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

^{1}

^{2}

^{3}

^{*}

## 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

## References

- Kodlin, D. A new response time distribution. Biometrics
**1967**, 23, 227–239. [Google Scholar] [CrossRef] [PubMed] - Rayleigh, F.R.S. On the resultant of a large number of vibrations of the same pitch and arbitrary phase. Lond. Edinb. Dublin Philos. Mag.
**1880**, 10, 73–78. [Google Scholar] [CrossRef] [Green Version] - Palovko, A.M. Fundamentals of Reliability Theory; Academic Press: New York, NY, USA, 1968. [Google Scholar]
- Gross, A.J.; Clark, V.A. Survival Distributions, Reliability Application in Biomedical Science; Wiley: New York, NY, USA, 1975. [Google Scholar]
- Lee, E.T.; Wang, J.W. Statistical Methods for Survival Data Analysis, 3rd ed.; Wiley & Sons Inc.: Hoboken, NJ, USA, 2003. [Google Scholar]
- Dyer, D.D.; Whisenand, C.W. Best linear unbiased estimator of the parameter of the Rayleigh distribution: Part-II optimum theory for selected order statistics. IEEE Trans. Reliab.
**1965**, 60, 229–231. [Google Scholar] [CrossRef] - Siddiqui, M.M. Some problems connected with Rayleigh distributions. J. Res. Natl. Bur. Stand.
**1962**, 66D, 167–174. [Google Scholar] [CrossRef] - Hirano, K. Rayleigh Distributions; Wiley: New York, NY, USA, 1986. [Google Scholar]
- Howlader, H.A.; Hossain, A. On Bayesian estimation and prediction from Rayleigh distribution based on type-II censored data. Commun. Stat. Theory Methods
**1995**, 24, 2251–2259. [Google Scholar] [CrossRef] - Rosen, K.; Van Buskirk, R.; Garbesi, K. Wind energy potential of coastal Eritrea: An analysis of sparse wind data. Sol. Energy
**1999**, 66, 201–213. [Google Scholar] [CrossRef] - Rehman, S.; Halawani, T.O.; Husain, T. Weibull parameters for wind speed distribution in Saudi Arabia. Sol. Energy
**1994**, 53, 473–479. [Google Scholar] [CrossRef] - Celik, A.N. Energy output estimation for small-scale wind power generators using Weibull-representative wind data. J. Wind Eng. Ind. Aerodyn.
**2003**, 91, 693–707. [Google Scholar] [CrossRef] - Pishgar-Komleh, S.H.; Keyhani, A.; Sefeedpari, P. Wind speed and power density analysis based on Weibull and Rayleigh distributions (a case study: Firouzkooh county of Iran). Renew. Sustain. Energy Rev.
**2015**, 42, 313–322. [Google Scholar] [CrossRef] - Sinha, S.K.; Howlader, H.A. Credible and HPD intervals of the parameter and reliability of Rayleigh distribution. IEEE Trans. Reliab.
**1983**, 32, 217–220. [Google Scholar] [CrossRef] - Ariyawansa, K.A.; Templeton, J.G.C. Structural inference on the parameter of the Rayleigh distribution from doubly censored samples. Stat. Hefte
**1984**, 25, 181–199. [Google Scholar] [CrossRef] - Howlader, H.A. HPD prediction intervals for Rayleigh distribution. IEEE Trans. Reliab.
**1985**, 34, 121–123. [Google Scholar] [CrossRef] - Lalitha, S.; Mishra, A. Modified maximum likelihood estimation for Rayleigh distribution. Commun. Stat. Theory Methods
**1996**, 25, 389–401. [Google Scholar] [CrossRef] - Elfattah, A.M.; AbdHassan, A.S.; Ziedan, D.M. Efficiency of maximum likelihood estimators under different censored sampling schemes for Rayleigh Distribution. Interstat Electron. J.
**2006**, 1, 1–16. [Google Scholar] - Dey, S.; Das, M.K. A note on prediction interval for a Rayleigh distribution: Bayesian approach. Am. J. Math. Manag. Sci.
**2007**, 27, 43–48. [Google Scholar] [CrossRef] - Dey, S. Comparison of Bayes estimators of the parameter and reliability function for Rayleigh distribution under different loss functions. Malays. J. Math. Sci.
**2009**, 3, 249–266. [Google Scholar] - Dey, S.; Dey, T. Bayesian estimation of the parameter and reliability of a Rayleigh distribution using records. Model Assist. Stat. Appl.
**2012**, 7, 81–90. [Google Scholar] [CrossRef] - Johnson, N.L.; Kotz, S.; Balakrishnan, N. Continuous Univariate Distributions, 2nd ed.; John Wiley & Sons: New York, NY, USA, 1994; Volume 1, ISBN 0-471-58495-9. [Google Scholar]
- Shannon, C.E. Mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] [Green Version] - Hall, P. Asymptotic theory and triple sampling of sequential estimation of a mean. Ann. Stat.
