# Evaluation of Bootstrap Confidence Intervals Using a New Non-Normal Process Capability Index

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Bootstrap Confidence Intervals

- Obtained a bootstrap sample of size $n$ i.e., ${x}_{1}{}^{*},{x}_{2}{}^{*},{x}_{3}{}^{*},\cdots ,{x}_{n}{}^{*}$ from original sample by putting $1/n$ as mass at each point.
- Let ${X}_{m}{}^{*}$ where $1\le m\le n$ be the mth bootstrap sample, then mth bootstrap estimator of $\theta $ is computed as$${\hat{\theta}}_{m}^{*}=\hat{\theta}\left({x}_{1}{}^{*},{x}_{2}{}^{*},{x}_{3}{}^{*},\cdots ,{x}_{n}{}^{*}\right)$$
- We get ${n}^{n}$ re-samples, for each sample a values of ${\hat{\theta}}_{m}^{*}$ are obtained. Each of these would be estimate of $\hat{\theta}$. From the set all these estimates would constitute an empirical bootstrap distribution of $\hat{\theta}$, see [1].

#### 2.1.1. Standard Bootstrap (SB) Confidence Interval

#### 2.1.2. Percentile Bootstrap (PB) Confidence Interval

#### 2.1.3. Bias-Corrected Percentile Bootstrap (BCPB) Confidence Interval

- The probability ${p}_{0}$ is calculated using the (ordered) distribution of ${\hat{\theta}}^{*}\left(i\right)$ as:$${p}_{0}=pr\left({\hat{\theta}}^{*}\le \hat{\theta}\right)$$
- Let $\varnothing $ and ${\varnothing}^{-1}$ represents the cumulative and inverse cumulative distribution functions ofa standard normal variable $z$, then:$${z}_{0}={\varnothing}^{-1}\left({p}_{0}\right)$$
- The percentiles of the ordered distribution of ${\hat{\theta}}^{*}$ is obtained as:$${P}_{L}=\varnothing \left(2{z}_{0}+{z}_{\frac{\alpha}{2}}\right)$$$${P}_{U}=\varnothing \left(2{z}_{0}+{z}_{1-\frac{\alpha}{2}}\right)$$

## 3. Simulation Study

## 4. Illustrative Examples

## 5. Comparison of Proposed C_{pkw} Index with the Existing C_{pk}

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Density plot and Q-Q plot of the fitted Weibull distribution for the breaking stress of carbon fibres.

**Figure 2.**Capability of Weibull distribution for proposed method using breaking stress of carbon fibers.

**Table 1.**The average widths 95% bootstrap confidence intervals and coverage probabilities of ${C}_{pkw}$ for Weibull distribution when η = 5.0.

n | β | True ${\mathit{C}}_{\mathit{N}\mathit{p}\mathit{k}}$ | Average Widths | Coverage Probabilities | ||||
---|---|---|---|---|---|---|---|---|

