# Inference about the Ratio of the Coefficients of Variation of Two Independent Symmetric or Asymmetric Populations

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## Abstract

**:**

## 1. Introduction

## 2. Asymptotic Results

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### 2.1. Constructing the Confidence Interval

**Theorem**

**2.2:**

**Proof.**

#### 2.2. Hypothesis Testing

#### 2.3. Normal Populations

## 3. Simulation Study

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Probability density function (PDF) of normal ($\mu ,{\sigma}^{2}$) distribution with different coefficient of variation (CV) values. Black: $\mu =1,\text{}\sigma =1,$ CV = 1; red: $\mu =1,\text{}\sigma =2,$ CV = 2; green: $\mu =1,\text{}\sigma =3,$ CV = 3; blue: $\mu =1,\text{}\sigma =5,$ CV = 5.

**Figure 2.**PDF of gamma ($\alpha ,\lambda $) distribution with different CV values. Black:$\text{}\alpha =1,\lambda =0.001,$ CV = 1; red: $\alpha =0.25,\lambda =0.001,$ CV = 2; green: $\alpha =0.11,\lambda =0.001,$ CV = 3; blue: $\alpha =0.04,\text{}\lambda =0.001,$ CV = 5.

**Figure 3.**PDF of beta ($\alpha ,\beta $) distribution with different CV values. Black: $\alpha =0.94,\beta =30.39,$ CV = 1; red: $\alpha =0.21,\beta =6.87,$ CV = 2; green: $\alpha =0.08,\beta =2.51,$ CV = 3; blue: $\alpha =0.009,\text{}\beta =0.285,$ CV = 5.

$\left(\mathit{m},\mathit{n}\right)$ | |||||||
---|---|---|---|---|---|---|---|

$\mathit{D}\mathit{i}\mathit{s}\mathit{t}\mathit{r}\mathit{i}\mathit{b}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}$ | $\left(\mathit{C}{\mathit{V}}_{\mathit{X}},\mathit{C}{\mathit{V}}_{\mathit{Y}}\right)$ | $\left(50,100\right)$ | $\left(75,100\right)$ | $\left(100,200\right)$ | $\left(200,300\right)$ | $\left(500,700\right)$ | $\left(700,1000\right)$ |

Normal | $\left(1,1\right)$ | $0.945$ | $0.947$ | $0.951$ | $0.953$ | $0.959$ | $0.960$ |

$\left(1,2\right)$ | $0.945$ | $0.948$ | $0.952$ | $0.953$ | $0.958$ | $0.959$ | |

$\left(2,3\right)$ | $0.944$ | $0.948$ | $0.953$ | $0.953$ | $0.959$ | $0.961$ | |

$\left(2,5\right)$ | $0.946$ | $0.950$ | $0.950$ | $0.955$ | $0.956$ | $0.960$ | |

Gamma | $\left(1,1\right)$ | $0.946$ | $0.948$ | $0.952$ | $0.956$ | $0.958$ | $0.961$ |

$\left(1,2\right)$ | $0.947$ | $0.949$ | $0.951$ | $0.954$ | $0.958$ | $0.961$ | |

$\left(2,3\right)$ | $0.947$ | $0.950$ | $0.952$ | $0.953$ | $0.959$ | $0.961$ | |

$\left(2,5\right)$ | $0.945$ | $0.949$ | $0.952$ | $0.956$ | $0.958$ | $0.962$ | |

Beta | $\left(1,1\right)$ | $0.944$ | $0.950$ | $0.950$ | $0.954$ | $0.958$ | $0.961$ |

$\left(1,2\right)$ | $0.946$ | $0.948$ | $0.952$ | $0.954$ | $0.957$ | $0.960$ | |

$\left(2,3\right)$ | $0.945$ | $0.947$ | $0.952$ | $0.954$ | $0.958$ | $0.959$ | |

$\left(2,5\right)$ | $0.945$ | $0.948$ | $0.951$ | $0.954$ | $0.956$ | $0.960$ |

$\left(\mathit{m},\mathit{n}\right)$ | |||||||
---|---|---|---|---|---|---|---|

$\mathit{D}\mathit{i}\mathit{s}\mathit{t}\mathit{r}\mathit{i}\mathit{b}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}$ | $\left(\mathit{C}{\mathit{V}}_{\mathit{X}},\mathit{C}{\mathit{V}}_{\mathit{Y}}\right)$ | $\left(50,100\right)$ | $\left(75,100\right)$ | $\left(100,200\right)$ | $\left(200,300\right)$ | $\left(500,700\right)$ | $\left(700,1000\right)$ |

Normal | $\left(1,1\right)$ | $8.64$ | $10.08$ | $14.08$ | $23.09$ | $51.92$ | $68.67$ |

