Lindley Approximation Method for Generalized Process Capability Indices

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Introduction

The gpciLindleyApprox package provides a generalized framework for computing, estimating, and validating Generalized Process Capability Indices (GPCIs) using the Lindley Approximation Method for uncensored process data under Bayesian inference.

Supported GPCIs include: - \(C_{py}\) (Process Capability Index based on Yield; Maiti, Saha & Nanda, 2010) - \(C_p, C_{pk}, C_{pu}, C_{pl}, C_{pm}, C_{pmk}\) - \(C_{pTk}\) (Saha, Dey & Maiti, 2019) - \(S_{pmk}\) (Dey & Saha, 2019) - \(C_{pc}\) (Saha, Dey & Nadarajah, 2022) - \(CN_{pk}\) (Saha, Dey & Maiti, 2018) - \(CN_{pmc}\) (Alotaibi, Dey & Saha, 2022) - \(CN_{pmkc}\) (Saha, Tripathi & Dey, 2024) - \(C_p(u,v)\) Vännman’s family (1995)

Basic Example with Custom PDF/CDF

Below is an example estimating GPCIs for uncensored data generated from a normal process using the high-level user interface function gpci_lindley():

set.seed(42)
data_obs <- rnorm(50, mean = 10, sd = 1)

# Fit GPCIs using Lindley approximation and Bootstrap CIs
fit_res <- gpci_lindley(
  data = data_obs,
  pdf = function(x, mean = 0, sd = 1) dnorm(x, mean = mean, sd = sd),
  cdf = function(x, mean = 0, sd = 1) pnorm(x, mean = mean, sd = sd),
  chain_length = 300,
  burn_in = 50,
  thinning = 1,
  USL = 13,
  LSL = 7,
  B = 100
)

