# Statistical Process Control for Unimodal Distribution Based on Maximum Entropy Distribution Approximation

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## Abstract

**:**

## 1. Introduction

## 2. ME Distribution

#### 2.1. Maximum Entropy Framework

#### 2.2. ME Distribution Density Determined by Moments

## 3. Control Chart Based on ME Distribution

#### 3.1. Statistic of the Simplified MELS

#### 3.2. Comparison Performance of the Three Charts

#### 3.3. Application Steps for the Proposed Method

## 4. Case Study

#### 4.1. Symmetric Unimodal Distribution

#### 4.2. Asymmetric Unimodal Distribution

#### 4.2.1. Comparison Based on the True Distribution

#### 4.2.2. Comparison Based on the ME Distribution Approximation

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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i | 0 | l | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

${m}_{i}$ | 1 | $-0.0357$ | $1.0069$ | $-0.1343$ | $2.6476$ | $0.1072$ |

${\lambda}_{i}$ | $-0.9313$ | $-0.0106$ | $-0.4721$ | $-0.0095$ | $-0.0048$ | $-0.0000$ |

LCL | UCL | UCL–LCL | |
---|---|---|---|

Shewhart | −1.9600 | 1.9600 | 3.9200 |

MELS | −1.9300 | 1.9500 | 3.8800 |

Quantile | −1.9621 | 1.9570 | 3.9191 |

LCL | UCL | UCL–LCL | |
---|---|---|---|

MELS | 0.2212 | 5.0007 | 4.7795 |

Quantile | 0.4500 | 5.4324 | 4.9824 |

i | 0 | l | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

${m}_{i}$ | 1 | 2.5083 | 7.9761 | 30.8721 | 128.1232 | 591.5907 |

${\lambda}_{i}$ | −2.6913 | 0.9143 | 0.3876 | −0.3339 | 0.0614 | −0.0036 |

LCL | UCL | UCL–LCL | |
---|---|---|---|

Shewhart | $-0.0617$ | 5.0735 | 4.5213 |

MELS | 0.0000 | 5.0180 | 5.0180 |

Quantile | 0.3150 | 5.7012 | 5.3862 |

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**MDPI and ACS Style**

Fang, X.; Song, M.; Chen, Y.
Statistical Process Control for Unimodal Distribution Based on Maximum Entropy Distribution Approximation. *Entropy* **2017**, *19*, 406.
https://doi.org/10.3390/e19080406

**AMA Style**

Fang X, Song M, Chen Y.
Statistical Process Control for Unimodal Distribution Based on Maximum Entropy Distribution Approximation. *Entropy*. 2017; 19(8):406.
https://doi.org/10.3390/e19080406

**Chicago/Turabian Style**

Fang, Xinghua, Mingshun Song, and Yizeng Chen.
2017. "Statistical Process Control for Unimodal Distribution Based on Maximum Entropy Distribution Approximation" *Entropy* 19, no. 8: 406.
https://doi.org/10.3390/e19080406