**1981**, 9, 1229–1238. [Google Scholar] [CrossRef] - Degroot, M.H. Optimal Statistical Decisions; McGraw-Hill: New York, NY, USA, 1970. [Google Scholar]
- Chow, Y.S.; Yu, K.F. The performance of a sequential procedure for the estimation of the mean. Ann. Stat.
**1981**, 9, 189–198. [Google Scholar] [CrossRef] - Martinsek, A.T. Negative regret, optional stopping and the elimination of outliers. J. Am. Stat. Assoc.
**1988**, 83, 160–163. [Google Scholar] [CrossRef] - Hamdy, H.I. Remarks on the asymptotic theory of triple stage estimation of the normal mean. Scand. J. Stat.
**1988**, 15, 303–310. [Google Scholar] - Dantzig, G.B. On the Non-existence of tests of student’s hypothesis having power function independent of σ. Ann. Math. Stat.
**1940**, 11, 186–192. [Google Scholar] [CrossRef] - Stein, C. A two-sample test for a linear hypothesis whose power is independent of the variance. Ann. Math. Stat.
**1945**, 16, 243–258. [Google Scholar] [CrossRef] - Stein, C. Some problems in sequential estimation (abstract). Econometrica
**1949**, 17, 77–78. [Google Scholar] - Seelbinder, B.M. On Stein’s two stage sampling scheme. Ann. Math. Stat.
**1953**, 24, 640–900. [Google Scholar] [CrossRef] - Cox, D.R. Estimation by double sampling. Biometrika
**1952**, 39, 217–227. [Google Scholar] [CrossRef] - Anscombe, F.J. Sequential estimation. J. R. Stat. Soc.
**1953**, 15, 1–21. [Google Scholar] [CrossRef] - Robbins, H. Sequential Estimation of the Mean of a Normal Population. Probability and Statistics (Harald Cramer Volume); Almquist and Wiksell: Uppsala, Sweden, 1959; pp. 235–245. [Google Scholar]
- Chow, Y.S.; Robbins, H. On the asymptotic theory of fixed-width sequential confidence intervals for the mean. Ann. Math. Stat.
**1965**, 36, 1203–1212. [Google Scholar] [CrossRef] - Mukhopadhyay, N. A note on three-stage and sequential point estimation procedures for a normal mean. J. Seq. Anal.
**1985**, 4, 311–319. [Google Scholar] [CrossRef] - Mukhopadhyay, N. Sequential estimation problems for negative exponential populations. Commun. Stat. Theory Methods
**1988**, 17, 2471–2506. [Google Scholar] [CrossRef] - Mukhopadhyay, N. Some properties of a three-stage procedure with applications in sequential analysis. Sankhya
**1990**, 52, 218–231. [Google Scholar] - Mukhopadhyay, N.; Hamdy, H.I.; AlMahmeed, M.; Costanza, M.C. Three stage point estimation procedures for a normal mean. J. Seq. Anal.
**1987**, 6, 21–36. [Google Scholar] [CrossRef] - Mukhopadhyay, N.; Mauromoustakos, A. Three stage estimation procedures of the negative exponential distribution. Metrika
**1987**, 34, 83–93. [Google Scholar] [CrossRef] - Hamdy, H.I.; Pallotta, W.J. Triple sampling procedure for estimating the scale parameter of Pareto distribution. Commun. Stat. Theory Methods
**1987**, 16, 2155–2164. [Google Scholar] [CrossRef] - Hamdy, H.I.; Mukhopadhyay, N.; Costanza, M.C.; Son, M.S. Triple stage point estimation for the exponential location parameter. Ann. Inst. Stat. Math.
**1988**, 40, 785–797. [Google Scholar] [CrossRef] - Hamdy, H.I.; AlMahmeed, M.; Nigm, A.; Son, M.S. Three stage estimation for the exponential location parameters. Metron
**1989**, 47, 279–294. [Google Scholar] - Mukhopadhyay, N.; Padmanabhan, A.R. A note on three-stage confidence intervals for the difference of locations: The exponential case. Metrika
**1993**, 40, 121–128. [Google Scholar] [CrossRef] - Takada, Y. Three stage estimation procedure of the multivariate normal mean. Sankhya
**1993**, 55, 124–129. [Google Scholar] - Hamdy, H.I. Performance of fixed width confidence intervals under type II errors: The exponential case. S. Afr. Stat. J.