# Display Summary Diagnostics Table
summary(fit_res)
#>     Index  MLE_Estimate Lindley_Estimate    Chain_Mean          Bias
#> 1     Cpy  8.768377e-01     8.462389e-01  8.517148e-01 -2.512291e-02
#> 2      Cp  1.000000e+04     1.000000e+04  1.000000e+04  0.000000e+00
#> 3     Cpk -2.333333e+04    -2.333333e+04 -2.333333e+04  0.000000e+00
#> 4     Cpu  4.333333e+04     4.333333e+04  4.333333e+04  0.000000e+00
#> 5     Cpl -2.333333e+04    -2.333333e+04 -2.333333e+04  0.000000e+00
#> 6     Cpm  1.000000e-01     1.000000e-01  1.000000e-01  0.000000e+00
#> 7    Cpmk -2.333333e-01    -2.333333e-01 -2.333333e-01  0.000000e+00
#> 8    CpTk  1.612155e-01     1.409581e-01  1.501106e-01 -1.110489e-02
#> 9    Spmk  1.452600e+03     1.229395e+03  1.348160e+03 -1.044404e+02
#> 10    Cpc  1.000000e+04     1.000000e+04  1.000000e+04  0.000000e+00
#> 11   CNpk -9.333830e-03    -1.420153e-02 -1.406346e-02 -4.729627e-03
#> 12  CNpmc  1.652457e-02     1.644815e-02  1.645452e-02 -7.004957e-05
#> 13 CNpmkc -1.551438e-04    -2.360532e-04 -2.337581e-04 -7.861429e-05
#> 14  Cp_uv -2.333333e-01    -2.333333e-01 -2.333333e-01  0.000000e+00
#>             MSE   Risk_Value   HPD90_Lower   HPD90_Upper   HPD95_Lower
#> 1  6.813539e-03 7.442066e-03  7.210851e-01  9.771918e-01  7.210851e-01
#> 2  0.000000e+00 0.000000e+00  1.000000e+04  1.000000e+04  1.000000e+04
#> 3  0.000000e+00 0.000000e+00 -2.333333e+04 -2.333333e+04 -2.333333e+04
#> 4  0.000000e+00 0.000000e+00  4.333333e+04  4.333333e+04  4.333333e+04
#> 5  0.000000e+00 0.000000e+00 -2.333333e+04 -2.333333e+04 -2.333333e+04
#> 6  0.000000e+00 0.000000e+00  1.000000e-01  1.000000e-01  1.000000e-01
#> 7  0.000000e+00 0.000000e+00 -2.333333e-01 -2.333333e-01 -2.333333e-01
#> 8  2.436033e-04 3.666940e-04  1.343623e-01  1.653620e-01  1.271329e-01
#> 9  5.056062e+04 5.097038e+04  9.809115e+02  1.622972e+03  9.139125e+02
#> 10 0.000000e+00 0.000000e+00  1.000000e+04  1.000000e+04  1.000000e+04
#> 11 1.269809e-04 1.493326e-04 -2.921677e-02 -1.430780e-03 -3.388581e-02
#> 12 2.150084e-08 2.640772e-08  1.626008e-02  1.663420e-02  1.620269e-02
#> 13 3.508226e-08 4.126238e-08 -4.856315e-04 -2.378194e-05 -5.632387e-04
#> 14 0.000000e+00 0.000000e+00 -2.333333e-01 -2.333333e-01 -2.333333e-01
#>      HPD95_Upper   HPD99_Lower   HPD99_Upper    HW_Stat HW_Pvalue HW_Passed
#> 1   1.009890e+00  6.936857e-01  1.038376e+00 0.06135820       0.5      TRUE
#> 2   1.000000e+04  1.000000e+04  1.000000e+04 0.00000000       0.5      TRUE
#> 3  -2.333333e+04 -2.333333e+04 -2.333333e+04 0.00000000       0.5      TRUE
#> 4   4.333333e+04  4.333333e+04  4.333333e+04 0.00000000       0.5      TRUE
#> 5  -2.333333e+04 -2.333333e+04 -2.333333e+04 0.00000000       0.5      TRUE
#> 6   1.000000e-01  1.000000e-01  1.000000e-01 0.00000000       0.5      TRUE
#> 7  -2.333333e-01 -2.333333e-01 -2.333333e-01 0.00000000       0.5      TRUE
#> 8   1.654190e-01  1.165332e-01  1.654190e-01 0.18050519       0.5      TRUE
#> 9   1.659205e+03  8.920865e+02  1.742970e+03 0.22773878       0.5      TRUE
#> 10  1.000000e+04  1.000000e+04  1.000000e+04 0.00000000       0.5      TRUE
#> 11 -1.379886e-03 -4.986817e-02 -6.254820e-04 0.10081918       0.5      TRUE
#> 12  1.663602e-02  1.606730e-02  1.665190e-02 0.08155822       0.5      TRUE
#> 13 -2.293601e-05 -8.288921e-04 -1.039655e-05 0.10081918       0.5      TRUE
#> 14 -2.333333e-01 -2.333333e-01 -2.333333e-01 0.00000000       0.5      TRUE
#>    Convergence_Prob  Boot95_Lower  Boot95_Upper
#> 1               0.5  7.354685e-01  1.049653e+00
#> 2               0.5  1.000000e+04  1.000000e+04
#> 3               0.5 -2.333333e+04 -2.333333e+04
#> 4               0.5  4.333333e+04  4.333333e+04
#> 5               0.5 -2.333333e+04 -2.333333e+04
#> 6               0.5  1.000000e-01  1.000000e-01
#> 7               0.5 -2.333333e-01 -2.333333e-01
#> 8               0.5  1.194694e-01  1.640426e-01
#> 9               0.5  8.386396e+02  1.632434e+03
#> 10              0.5  1.000000e+04  1.000000e+04
#> 11              0.5 -3.407695e-02 -1.255760e-03
#> 12              0.5  1.620628e-02  1.663931e-02
#> 13              0.5 -5.664156e-04 -2.087283e-05
#> 14              0.5 -2.333333e-01 -2.333333e-01