SB | PB | BCPB | SB | PB | BCPB | |||

10 | 2.00 | 1.4114 | 1.3825 | 1.1175 | 0.9672 | 0.8584 | 0.9222 | |

15 | 2.00 | 1.0081 | 0.9958 | 0.8708 | 0.9538 | 0.8922 | 0.9306 | |

20 | 2.00 | 0.8208 | 0.8129 | 0.7362 | 0.9554 | 0.9034 | 0.9422 | |

25 | 2.00 | 0.6866 | 0.7115 | 0.7054 | 0.6518 | 0.9528 | 0.9166 | 0.9430 |

30 | 2.00 | 0.6341 | 0.6291 | 0.5894 | 0.9512 | 0.9158 | 0.9400 | |

35 | 2.00 | 0.5751 | 0.5709 | 0.5399 | 0.9520 | 0.9234 | 0.9430 | |

40 | 2.00 | 0.5322 | 0.5288 | 0.5033 | 0.9528 | 0.9282 | 0.9434 | |

10 | 2.50 | 1.7474 | 1.7088 | 1.3516 | 0.9682 | 0.8530 | 0.9196 | |

15 | 2.50 | 1.2333 | 1.2175 | 1.0497 | 0.9556 | 0.8874 | 0.9316 | |

20 | 2.50 | 0.9976 | 0.9877 | 0.8853 | 0.9558 | 0.8998 | 0.9412 | |

25 | 2.50 | 0.8957 | 0.8616 | 0.8544 | 0.7829 | 0.9530 | 0.9120 | 0.9404 |

30 | 2.50 | 0.7660 | 0.7600 | 0.7073 | 0.9512 | 0.9132 | 0.9408 | |

35 | 2.50 | 0.6936 | 0.6882 | 0.6475 | 0.9518 | 0.9206 | 0.9412 | |

40 | 2.50 | 0.6411 | 0.6372 | 0.6034 | 0.9538 | 0.9274 | 0.9432 | |

10 | 3.00 | 2.0854 | 2.0376 | 1.5885 | 0.9704 | 0.8480 | 0.9178 | |

15 | 3.00 | 1.4601 | 1.4409 | 1.2302 | 0.9574 | 0.8824 | 0.9318 | |

20 | 3.00 | 1.1759 | 1.1642 | 1.0361 | 0.9574 | 0.8984 | 0.9392 | |

25 | 3.00 | 1.1049 | 1.0131 | 1.0046 | 0.9157 | 0.9538 | 0.9106 | 0.9400 |

30 | 3.00 | 0.8993 | 0.8923 | 0.8268 | 0.9514 | 0.9120 | 0.9394 | |

35 | 3.00 | 0.8135 | 0.8070 | 0.7567 | 0.9510 | 0.9190 | 0.9390 | |

40 | 3.00 | 0.7515 | 0.7467 | 0.7051 | 0.9538 | 0.9254 | 0.9428 | |

10 | 3.50 | 2.4241 | 2.3668 | 1.8262 | 0.9726 | 0.8452 | 0.9178 | |

15 | 3.50 | 1.6876 | 1.6646 | 1.4118 | 0.9584 | 0.8798 | 0.9314 | |

20 | 3.50 | 1.3550 | 1.3413 | 1.1881 | 0.9580 | 0.8956 | 0.9374 | |

25 | 3.50 | 1.3140 | 1.1654 | 1.1552 | 1.0494 | 0.9538 | 0.9078 | 0.9382 |

30 | 3.50 | 1.0335 | 1.0254 | 0.9474 | 0.9516 | 0.9114 | 0.9390 | |

35 | 3.50 | 0.9342 | 0.9269 | 0.8667 | 0.9508 | 0.9170 | 0.9390 | |

40 | 3.50 | 0.8626 | 0.8569 | 0.8074 | 0.9530 | 0.9248 | 0.9426 |

**Table 2.**The average widths 95% bootstrap confidence intervals and coverage probabilities of ${C}_{pkw}$ for Weibull distribution when η = 5.5.

n | β | True ${\mathit{C}}_{\mathit{N}\mathit{p}\mathit{k}}$ | Average Widths | Coverage Probabilities | ||||
---|---|---|---|---|---|---|---|---|

