Lindley Approximation Method for Generalized Process Capability Indices under Progressive Type-II Censoring

Shikhar Tyagi, Sumit Kumar, Arvind Pandey, Bhupendra Singh, Vrijesh Tripathi

Introduction

The gpciLindApproxProgII package provides a Bayesian statistical framework for computing Generalized Process Capability Indices (GPCIs) under Progressive Type-II Censored Data using Lindley’s 3rd-order Approximation Method.

Progressive Type-II Censoring Model

Let \(n\) units be placed on test and \(m\) failure times \(X = (x_1, x_2, \dots, x_m)\) be observed under progressive removal scheme \(R = (R_1, R_2, \dots, R_m)\). The progressive log-likelihood function is:

\[\ell(\theta) = \sum_{i=1}^m \log f(x_i; \theta) + \sum_{i=1}^m R_i \log S(x_i; \theta)\]

Example Analysis

Below is a demonstration of fitting progressive Type-II failure data with custom user functions or built-in distributions.

# Failure times and progressive removal scheme
x <- c(0.5, 1.2, 2.1, 3.4, 4.8)
r <- c(1, 0, 2, 0, 1)

# Fit model using Lindley approximation and chain generation
fit <- lindley_prog_gpci(
  x = x,
  r_removals = r,
  distribution = dist_weibull(),
  USL = 6, LSL = 0,
  chain_length = 200,
  burn_in = 50,
  thinning = 1,
  B = 50
)

# Print Summary Table
summary(fit)
#>     Index MLE_Estimate Lindley_Estimate    Chain_Mean          Bias
#> 1     Cpy 4.953015e-01     4.247116e-01  4.336701e-01 -6.163136e-02
#> 2      Cp 1.000000e+04     1.000000e+04  7.664631e+03 -2.335369e+03
#> 3     Cpk 2.977036e-73    -3.832680e-43  2.148941e-03  2.148941e-03
#> 4     Cpu 2.000000e+04     2.000000e+04  1.532922e+04 -4.670785e+03
#> 5     Cpl 2.977036e-73    -3.832680e-43  4.733695e-02  4.733695e-02
#> 6     Cpm 3.333333e-01     3.333333e-01  2.899036e-01 -4.342977e-02
#> 7    Cpmk 9.923452e-78    -1.277560e-47  2.205904e-03  2.205904e-03
#> 8    CpTk 1.249103e-01     1.102875e-01  9.896926e-02 -2.594101e-02
#> 9    Spmk 2.141650e+03     1.977308e+03  1.195595e+03 -9.460549e+02
#> 10    Cpc 1.000000e+04     1.000000e+04  7.666467e+03 -2.333533e+03
#> 11   CNpk 0.000000e+00     0.000000e+00 -1.018422e-03 -1.018422e-03
#> 12  CNpmc 4.792753e-02     4.136523e-02  3.793141e-02 -9.996121e-03
#> 13 CNpmkc 0.000000e+00     0.000000e+00  2.175997e-04  2.175997e-04
#> 14  Cp_uv 0.000000e+00     0.000000e+00  2.205904e-03  2.205904e-03
#>             MSE   Risk_Value   HPD90_Lower  HPD90_Upper   HPD95_Lower
#> 1  2.975805e-02 3.351775e-02  2.288093e-01 6.083192e-01  2.148351e-01
#> 2  2.288477e+07 2.289410e+07  1.476244e-01 1.000000e+04  1.140410e-01
#> 3  8.481060e-04 8.527256e-04 -4.559086e-02 1.596440e-02 -7.412248e-02
#> 4  9.154097e+07 9.155965e+07  1.596440e-02 2.000000e+04 -3.474110e-02
#> 5  1.374923e-02 1.600780e-02  0.000000e+00 2.789130e-01  0.000000e+00
#> 6  9.320879e-03 1.119344e-02  1.319767e-01 3.333333e-01  1.033579e-01
#> 7  7.933548e-04 7.982226e-04 -3.922170e-02 1.496891e-02 -7.055094e-02
#> 8  1.396455e-03 2.066491e-03  6.034201e-02 1.442729e-01  4.814858e-02
#> 9  1.556635e+06 1.560411e+06  3.704777e-05 2.259423e+03  3.704777e-05
#> 10 2.285828e+07 2.286760e+07  1.473944e-01 1.000000e+04  1.211196e-01
#> 11 4.470092e-03 4.471129e-03 -1.076143e-01 3.449242e-02 -1.147313e-01
#> 12 2.850084e-04 3.847646e-04  1.361870e-02 5.463672e-02  1.094069e-02
#> 13 2.314370e-06 2.361721e-06 -1.752894e-03 8.733899e-05 -1.941208e-03
#> 14 7.933548e-04 7.982226e-04 -3.922170e-02 1.496891e-02 -7.055094e-02
#>     HPD95_Upper   HPD99_Lower  HPD99_Upper    HW_Stat HW_Pvalue HW_Passed
#> 1  7.374006e-01  2.148351e-01 9.766824e-01 0.12897949       0.5      TRUE
#> 2  1.000000e+04  3.283705e-02 1.000000e+04 0.12875664       0.5      TRUE
#> 3  3.836402e-02 -7.412248e-02 1.694900e-01 0.05149238       0.5      TRUE
#> 4  2.000000e+04 -7.219118e-02 2.000000e+04 0.12875530       0.5      TRUE
#> 5  3.159760e-01  0.000000e+00 3.494605e-01 0.05204297       0.5      TRUE
#> 6  3.333333e-01  3.099694e-02 3.333333e-01 0.08028526       0.5      TRUE
#> 7  3.560926e-02 -7.055094e-02 1.668443e-01 0.05279078       0.5      TRUE
#> 8  1.478136e-01  3.898187e-02 1.658662e-01 0.16286150       0.5      TRUE
#> 9  2.441432e+03  3.704777e-05 2.854394e+03 0.11041115       0.5      TRUE
#> 10 1.000000e+04  3.099694e-02 1.000000e+04 0.12823249       0.5      TRUE
#> 11 1.548203e-01 -2.981325e-01 2.828334e-01 0.06465875       0.5      TRUE
#> 12 5.555551e-02  3.140615e-03 5.555551e-02 0.07473083       0.5      TRUE
#> 13 2.376127e-03 -2.338032e-03 8.973482e-03 0.05086908       0.5      TRUE
#> 14 3.560926e-02 -7.055094e-02 1.668443e-01 0.05279078       0.5      TRUE
#>    Convergence_Prob Boot95_Lower Boot95_Upper
#> 1               0.5 3.222651e-01 1.584475e+00
#> 2               0.5 2.250152e+03 1.000000e+04
#> 3               0.5 0.000000e+00 7.402421e-08
#> 4               0.5 4.500253e+03 2.000000e+04
#> 5               0.5 0.000000e+00 7.402421e-08
#> 6               0.5 2.220331e-01 3.333333e-01
#> 7               0.5 0.000000e+00 2.139833e-10
#> 8               0.5 2.368319e-02 1.576375e-01
#> 9               0.5 5.754798e+00 2.472119e+03
#> 10              0.5 2.250173e+03 1.000000e+04
#> 11              0.5 0.000000e+00 7.032465e-10
#> 12              0.5 2.391953e-02 5.555556e-02
#> 13              0.5 0.000000e+00 1.711670e-11
#> 14              0.5 0.000000e+00 2.139833e-10

Plotting Results

plot(fit)