Using Built-in Distribution Objects

The package provides pre-defined distribution objects such as dist_normal(), dist_weibull(), dist_gamma(), dist_logistic_exponential(), and dist_exponentiated_exponential():

dist_weib <- dist_weibull()

fit_weib <- lindley_gpci(
  data = rweibull(50, shape = 2, scale = 5),
  distribution = dist_weib,
  chain_length = 300,
  burn_in = 50,
  thinning = 1,
  USL = 8,
  LSL = 1,
  B = 100
)

summary(fit_weib)
#>     Index  MLE_Estimate Lindley_Estimate    Chain_Mean          Bias
#> 1     Cpy  6.107152e-01     5.998779e-01  6.002447e-01 -1.047052e-02
#> 2      Cp  1.166667e+04     1.166667e+04  1.166667e+04  0.000000e+00
#> 3     Cpk -3.333333e+03    -3.333333e+03 -3.333333e+03  0.000000e+00
#> 4     Cpu  2.666667e+04     2.666667e+04  2.666667e+04  0.000000e+00
#> 5     Cpl -3.333333e+03    -3.333333e+03 -3.333333e+03  0.000000e+00
#> 6     Cpm  2.592593e-01     2.592593e-01  2.592593e-01  0.000000e+00
#> 7    Cpmk -7.407407e-02    -7.407407e-02 -7.407407e-02  0.000000e+00
#> 8    CpTk  1.305948e-01     1.306411e-01  1.295821e-01 -1.012621e-03
#> 9    Spmk  2.230839e+03     2.328966e+03  2.207974e+03 -2.286572e+01
#> 10    Cpc  1.166667e+04     1.166667e+04  1.166667e+04  0.000000e+00
#> 11   CNpk -5.048286e-02    -5.504173e-02 -5.498003e-02 -4.497162e-03
#> 12  CNpmc  3.455816e-02     3.421865e-02  3.421933e-02 -3.388242e-04
#> 13 CNpmkc -1.864687e-03    -2.033079e-03 -2.030799e-03 -1.661118e-04
#> 14  Cp_uv -7.407407e-02    -7.407407e-02 -7.407407e-02  0.000000e+00
#>             MSE   Risk_Value   HPD90_Lower   HPD90_Upper   HPD95_Lower
#> 1  3.451728e-03 3.561168e-03  5.136530e-01  6.910673e-01  4.914843e-01
#> 2  0.000000e+00 0.000000e+00  1.166667e+04  1.166667e+04  1.166667e+04
#> 3  0.000000e+00 0.000000e+00 -3.333333e+03 -3.333333e+03 -3.333333e+03
#> 4  0.000000e+00 0.000000e+00  2.666667e+04  2.666667e+04  2.666667e+04
#> 5  0.000000e+00 0.000000e+00 -3.333333e+03 -3.333333e+03 -3.333333e+03
#> 6  0.000000e+00 0.000000e+00  2.592593e-01  2.592593e-01  2.592593e-01
#> 7  0.000000e+00 0.000000e+00 -7.407407e-02 -7.407407e-02 -7.407407e-02
#> 8  2.159494e-04 2.169746e-04  1.054837e-01  1.515815e-01  1.026981e-01
#> 9  1.386249e+05 1.387084e+05  1.482967e+03  2.703227e+03  1.374317e+03
#> 10 0.000000e+00 0.000000e+00  1.166667e+04  1.166667e+04  1.166667e+04
#> 11 7.179439e-04 7.381532e-04 -9.445815e-02 -1.907772e-02 -1.037342e-01
#> 12 1.333595e-06 1.448391e-06  3.267486e-02  3.580915e-02  3.183853e-02
#> 13 9.795230e-07 1.007115e-06 -3.489004e-03 -7.046744e-04 -3.831634e-03
#> 14 0.000000e+00 0.000000e+00 -7.407407e-02 -7.407407e-02 -7.407407e-02
#>      HPD95_Upper   HPD99_Lower   HPD99_Upper    HW_Stat HW_Pvalue HW_Passed
#> 1   7.100755e-01  4.554290e-01  7.548129e-01 0.11658476       0.5      TRUE
#> 2   1.166667e+04  1.166667e+04  1.166667e+04 0.00000000       0.5      TRUE
#> 3  -3.333333e+03 -3.333333e+03 -3.333333e+03 0.00000000       0.5      TRUE
#> 4   2.666667e+04  2.666667e+04  2.666667e+04 0.00000000       0.5      TRUE
#> 5  -3.333333e+03 -3.333333e+03 -3.333333e+03 0.00000000       0.5      TRUE
#> 6   2.592593e-01  2.592593e-01  2.592593e-01 0.00000000       0.5      TRUE
#> 7  -7.407407e-02 -7.407407e-02 -7.407407e-02 0.00000000       0.5      TRUE
#> 8   1.535985e-01  9.953666e-02  1.607303e-01 0.35399092       0.1      TRUE
#> 9   2.813694e+03  1.202666e+03  2.990766e+03 0.03601605       0.5      TRUE
#> 10  1.166667e+04  1.166667e+04  1.166667e+04 0.00000000       0.5      TRUE
#> 11 -1.169827e-02 -1.479361e-01 -1.169827e-02 0.28295145       0.5      TRUE
#> 12  3.580915e-02  3.067226e-02  3.616492e-02 0.16195705       0.5      TRUE
#> 13 -4.320995e-04 -5.464320e-03 -4.320995e-04 0.28295145       0.5      TRUE
#> 14 -7.407407e-02 -7.407407e-02 -7.407407e-02 0.00000000       0.5      TRUE
#>    Convergence_Prob  Boot95_Lower  Boot95_Upper
#> 1               0.5  5.182434e-01  7.236522e-01
#> 2               0.5  1.166667e+04  1.166667e+04
#> 3               0.5 -3.333333e+03 -3.333333e+03
#> 4               0.5  2.666667e+04  2.666667e+04
#> 5               0.5 -3.333333e+03 -3.333333e+03
#> 6               0.5  2.592593e-01  2.592593e-01
#> 7               0.5 -7.407407e-02 -7.407407e-02
#> 8               0.1  9.887280e-02  1.547374e-01
#> 9               0.5  1.319628e+03  2.724054e+03
#> 10              0.5  1.166667e+04  1.166667e+04
#> 11              0.5 -9.886863e-02 -2.096734e-02
#> 12              0.5  3.259221e-02  3.592499e-02
#> 13              0.5 -3.651914e-03 -7.744714e-04
#> 14              0.5 -7.407407e-02 -7.407407e-02