**1997**, 31, 259–269. [Google Scholar] - AlMahmeed, M.; Hamdy, H.I. Sequential estimation of linear models in three stages. Metrika
**1990**, 37, 19–36. [Google Scholar] [CrossRef] - AlMahmeed, M.; AlHessainan, A.; Son, M.S.; Hamdy, H.I. Three stage estimation for the mean of a one-parameter exponential family. Korean Commun. Stat.
**1998**, 5, 539–557. [Google Scholar] - Costanza, M.C.; Hamdy, H.I.; Haugh, L.D.; Son, M.S. Type II error performance of triple sampling fixed precision confidence intervals for the normal mean. Metron
**1995**, 53, 69–82. [Google Scholar] - Yousef, A.; Kimber, A.; Hamdy, H.I. Sensitivity of Normal-Based Triple sampling sequential point estimation to the normality assumption. J. Stat. Plan. Inference
**2013**, 143, 1606–1618. [Google Scholar] [CrossRef] [Green Version] - Yousef, A.S. Construction a three-stage asymptotic coverage probability for the mean using Edgeworth second order approximation. In Selected Papers on the International Conference on Mathematical Sciences and Statistics 2013; Springer: Singapore, 2014; pp. 53–67. [Google Scholar] [CrossRef]
- Hamdy, H.I.; Son, S.M.; Yousef, S.A. Sensitivity analysis of multistage sampling to departure of an underlying distribution from normality with computer simulations. J. Seq. Anal.
**2015**, 34, 532–558. [Google Scholar] [CrossRef] - Yousef, A. A Note on a three-stage sequential confidence interval for the mean when the underlying distribution departs away from normality. Int. Appl. Math. Stat.
**2018**, 57, 57–69. [Google Scholar] - Yousef, A.; Hamdy, H. Three-stage estimation for the mean and variance of the normal distribution with application to inverse coefficient of variation. Mathematics
**2019**, 7, 831. [Google Scholar] [CrossRef] [Green Version] - Yousef, A.; Hamdy, H. Three-stage sequential estimation of the inverse coefficient of variation of the normal distribution. Computation
**2019**, 7, 69. [Google Scholar] [CrossRef] [Green Version] - Yousef, A. Performance of three-stage sequential estimation of the inverse coefficient of variation under type II error probability: A Monte Carlo simulation study. Front. Phys.
**2020**. [Google Scholar] [CrossRef] - Liu, W. A k-stage sequential sampling procedure for estimation of a normal mean. J. Stat. Plan. Inference
**1995**, 65, 109–127. [Google Scholar] [CrossRef] - Son, M.S.; Haugh, H.I.; Hamdy, H.I.; Costanza, M.C. Controlling type II error while constructing triple sampling fixed precision confidence intervals for the normal mean. Ann. Inst. Stat. Math.
**1997**, 49, 681–692. [Google Scholar] [CrossRef] - Tahir, M. Sequential estimation of the square of the Rayleigh parameter. J. Math. Stat.
**2014**, 10, 275–280. [Google Scholar] [CrossRef] - Wald, A. Sequential Analysis; Wiley: New York, NY, USA, 1947. [Google Scholar]
- Anscombe, F.J. Large sample theory of sequential estimation. Math. Proc. Camb. Philos. Soc.
**1952**, 48, 600–607. [Google Scholar] [CrossRef] - Ghosh, M.; Mukhopadhyay, N. Consistency and asymptotic efficiency of two-stage and sequential estimation procedures. Sankhya
**1981**, 43, 220–227. [Google Scholar] - Simon, G. On the cost of not knowing the variance when making a fixed-width confidence interval for the mean. Ann. Math. Stat.
**1968**, 39, 1946–1952. [Google Scholar] [CrossRef]

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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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