Estimation of SSR Model Parameters with MWD Marginals via Clayton Copula — fit.SSR.ClaytonMWD • SSReliabilityClaytonMWD

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).

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.

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.

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).

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)
#> Warning: executing %dopar% sequentially: no parallel backend registered
print(fit.SSR)
#> 
#> Fitting Results of Stress-Strength Reliability Model with MWD Marginals via Clayton Copula 
#> 
#> 
#> Table: Parameter Estimates
#> 
#> |     |      a1|      b1| lambda1|      a2|      b2| lambda2|   theta|       R|
#> |:----|-------:|-------:|-------:|-------:|-------:|-------:|-------:|-------:|
#> |mle  | 0.98542| 2.66095| 2.11322| 0.02428| 0.45908| 6.33059| 0.50551| 0.50428|
#> |lse  | 0.07196| 1.72091| 5.71667| 0.01166| 0.00001| 7.06623| 1.09884| 0.49349|
#> |wlse | 0.04261| 1.29079| 6.20629| 0.01154| 0.00001| 7.10656| 1.23631| 0.49824|
#> |mps  | 0.68997| 2.31111| 2.39000| 0.02043| 0.26258| 6.43699| 0.92547| 0.51110|
#> 
#> 
#> Table: 95% Asymptotic Confidence Intervals (MLE)
#> 
#> |       |      a1|      b1| lambda1|      a2|      b2| lambda2|   theta|       R|
#> |:------|-------:|-------:|-------:|-------:|-------:|-------:|-------:|-------:|
#> |lower  | 0.00000| 0.43992| 0.00000| 0.00000| 0.00000| 4.21722| 0.14734| 0.40310|
#> |upper  | 4.33682| 4.88197| 5.79639| 0.06723| 1.39756| 8.44395| 0.86369| 0.60545|
#> |length | 4.33682| 4.44205| 5.79639| 0.06723| 1.39756| 4.22673| 0.71636| 0.20235|
#> 
#> 
#> Table: 95% Bootstrap CIs (MLE)
#> 
#> |       |      a1|      b1| lambda1|      a2|      b2| lambda2|   theta|       R|
#> |:------|-------:|-------:|-------:|-------:|-------:|-------:|-------:|-------:|
#> |lower  | 0.06764| 1.11748| 0.24777| 0.03399| 0.35264| 3.06021| 0.32101| 0.46058|
#> |upper  | 6.38821| 3.86279| 5.14100| 0.64698| 2.66466| 6.31054| 0.76780| 0.57723|
#> |length | 6.32057| 2.74531| 4.89323| 0.61300| 2.31202| 3.25033| 0.44679| 0.11665|
#> 
#> 
#> Table: 95% Bootstrap CIs (LSE)
#> 
#> |       |       a1|      b1| lambda1|      a2|      b2| lambda2|   theta|       R|
#> |:------|--------:|-------:|-------:|-------:|-------:|-------:|-------:|-------:|
#> |lower  |  0.00541| 0.01570| 0.20336| 0.00731| 0.00001| 3.20366| 0.76563| 0.40414|
#> |upper  | 14.54734| 5.14369| 8.55168| 0.52132| 3.04501| 7.79835| 1.16978| 0.49216|
#> |length | 14.54193| 5.12799| 8.34832| 0.51400| 3.04500| 4.59469| 0.40415| 0.08802|
#> 
#> 
#> Table: 95% Bootstrap CIs (WLSE)
#> 
#> |       |      a1|      b1| lambda1|      a2|      b2| lambda2|   theta|       R|
#> |:------|-------:|-------:|-------:|-------:|-------:|-------:|-------:|-------:|
#> |lower  | 0.00562| 0.03705| 2.04712| 0.01145| 0.00402| 4.57018| 1.02749| 0.40953|
#> |upper  | 2.01656| 3.85187| 8.41870| 0.14418| 2.20315| 7.24552| 2.83335| 0.49906|
#> |length | 2.01094| 3.81482| 6.37159| 0.13273| 2.19913| 2.67535| 1.80586| 0.08953|
#> 
#> 
#> Table: 95% Bootstrap CIs (MPS)
#> 
#> |       |      a1|      b1| lambda1|      a2|      b2| lambda2|   theta|       R|
#> |:------|-------:|-------:|-------:|-------:|-------:|-------:|-------:|-------:|
#> |lower  | 0.05764| 0.85330| 0.15124| 0.00905| 0.01090| 3.83919| 0.37910| 0.46552|
#> |upper  | 5.12297| 3.58393| 4.96806| 0.25021| 1.89687| 7.46643| 5.52486| 0.61246|
#> |length | 5.06533| 2.73063| 4.81682| 0.24116| 1.88596| 3.62724| 5.14576| 0.14695|
#> 
#> Kendall's tau estimate for theta: 1.25912