References

Lindley, D. V. (1980). Approximate Bayesian methods. Trabajos de Estadística y de Investigación Operativa, 31(1), 223-245.
Maiti, S. S., Saha, M., & Nanda, A. K. (2010). On generalizing process capability indices. Quality Technology & Quantitative Management, 7(3), 279-289.
Saha, M., Dey, S., & Maiti, S. S. (2018). Parametric and non-parametric bootstrap confidence intervals of CNpk for exponential power distribution. Journal of Industrial and Production Engineering, 35(3), 160-169.
Dey, S., & Saha, M. (2019). Assessing the process capability index Spmk using improved estimators. Life Cycle Reliability and Safety Engineering, 8, 81-88.
Saha, M., Dey, S., & Maiti, S. S. (2019). Bootstrap confidence intervals of CpTk for two parameter logistic exponential distribution with applications. International Journal of System Assurance Engineering and Management.
Alotaibi, R., Dey, S., & Saha, M. (2022). Estimation and confidence intervals of a new PCI CNpmc for logistic-exponential process distribution. Journal of Mathematics, 2022, 3135264.
Saha, M., Dey, S., & Nadarajah, S. (2022). Parametric inference of the process capability index Cpc for exponentiated exponential distribution. Journal of Applied Statistics, 49(16), 4097-4121.
Saha, M., Tripathi, V., & Dey, S. (2024). Classical inference of a new PCI CNpmkc for logistic-exponential process distribution. International Journal of Reliability, Quality and Safety Engineering, 31(3), 2450013.

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