$\left(1,2\right)$ | $8.72$ | $10.29$ | $16.41$ | $21.85$ | $52.19$ | $74.17$ | |

$\left(2,3\right)$ | $9.52$ | $9.50$ | $15.42$ | $21.10$ | $51.05$ | $65.87$ | |

$\left(2,5\right)$ | $9.35$ | $10.90$ | $15.25$ | $24.31$ | $49.97$ | $74.90$ | |

Gamma | $\left(1,1\right)$ | $9.45$ | $9.02$ | $15.16$ | $22.13$ | $47.05$ | $74.92$ |

$\left(1,2\right)$ | $8.00$ | $9.58$ | $14.65$ | $24.87$ | $49.96$ | $66.20$ | |

$\left(2,3\right)$ | $9.63$ | $9.29$ | $14.47$ | $21.91$ | $52.52$ | $66.84$ | |

$\left(2,5\right)$ | $8.69$ | $9.83$ | $16.29$ | $24.27$ | $50.68$ | $66.11$ | |

Beta | $\left(1,1\right)$ | $9.53$ | $10.57$ | $14.19$ | $21.47$ | $53.26$ | $66.58$ |

$\left(1,2\right)$ | $9.20$ | $9.50$ | $14.15$ | $24.85$ | $48.67$ | $75.00$ | |

$\left(2,3\right)$ | $9.02$ | $9.63$ | $14.25$ | $23.52$ | $50.29$ | $69.67$ | |

$\left(2,5\right)$ | $8.75$ | $9.17$ | $15.73$ | $22.89$ | $50.79$ | $73.95$ |

$\left(\mathit{m},\mathit{n}\right)$ | |||||||
---|---|---|---|---|---|---|---|

$\mathit{D}\mathit{i}\mathit{s}\mathit{t}\mathit{r}\mathit{i}\mathit{b}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}$ | $\left(\mathit{C}{\mathit{V}}_{\mathit{X}},\mathit{C}{\mathit{V}}_{\mathit{Y}}\right)$ | $\left(50,100\right)$ | $\left(75,100\right)$ | $\left(100,200\right)$ | $\left(200,300\right)$ | $\left(500,700\right)$ | $\left(700,1000\right)$ |

Normal | $\left(1,1\right)$ | $0.444$ | $0.551$ | $0.662$ | $0.701$ | $0.899$ | $0.977$ |

$\left(1,2\right)$ | $0.432$ | $0.580$ | $0.656$ | $0.795$ | $0.860$ | $0.982$ | |

$\left(2,3\right)$ | $0.408$ | $0.600$ | $0.602$ | $0.718$ | $0.859$ | $0.943$ | |

$\left(2,5\right)$ | $0.481$ | $0.569$ | $0.681$ | $0.740$ | $0.848$ | $0.955$ | |

Gamma | $\left(1,1\right)$ | $0.428$ | $0.545$ | $0.677$ | $0.760$ | $0.851$ | $0.905$ |

$\left(1,2\right)$ | $0.407$ | $0.544$ | $0.612$ | $0.775$ | $0.880$ | $0,909$ | |

$\left(2,3\right)$ | $0.484$ | $0.508$ | $0.611$ | $0.708$ | $0.855$ | $0.940$ | |

$\left(2,5\right)$ | $0.494$ | $0.556$ | $0.647$ | $0.754$ | $0.800$ | $0.978$ | |

Beta | $\left(1,1\right)$ | $0.411$ | $0.599$ | $0.657$ | $0.709$ | $0.870$ | $0.946$ |

$\left(1,2\right)$ | $0.489$ | $0.585$ | $0.652$ | $0.763$ | $0.841$ | $0.978$ | |

$\left(2,3\right)$ | $0.411$ | $0.505$ | $0.606$ | $0.724$ | $0.874$ | $0.908$ | |

$\left(2,5\right)$ | $0.461$ | $0.527$ | $0.671$ | $0.757$ | $0.847$ | $0.933$ |

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**MDPI and ACS Style**

Yue, Z.; Baleanu, D.
Inference about the Ratio of the Coefficients of Variation of Two Independent Symmetric or Asymmetric Populations. *Symmetry* **2019**, *11*, 824.
https://doi.org/10.3390/sym11060824

**AMA Style**

Yue Z, Baleanu D.
Inference about the Ratio of the Coefficients of Variation of Two Independent Symmetric or Asymmetric Populations. *Symmetry*. 2019; 11(6):824.
https://doi.org/10.3390/sym11060824

**Chicago/Turabian Style**

Yue, Zhang, and Dumitru Baleanu.
2019. "Inference about the Ratio of the Coefficients of Variation of Two Independent Symmetric or Asymmetric Populations" *Symmetry* 11, no. 6: 824.
https://doi.org/10.3390/sym11060824