# Multistage Estimation of the Scale Parameter of Rayleigh Distribution with Simulation

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## Abstract

**:**

**1981**, 9, 1229–1238). Both point and confidence interval estimation are considered via a unified optimal decision framework, which enables one to make the maximum use of the available data and, at the same time, reduces the number of sampling operations by using bulk samples. The asymptotic characteristics of the proposed sampling procedure are fully discussed for both point and confidence interval estimation. Since the results are asymptotic, Monte Carlo simulation studies are conducted to provide the feel of small, moderate, and large sample size performance in typical situations using the Microsoft Developer Studio software. The procedure enjoys several interesting asymptotic characteristics illustrated by the asymptotic results and supported by simulation.

## 1. Introduction

## 2. Estimation Problems

#### 2.1. Minimum Risk Point Estimation

#### 2.2. Fixed-Width Confidence Interval

#### 2.3. A Unified Decision Framework

## 3. Multistage Sampling

**Assumption A:**

**Theorem**

**1.**

- (i)
- $E({\overline{T}}_{{N}_{1}})=\sigma -\frac{2(4-\pi )\sigma}{\pi}{(\delta {n}^{*})}^{-1}+o({d}^{2})$,
- (ii)
- $E({\overline{T}}_{{N}_{1}}^{2})={\sigma}^{2}-\frac{3(4-\pi )}{\pi}{\sigma}^{2}{(\delta {n}^{*})}^{-1}+o({d}^{2})$,
- (iii)
- $Var({\overline{T}}_{{N}_{1}})=\frac{(4-\pi )}{\pi}{\sigma}^{2}{(\delta {n}^{*})}^{-1}+o({d}^{2})$,
- (iv)
- $E({\overline{T}}_{{N}_{1}}^{4})={\sigma}^{4}-\frac{2(4-\pi )}{\pi}{\sigma}^{4}{(\delta {n}^{*})}^{-1}+o({d}^{2})$,
- (v)
- $Var({\overline{T}}_{{N}_{1}}^{2})=\frac{4(4-\pi )}{\pi}{\sigma}^{4}{(\delta {n}^{*})}^{-1}+o({d}^{2})$.

**Proof.**

**Theorem**

**2.**

- (i)
- $E(N)={n}^{*}-\frac{3(4-\pi )}{\pi}{\delta}^{-1}+\frac{1}{2}+o(1)$,
- (ii)
- $Var(N)=\frac{4(4-\pi )}{\pi}{\delta}^{-1}{n}^{*}+o(d)$.

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

- (i)
- $E({\overline{T}}_{N})=\sigma -\frac{2(4-\pi )\sigma}{\pi {n}^{*}}+o({d}^{2}),$
- (ii)
- $E({\overline{T}}_{N}^{2})={\sigma}^{2}+(\delta -4)\frac{(4-\pi ){\sigma}^{2}}{\pi {n}^{*}}+o({d}^{2})$,
- (iii)
- $Var({\overline{T}}_{N})=\frac{\delta (4-\pi ){\sigma}^{2}}{\pi {n}^{*}}+o({d}^{2}).$

**Proof.**

**Theorem**

**5.**

**Proof.**

#### 3.1. Three-Stage Minimum Risk Point Estimation

#### 3.2. Three-Stage Fixed-Width Confidence Interval

## 4. Simulation Study

**First.**Generate an initial sample of size $m(\ge 2),$ say ${T}_{1,i},{T}_{2,i},\dots ,{T}_{m,i}$ from Rayleigh distribution with scale parameter $\sigma $ and calculate ${\overline{T}}_{m}$ as an initial estimate of $\sigma $.

**Second**. Apply the three-stage sampling procedure as presented in (7) and (8) to determine the stopping sample size at this iteration, whether in the first or second stage ${N}_{i}^{*}$.