SB | PB | BCPB | SB | PB | BCPB | |||

10 | 2.00 | 1.4091 | 1.3832 | 1.1300 | 0.9620 | 0.8576 | 0.9202 | |

15 | 2.00 | 1.0138 | 1.0025 | 0.8852 | 0.9468 | 0.8914 | 0.9244 | |

20 | 2.00 | 0.8329 | 0.8251 | 0.7529 | 0.9464 | 0.9030 | 0.9354 | |

25 | 2.00 | 0.7361 | 0.7264 | 0.7201 | 0.6692 | 0.9464 | 0.9152 | 0.9380 |

30 | 2.00 | 0.6510 | 0.6454 | 0.6073 | 0.9436 | 0.9146 | 0.9350 | |

35 | 2.00 | 0.5933 | 0.5885 | 0.5584 | 0.9448 | 0.9232 | 0.9394 | |

40 | 2.00 | 0.5512 | 0.5472 | 0.5221 | 0.9480 | 0.9272 | 0.9410 | |

10 | 2.50 | 1.7492 | 1.7141 | 1.3717 | 0.9654 | 0.8522 | 0.9180 | |

15 | 2.50 | 1.2442 | 1.2296 | 1.0711 | 0.9488 | 0.8854 | 0.9264 | |

20 | 2.50 | 1.0157 | 1.0061 | 0.9090 | 0.9486 | 0.8998 | 0.9356 | |

25 | 2.50 | 0.9576 | 0.8826 | 0.8750 | 0.8070 | 0.9482 | 0.9114 | 0.9372 |

30 | 2.50 | 0.7890 | 0.7824 | 0.7316 | 0.9452 | 0.9130 | 0.9362 | |

35 | 2.50 | 0.7179 | 0.7119 | 0.6721 | 0.9450 | 0.9190 | 0.9390 | |

40 | 2.50 | 0.6661 | 0.6615 | 0.6281 | 0.9498 | 0.9270 | 0.9424 | |

10 | 3.00 | 2.0925 | 2.0484 | 1.6163 | 0.9676 | 0.8474 | 0.9160 | |

15 | 3.00 | 1.4770 | 1.4589 | 1.2595 | 0.9516 | 0.8820 | 0.9272 | |

20 | 3.00 | 1.2005 | 1.1890 | 1.0670 | 0.9518 | 0.8970 | 0.9336 | |

25 | 3.00 | 1.1792 | 1.0405 | 1.0313 | 0.9467 | 0.9480 | 0.9092 | 0.9370 |

30 | 3.00 | 0.9286 | 0.9208 | 0.8575 | 0.9464 | 0.9106 | 0.9364 | |

35 | 3.00 | 0.8438 | 0.8369 | 0.7872 | 0.9456 | 0.9178 | 0.9376 | |

40 | 3.00 | 0.7823 | 0.7767 | 0.7355 | 0.9504 | 0.9244 | 0.9422 | |

10 | 3.50 | 2.4375 | 2.3839 | 1.8629 | 0.9688 | 0.8460 | 0.9142 | |

15 | 3.50 | 1.7111 | 1.6895 | 1.4488 | 0.9528 | 0.8790 | 0.9264 | |

20 | 3.50 | 1.3865 | 1.3730 | 1.2264 | 0.9526 | 0.8954 | 0.9342 | |

25 | 3.50 | 1.4007 | 1.1994 | 1.1887 | 1.0869 | 0.9480 | 0.9076 | 0.9362 |

30 | 3.50 | 1.0691 | 1.0602 | 0.9846 | 0.9478 | 0.9106 | 0.9362 | |

35 | 3.50 | 0.9707 | 0.9629 | 0.9033 | 0.9448 | 0.9164 | 0.9376 | |

40 | 3.50 | 0.8994 | 0.8928 | 0.8437 | 0.9508 | 0.9248 | 0.9416 |

**Table 3.**The average widths 95% bootstrap confidence intervals and coverage probabilities of ${C}_{pkw}$ for Weibull distribution when η = 6.0.

n | β | True ${\mathit{C}}_{\mathit{N}\mathit{p}\mathit{k}}$ | Average Widths | Coverage Probabilities | ||||
---|---|---|---|---|---|---|---|---|

