# On Efficient Estimation of Process Variability

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Proposed Estimator

_{i}is obtained by minimizing Equation (23) and is given as

#### 3.1. Special Cases

#### 3.2. Efficiency Comparison of the Generalized Exponential Estimator

## 4. Results

#### 4.1. Numerical Study

#### 4.2. Simulation Results

## 5. Conclusions and Recommendations

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Estimator | $\mathit{\beta}$ | $\mathit{a}$ | g |
---|---|---|---|

${t}_{o}={s}_{y}^{2}$ | 0 | a | g |

${t}_{eR1}={s}_{y}^{2}\mathrm{exp}\left(\frac{{s}_{x}^{*2}-{S}_{x}^{2}}{{S}_{x}^{2}+\left(a-1\right){s}_{x}^{*2}}\right)$ | 1 | a | g |

${t}_{eR2}={s}_{y}^{2}\mathrm{exp}\left(\frac{{s}_{x}^{*2}-{S}_{x}^{2}}{{S}_{x}^{2}}\right)$ | 1 | 1 | g |

${t}_{eR3}={s}_{y}^{2}\mathrm{exp}\left(\frac{{s}_{x}^{*2}-{S}_{x}^{2}}{{S}_{x}^{2}+{s}_{x}^{*2}}\right)={t}_{er}$ | 1 | 2 | g |

Measure | Source | N | n | $\overline{\mathit{Y}}$ | $\overline{\mathit{X}}$ | ${\mathit{C}}_{\mathit{y}}$ | ${\mathit{C}}_{\mathit{x}}$ |

Population I | [12] | 142 | 20 | 4015.218 | 2900.387 | 2.112 | 2.197 |

Population II | [13] | 80 | 25 | 5182.638 | 283.875 | 0.352 | 0.943 |

Population III | [14] | 64 | 8 | 141.500 | 51.187 | 0.537 | 0.509 |

Population IV | [15] | 51 | 7 | 13.067 | 543.373 | 0.323 | 0.684 |

Population V | [16] | 58 | 12 | 85.948 | 93.000 | 1.121 | 1.127 |

Population VI | [16] | 80 | 23 | 90.813 | 104.575 | 0.392 | 0.379 |

Population VII | [17] | 53 | 10 | 917.019 | 1417.245 | 0.402 | 0.682 |

Population VIII | [17] | 66 | 14 | 974.424 | 1716.136 | 0.512 | 0.612 |

Population IX | [18] | 71 | 12 | 4137.803 | 241.944 | 0.306 | 0.557 |

Population X | [18] | 75 | 12 | 1.377 | 6.347 | 1.748 | 0.418 |

Measure | Source | ${\mathit{\rho}}_{\mathit{x}\mathit{y}}$ | ${\mathit{\beta}}_{\mathbf{2}}\left(\mathit{y}\right)$ | ${\mathit{\beta}}_{\mathbf{2}}\left(\mathit{x}\right)$ | ${\mathit{\beta}}_{\mathbf{1}}\left(\mathit{x}\right)$ | $\mathit{h}$ | ${\mathit{\lambda}}_{\mathbf{21}}$ |

Population I | [12] | 0.995 | 40.854 | 48.157 | 40.218 | 43.762 | 5.979 |

Population II | [13] | 0.914 | 2.267 | 3.650 | 1.295 | 2.337 | 0.548 |

Population III | [14] | −0.818 | 2.378 | 1.658 | 0.006 | 1.438 | −0.146 |

Population IV | [15] | 0.446 | 3.653 | 15.231 | 7.636 | 4.472 | 0.942 |

Population V | [16] | 0.978 | 24.747 | 6.175 | 3.333 | 5.167 | 155.121 |

Population VI | [16] | 0.628 | 4.573 | 5.701 | 0.629 | 10.772 | 96.058 |

Population VII | [17] | 0.775 | 5.355 | 8.526 | 5.668 | 4.839 | 562.914 |

Population VIII | [17] | 0.776 | 9.128 | 19.038 | 11.960 | 11.210 | 1142.321 |

Population IX | [18] | −0.382 | 4.458 | 2.635 | 0.255 | 2.181 | 247.516 |

Population X | [18] | 0.222 | 31.544 | 2.019 | 0.004 | 4.271 | 4.027 |

Population Number | Estimator | |||||||
---|---|---|---|---|---|---|---|---|

${\mathit{t}}_{0}$ | ${\mathit{t}}_{\mathit{I}}$ | ${\mathit{t}}_{\mathit{S}\mathit{R}}$ | ${\mathit{t}}_{\mathit{A}}$ | ${\mathit{t}}_{\mathit{Y}\mathit{K}2}$ | ${\mathit{t}}_{\mathit{r}\mathit{P}}$ | ${\mathit{t}}_{\mathit{e}\mathit{R}}$ | ${\mathit{t}}_{\mathit{e}\mathit{R}\mathit{G}}$ | |