**Third.**Record the resultant values of stage ${N}_{i}^{*}$ and ${T}_{i}^{*}$.Hence, for each experimental combination we have two vectors of size $s=$50,000 $({N}_{1}^{*},{N}_{2}^{*},\dots ,{N}_{s}^{*})$ and $({\overline{T}}_{1}^{*},{\overline{T}}_{2}^{*},\dots ,{\overline{T}}_{s}^{*})$. Define:

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Distribution Characteristic | Mathematical Representation | Three Stage Point Estimate |
---|---|---|

The Mode | $\sigma $ | ${\overline{T}}_{N}$ |

The Median | $\sigma \sqrt{2ln(2)}$ | ${\overline{T}}_{N}\sqrt{2ln(2)}$ |

Reliability at time ${T}_{0}$ | ${e}^{-({T}_{0}/2{T}_{N}^{2})}$ | ${e}^{-({T}_{0}/2{T}_{N}^{2})}$ |

Hazard Function at time ${T}_{0}$ | ${T}_{0}/{\sigma}^{2}$ | ${T}_{0}/{\overline{T}}_{N}^{2}$ |

Entropy, $\gamma =0.5772$ | $1+log(\sigma /\sqrt{2})+\gamma /2$ | $1+log({\overline{T}}_{N}/\sqrt{2})+\gamma /2$ |

${\mathit{n}}^{*}$ | 25 | 50 | 100 | 150 | 200 | 250 | 300 | 400 | 500 |
---|---|---|---|---|---|---|---|---|---|

$\overline{N}$ | 22.79 | 48.84 | 98.95 | 148.75 | 198.82 | 248.68 | 298.89 | 399.10 | 499.04 |

$S(\overline{N})$ | 0.040 | 0.049 | 0.069 | 0.085 | 0.099 | 0.110 | 0.121 | 0.141 | 0.156 |

$\widehat{\sigma}$ | 9.588 | 9.859 | 9.944 | 9.962 | 9.970 | 9.975 | 9.981 | 9.987 | 9.989 |

$S(\widehat{\sigma})$ | 0.006 | 0.004 | 0.002 | 0.002 | 0.002 | 0.002 | 0.001 | 0.001 | 0.001 |

$\widehat{\mu}$ | 12.537 | 12.537 | 12.537 | 12.485 | 12.495 | 12.519 | 12.521 | 12.527 | 12.529 |

$S(\widehat{\mu})$ | 0.010 | 0.010 | 0.010 | 0.004 | 0.003 | 0.003 | 0.003 | 0.002 | 0.002 |

$\widehat{var}(x)$ | 6.281 | 6.459 | 6.515 | 6.526 | 6.532 | 6.535 | 6.539 | 6.543 | 6.544 |

$\widehat{Svar}(x)$ | 0.004 | 0.003 | 0.002 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 |

$\widehat{med}(x)$ | 10.795 | 11.111 | 11.209 | 11.228 | 11.230 | 11.232 | 11.225 | 11.212 | 11.188 |

$S\widehat{med}(x)$ | 0.007 | 0.005 | 0.003 | 0.003 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 |

$\widehat{Ent}$ | 3.1932 | 3.2262 | 3.237 | 3.240 | 3.241 | 3.241 | 3.242 | 3.243 | 3.243 |

$\widehat{SEnt}$ | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

$\widehat{\omega}$ | −27.21 | −51.16 | −101.1 | −80.27 | −148.79 | −85.89 | −165.37 | −169.95 | −178.12 |

$1-\widehat{\alpha}$ | 0.8823 | 0.927 | 0.940 | 0.942 | 0.944 | 0.946 | 0.948 | 0.945 | 0.949 |

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**MDPI and ACS Style**

Yousef, A.; Hassan, E.E.H.; Amin, A.A.; Hamdy, H.I. Multistage Estimation of the Scale Parameter of Rayleigh Distribution with Simulation. *Symmetry* **2020**, *12*, 1925.
https://doi.org/10.3390/sym12111925

**AMA Style**

Yousef A, Hassan EEH, Amin AA, Hamdy HI. Multistage Estimation of the Scale Parameter of Rayleigh Distribution with Simulation. *Symmetry*. 2020; 12(11):1925.
https://doi.org/10.3390/sym12111925

**Chicago/Turabian Style**

Yousef, Ali, Emad E. H. Hassan, Ayman A. Amin, and Hosny I. Hamdy. 2020. "Multistage Estimation of the Scale Parameter of Rayleigh Distribution with Simulation" *Symmetry* 12, no. 11: 1925.
https://doi.org/10.3390/sym12111925