SB | PB | BCPB | SB | PB | BCPB | |||

10 | 2.00 | 1.3752 | 1.3523 | 1.1176 | 0.9630 | 0.8610 | 0.9204 | |

15 | 2.00 | 0.9894 | 0.9811 | 0.8771 | 0.9448 | 0.8924 | 0.9246 | |

20 | 2.00 | 0.8162 | 0.8101 | 0.7484 | 0.9434 | 0.9040 | 0.9320 | |

25 | 2.00 | 0.7813 | 0.7141 | 0.7085 | 0.6664 | 0.9418 | 0.9160 | 0.9302 |

30 | 2.00 | 0.6430 | 0.6380 | 0.6065 | 0.9374 | 0.9146 | 0.9292 | |

35 | 2.00 | 0.5894 | 0.5845 | 0.5597 | 0.9378 | 0.9220 | 0.9286 | |

40 | 2.00 | 0.5504 | 0.5461 | 0.5252 | 0.9402 | 0.9284 | 0.9324 | |

10 | 2.50 | 1.7010 | 1.6702 | 1.3518 | 0.9656 | 0.8584 | 0.9186 | |

15 | 2.50 | 1.2077 | 1.1971 | 1.0573 | 0.9478 | 0.9018 | 0.9282 | |

20 | 2.50 | 0.9886 | 0.9816 | 0.9000 | 0.9476 | 0.9018 | 0.9326 | |

25 | 2.50 | 1.0142 | 0.8608 | 0.8545 | 0.7997 | 0.9412 | 0.9126 | 0.9328 |

30 | 2.50 | 0.7727 | 0.7673 | 0.7268 | 0.9402 | 0.9142 | 0.9298 | |

35 | 2.50 | 0.7068 | 0.7016 | 0.6701 | 0.9394 | 0.9198 | 0.9316 | |

40 | 2.50 | 0.6590 | 0.6544 | 0.6282 | 0.9424 | 0.9272 | 0.9322 | |

10 | 3.00 | 2.0275 | 1.9882 | 1.5862 | 0.9686 | 0.8550 | 0.9188 | |

15 | 3.00 | 1.4260 | 1.4129 | 1.2372 | 0.9514 | 0.8858 | 0.9298 | |

20 | 3.00 | 1.1603 | 1.1524 | 1.0509 | 0.9496 | 0.9006 | 0.9334 | |

25 | 3.00 | 1.2470 | 1.0064 | 0.9993 | 0.9322 | 0.9436 | 0.9106 | 0.9344 |

30 | 3.00 | 0.9010 | 0.8952 | 0.8462 | 0.9414 | 0.9142 | 0.9302 | |

35 | 3.00 | 0.8225 | 0.8173 | 0.7795 | 0.9402 | 0.9180 | 0.9338 | |

40 | 3.00 | 0.7657 | 0.7610 | 0.7300 | 0.9428 | 0.9252 | 0.9332 | |

10 | 3.50 | 2.3538 | 2.3066 | 1.8190 | 0.9702 | 0.8548 | 0.9174 | |

15 | 3.50 | 1.6434 | 1.6276 | 1.4158 | 0.9540 | 0.8854 | 0.9300 | |

20 | 3.50 | 1.3308 | 1.3220 | 1.2002 | 0.9504 | 0.9002 | 0.9332 | |

25 | 3.50 | 1.4798 | 1.1504 | 1.1430 | 1.0629 | 0.9460 | 0.9098 | 0.9338 |

30 | 3.50 | 1.0274 | 1.0215 | 0.8866 | 0.9420 | 0.9144 | 0.9352 | |

35 | 3.50 | 0.9362 | 0.9309 | 0.8866 | 0.9420 | 0.9194 | 0.9340 | |

40 | 3.50 | 0.8700 | 0.8653 | 0.8298 | 0.9448 | 0.9240 | 0.9366 |

**Table 4.**Bootstrap confidence intervals and widths of the new ${C}_{pkw}$ and Clements traditional ${C}_{pk}$.

Weibull Distribution | ||||
---|---|---|---|---|

Methods | Bootstrap New ${\mathit{C}}_{\mathit{p}\mathit{k}\mathit{w}}$ | Traditional ${\mathit{C}}_{\mathit{p}\mathit{k}}$ | ||

Confidence Intervals | Widths | Confidence Intervals | Widths | |

SB | (0.5654, 0.7513) | 0.1859 | (1.0322, 1.3014) | 0.2692 |

PB | (0.5948, 0.7829) | 0.1881 | (0.8974, 1.1605) | 0.2631 |

BCPB | (0.5805, 0.7575) | 0.1770 | (0.9251, 1.1736) | 0.2485 |

Exponential Weibull Distribution | ||||

Methods | Bootstrap${\mathit{C}}_{\mathit{p}\mathit{k}\mathit{e}\mathit{w}}$ | Traditional${\mathit{C}}_{\mathit{p}\mathit{k}}$ | ||

Confidence Intervals | Widths | Confidence Intervals | Widths | |

SB | (0.5672, 0.8748) | 0.3076 | (0.7204, 1.0398) | 0.3194 |

PB | (0.5953, 0.9056) | 0.3103 | (0.6633, 0.9707) | 0.3074 |

BCPB | (0.5739, 0.8816) | 0.3077 | (0.6364, 0.9589) | 0.3225 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rao, G.S.; Albassam, M.; Aslam, M.
Evaluation of Bootstrap Confidence Intervals Using a New Non-Normal Process Capability Index. *Symmetry* **2019**, *11*, 484.
https://doi.org/10.3390/sym11040484

**AMA Style**

Rao GS, Albassam M, Aslam M.
Evaluation of Bootstrap Confidence Intervals Using a New Non-Normal Process Capability Index. *Symmetry*. 2019; 11(4):484.
https://doi.org/10.3390/sym11040484

**Chicago/Turabian Style**

Rao, Gadde Srinivasa, Mohammed Albassam, and Muhammad Aslam.
2019. "Evaluation of Bootstrap Confidence Intervals Using a New Non-Normal Process Capability Index" *Symmetry* 11, no. 4: 484.
https://doi.org/10.3390/sym11040484