# Multistage Estimation of the Rayleigh Distribution Variance

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Estimation Problems

#### 2.1. Minimum Risk Point Estimation for the Parameter $\theta $

#### 2.2. Fixed-Width Confidence Interval Estimation for the Parameter $\theta $

#### 2.3. A Unified Decision Framework for Point and Interval Estimation

## 3. Three-Stage Sequential Procedure for Inference

**Assumption**

**A**.

#### 3.1. Asymptotic Characteristics for the Main Study Phase

**Theorem**

**1.**

**Proof**.

- If $m\le {N}_{1}<\nu <\delta {n}^{*}$, we have:$$E\left(IV\right)={(\delta {\lambda}^{2})}^{2}mE\left({\overline{W}}_{m}-\theta \right){\left(g\left({\overline{W}}_{m}\right)-g\left(\theta \right)\right)}^{2}{\nu}^{-3}\le \text{}M\frac{\delta {\lambda}^{2}mE\left({\overline{W}}_{m}-\theta \right)}{{m}^{3}}=0.$$

- If $\delta {n}^{*}<\nu <{N}_{1}$, $E\left(IV\right)={(\delta {\lambda}^{2})}^{2}mE(\left({\overline{W}}_{m}-\theta \right){\left(g\left({\overline{W}}_{m}\right)\text{}-\text{}g\left(\theta \right)\right)}^{2}{\nu}^{-3}\le \text{}M$ ${(\delta {\lambda}^{2})}^{2}mE\left({\overline{W}}_{m}-\theta \right)/{\left(\delta {n}^{*}\right)}^{3}=0$.

**Theorem**

**2**.

**Proof**.

#### 3.2. Asymptotic Characteristics for the Fine-Tuning Phase

**Theorem**

**3**.

**Proof**.

**Theorem**

**4**.

**Proof**.

**Lemma 1**.

**Proof**.

**Theorem**

**5**.

**Proof**.

#### 3.3. The Asymptotic Regret

#### 3.4. Three-Stage Asymptotic Coverage Probability for the Parameter $\theta $

## 4. Simulation Study

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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${\mathit{n}}^{*}$ | $\overline{\mathit{N}}$ | $\mathit{S}\left(\overline{\mathit{N}}\right)$ | $\widehat{\mathit{\theta}}$ | $\mathit{S}\left(\widehat{\mathit{\theta}}\right)$ | $\widehat{\mathit{v}\mathit{a}\mathit{r}}\left(\mathit{x}\right)$ | $1-\widehat{\mathit{\alpha}}$ |
---|---|---|---|---|---|---|

24 | 21.42 | 0.055 | 4.5220 | 0.0048 | 1.94086 | 0.9096 |

43 | 39.26 | 0.088 | 4.5909 | 0.0046 | 1.97043 | 0.8066 |

61 | 57.14 | 0.113 | 4.6970 | 0.0040 | 2.01597 | 0.8618 |

76 | 72.36 | 0.130 | 4.7639 | 0.0036 | 2.04468 | 0.8830 |

96 | 92.66 | 0.154 | 4.8196 | 0.0031 | 2.06859 | 0.8977 |

125 | 122.50 | 0.183 | 4.8749 | 0.0026 | 2.09232 | 0.9138 |

171 | 169.57 | 0.223 | 4.9197 | 0.0021 | 2.11155 | 0.9241 |

246 | 246.55 | 0.284 | 4.9503 | 0.0016 | 2.12469 | 0.9324 |

500 | 509.36 | 0.468 | 4.9790 | 0.0010 | 2.13701 | 0.9428 |

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**MDPI and ACS Style**

Yousef, A.; Amin, A.A.; Hassan, E.E.; Hamdy, H.I.
Multistage Estimation of the Rayleigh Distribution Variance. *Symmetry* **2020**, *12*, 2084.
https://doi.org/10.3390/sym12122084

**AMA Style**

Yousef A, Amin AA, Hassan EE, Hamdy HI.
Multistage Estimation of the Rayleigh Distribution Variance. *Symmetry*. 2020; 12(12):2084.
https://doi.org/10.3390/sym12122084

**Chicago/Turabian Style**

Yousef, Ali, Ayman A. Amin, Emad E. Hassan, and Hosny I. Hamdy.
2020. "Multistage Estimation of the Rayleigh Distribution Variance" *Symmetry* 12, no. 12: 2084.
https://doi.org/10.3390/sym12122084