in Kizilaslan (2026).

Value

A list containing:

R	Approximated reliability value based on $N$ generated random samples.
R_MCsample	Monte Carlo approximate of the reliability $R$ based on $N$ generated samples.
sample	Generated random sample from $MWD(a_1, b_1, \lambda_1)$ with $N$ size.

References

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

Examples

# example code
a1 <- 0.75; b1 <- 1.5; lambda1 <- 0.6
a2 <- 1.2; b2 <- 0.5; lambda2 <- 0.9
theta <- 1 # 2, 3, 4, 5
R <- Reliability_Clayton_MWD(a1, b1, lambda1, a2, b2, lambda2, theta)
R$value
# approximated reliability R based on MC method
R_MC <- Reliability_Clayton_MWD_MC(a1, b1, lambda1, a2, b2, lambda2, theta, N = 50000)
R_MC$R

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Random Generation for MWD Marginals via Clayton Copula

Description

Generates bivariate random samples from a dependent stress–strength model where both marginals follow the Modified Weibull Distribution (MWD), and the dependence structure between the variables is modeled using a Clayton copula.

Generates bivariate random samples from a dependent stress–strength model where both marginals follow the Modified Weibull Distribution (MWD), and dependence between variables is modeled using a Clayton copula.

Usage

rMweibull_Clayton(n, a1, b1, lambda1, a2, b2, lambda2, theta)

Arguments

n	Integer. Number of observations to be generated.
a1, b1, lambda1	Parameters of the strength variable $X$, with $a_1 > 0$, $b_1 \ge 0$, and $\lambda_1 \ge 0$.
a2, b2, lambda2	Parameters of the stress variable $Y$, with $a_2 > 0$, $b_2 \ge 0$, and $\lambda_2 \ge 0$.
theta	Clayton copula dependence parameter, $\theta > 0$.

Details

Bivariate Random Data Generation under Clayton Copula with MWD Marginals

This function generates dependent uniform variables using the Clayton copula, which are then transformed via inverse CDFs of the Modified Weibull marginals to obtain $(X, Y)$.

Further details are provided in Kizilaslan (2026).

Value

A list containing:

A list containing:

U	Uniform samples used in the copula construction.
V	Dependent uniform samples generated via the Clayton copula.
X	Simulated observations from $X \sim \mathrm{MWD}(a_1, b_1, \lambda_1)$ obtained by transforming $U$.
Y	Simulated observations from $Y \sim \mathrm{MWD}(a_2, b_2, \lambda_2)$ obtained by transforming $V$.

References

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

Examples

set.seed(123)
n <- 50
a1 <- 0.75; b1 <- 1.5; lambda1 <- 0.6
a2 <- 1.2; b2 <- 0.5; lambda2 <- 0.9
theta <- 1 # 2, 3, 4, 5
# data generation
dat <- rMweibull_Clayton(n, a1, b1, lambda1, a2, b2, lambda2, theta)
str(dat)

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Terkos Dam Data

Description

Terkos Dam is one of the largest dams supplying Istanbul, Türkiye, and is located on the European side. The dataset consists of daily occupancy rates of Istanbul’s dams, retrieved in March 2026 from Istanbul Metropolitan Municipality datasets website https://data.ibb.gov.tr/en.

Usage

TerkosDam

Format

A numeric vector of length 95, representing monthly average occupancy rates.

Details

The data span the period from late October 2000 to mid-February 2024. Monthly average occupancy rates are computed based on the daily data for the period September-December of each year, resulting in a total of 95 observations.

Source

Istanbul Metropolitan Municipality Open Data Portal https://data.ibb.gov.tr/en. Licensed under CC BY 4.0.

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Kendall's Tau-based Estimation of the Clayton Copula Parameter

Description

Estimates the dependence parameter $\theta$ of the Clayton copula using Kendall's tau-based moment estimator.

Usage

theta_Ktau_estimate(data)

Arguments

data	A list containing two numeric vectors: X (strength) and Y (stress).

Details

Kendall's Tau Estimator for the Clayton Copula Parameter

The estimator is derived from the relationship between Kendall's tau and the Clayton copula parameter: $\tau = \theta / (\theta + 2)$.

Value

A numeric scalar giving the estimate of $\theta$ based on Kendall's tau ($\tau$).

Examples

set.seed(123)
n <- 50
a1 <- 0.75; b1 <- 1.5; lambda1 <- 0.6
a2 <- 1.2; b2 <- 0.5; lambda2 <- 0.9
theta <- 5 # 1, 2, 3, 4
# data generation
dat <- SSReliabilityClaytonMWD::rMweibull_Clayton(n, a1, b1, lambda1, a2, b2, lambda2, theta)
theta_Ktau_estimate(dat)

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

Description

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.

Usage

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.

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).

Value

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

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.

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