1 | 100 | 33.731 | 22.284 | 135.988 | 81.601 | 36.427 | 83.205 | 21.703 |

2 | 100 | 98.106 | 46.764 | 158.333 | 56.739 | 96.203 | 56.552 | 41.038 |

3 | 100 | 84.202 | 80.172 | 99.312 | 110.281 | 95.337 | 82.673 | 68.061 |

4 | 100 | 174.642 | 103.215 | 128.675 | 86.316 | 78.946 | 73.917 | 59.606 |

5 | 100 | 31.169 | 81.740 | 77.460 | 81.330 | 82.980 | 74.525 | 10.052 |

6 | 100 | 37.034 | 63.532 | 85.541 | 83.752 | 57.028 | 77.202 | 11.427 |

7 | 100 | 96.512 | 55.054 | 89.234 | 82.016 | 86.182 | 72.476 | 48.790 |

8 | 100 | 70.693 | 29.870 | 87.055 | 58.140 | 84.745 | 64.014 | 25.314 |

9 | 100 | 78.955 | 77.656 | 74.898 | 72.663 | 75.669 | 82.562 | 65.989 |

10 | 100 | 81.918 | 90.125 | 105.364 | 92.336 | 100.016 | 84.913 | 54.820 |

Coefficient | Estimator | ||||||
---|---|---|---|---|---|---|---|

${\mathit{t}}_{\mathit{I}}$ | ${\mathit{t}}_{\mathit{S}\mathit{R}}$ | ${\mathit{t}}_{\mathit{A}}$ | ${\mathit{t}}_{\mathit{Y}\mathit{K}2}$ | ${\mathit{t}}_{\mathit{r}\mathit{P}}$ | ${\mathit{t}}_{\mathit{e}\mathit{R}}$ | ${\mathit{t}}_{\mathit{e}\mathit{R}\mathit{G}}$ | |

${\widehat{\beta}}_{0}$ | 88.449 (0.003) | 82.312 (0.000) | 94.493 (0.000) | 86.106 (0.000) | 90.983 (0.000) | 76.948 (0.000) | 53.468 (0.000) |

${\widehat{\beta}}_{1}$ | −11.353 (0.700) | −9.407 (0.474) | 7.131 (0.701) | −16.670 (0.097) | 0.795 (0.930) | −10.860 (0.058) | −26.908 (0.030) |

${\widehat{\beta}}_{2}$ | −0.408 (0.748) | −0.798 (0.181) | 0.573 (0.482) | 0.175 (0.656) | −1.063 (0.026) | 0.282 (0.217) | −0.052 (0.907) |

F-ratio | 0.251 (0.785) | 2.420 (0.159) | 0.624 (0.482) | 1.946 (0.213) | 4.898 (0.047) | 2.618 (0.142) | 4.849 (0.048) |

$\mathit{\rho}$ | Estimator | |||||||
---|---|---|---|---|---|---|---|---|

${\mathit{t}}_{0}$ | ${\mathit{t}}_{\mathit{I}}$ | ${\mathit{t}}_{\mathit{S}\mathit{R}}$ | ${\mathit{t}}_{\mathit{A}}$ | ${\mathit{t}}_{\mathit{Y}\mathit{K}2}$ | ${\mathit{t}}_{\mathit{r}\mathit{P}}$ | ${\mathit{t}}_{\mathit{e}\mathit{R}}$ | ${\mathit{t}}_{\mathit{e}\mathit{R}\mathit{G}}$ | |

−0.9 | 100 | 189.134 | 119.577 | 100.037 | 100.031 | 109.366 | 91.881 | 84.899 |

−0.7 | 100 | 193.632 | 121.793 | 100.038 | 100.518 | 109.607 | 88.730 | 82.266 |

−0.5 | 100 | 196.582 | 123.296 | 100.042 | 100.857 | 109.711 | 87.862 | 86.497 |

−0.3 | 100 | 198.862 | 124.449 | 100.047 | 101.115 | 109.763 | 91.445 | 85.654 |

−0.1 | 100 | 199.721 | 124.892 | 100.040 | 101.216 | 109.776 | 92.079 | 87.677 |

0.1 | 100 | 200.035 | 124.974 | 100.038 | 101.221 | 109.772 | 87.667 | 84.565 |

0.3 | 100 | 198.773 | 124.377 | 100.040 | 101.094 | 109.762 | 90.272 | 88.006 |

0.5 | 100 | 196.642 | 123.334 | 100.043 | 100.866 | 109.715 | 88.728 | 83.709 |

0.7 | 100 | 192.997 | 121.545 | 100.045 | 100.475 | 109.584 | 92.261 | 82.935 |

0.9 | 100 | 189.171 | 119.600 | 100.043 | 100.037 | 109.366 | 89.811 | 86.255 |

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**MDPI and ACS Style**

Akhlaq, T.; Ismail, M.; Shahbaz, M.Q.
On Efficient Estimation of Process Variability. *Symmetry* **2019**, *11*, 554.
https://doi.org/10.3390/sym11040554

**AMA Style**

Akhlaq T, Ismail M, Shahbaz MQ.
On Efficient Estimation of Process Variability. *Symmetry*. 2019; 11(4):554.
https://doi.org/10.3390/sym11040554

**Chicago/Turabian Style**

Akhlaq, Tanveer, Muhammad Ismail, and Muhammad Qaiser Shahbaz.
2019. "On Efficient Estimation of Process Variability" *Symmetry* 11, no. 4: 554.
https://doi.org/10.3390/sym11040554