# Interval Estimation for the Two-Parameter Exponential Distribution Based on the Upper Record Values

## Abstract

**:**

## 1. Introduction

## 2. Interval Estimation and Confidence Region for Two Parameters

_{1}and v

_{2}as F(v

_{1},v

_{2}). Furthermore, the second set of pivotal quantities consists of ${h}_{2}(\mu )=\frac{{h}_{1}/2}{{g}_{1}/(2n)}={\frac{{Z}_{0}/2}{{\displaystyle \sum _{i=1}^{n}{Z}_{i}}/(2n)}}_{}=n(\frac{{R}_{0}-\mu}{{R}_{n}-{R}_{0}})$ and ${g}_{2}(\mu ,\theta )=2{\displaystyle \sum _{i=0}^{n}{Z}_{i}}=2{Y}_{n}=2\frac{{R}_{n}-\mu}{\theta}$. The distributions of these two pivotal quantities are ${h}_{2}(\mu )~F(2,2n)$ and ${g}_{2}(\mu ,\theta )~{\chi}^{2}(2(n+1))$. The distributions of all pivotal quantities are not a function of parameters. Utilizing the pivotal quantity ${g}_{1}(\theta )$, we can build the confidence interval for the scale parameter θ as follows:

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**1.**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

_{1}is obtained as follows:

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

_{2}is obtained as follows:

## 3. Simulation Study

## 4. A Biometrical Example

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Wu, S.F. A modified one stage multiple comparison procedure of exponential location parameters with the control under heteroscedasticity. Commun. Stat.-Theory Methods
**2021**. [Google Scholar] [CrossRef] - Johnson, N.L.; Kotz, S. Continuous Univariate Distributions; John Wiley & Sons, Inc.: New York, NY, USA, 1970. [Google Scholar]
- Bain, L.J. Statistical Analysis of Reliability and Life Testing Models; Marcer Dekker: New York, NY, USA, 1978. [Google Scholar]
- Lawless, J.F. Statistical Models and Methods for Lifetime Data; John Wiley & Sons, Inc.: New York, NY, USA, 1982. [Google Scholar]
- Zelen, M. Application of exponential models to problems in cancer research. J. R. Stat. Soc.
**1966**, 129, 368–398. [Google Scholar] [CrossRef] - Shafiq, A.; Lone, S.A.; Sindhu, T.N.; Khatib, Y.E.; Al-Mdallal, Q.M.; Muhammad, T. A new modified Kies Fréchet distribution: Applications of mortality rate of COVID-19. Results Physics
**2021**, 28, 104638. [Google Scholar] [CrossRef] [PubMed] - El-Khatib, Y.; Hatemi-J, A. Option valuation and hedging in markets with a crunch. J. Econ. Stud.
**2017**, 44, 801–815. [Google Scholar] [CrossRef] - Alzaatreh, A.; Famoye, F.; Lee, C. Weibull-Pareto Distribution and Its Applications. Commun. Stat.-Theory Methods
**2013**, 42, 1673–1691. [Google Scholar] [CrossRef] - Al-Hussaini, E.K.; Ahmad, A.A. On Bayesian Interval Prediction of Future Records. Test
**2003**, 12, 79–99. [Google Scholar] [CrossRef] - Asgharzadeh, A.; Abdi, M.; Kuş, C. Interval estimation for the two-parameter Pareto distribution based on record. Selçuk J. Appl. Math.
**2011**, 149–161. [Google Scholar] - Jana, N.; Bera, S. Interval estimation of multicomponent stress–strength reliability based on inverse Weibull distribution. Math. Comput. Simul.
**2022**, 191, 95–119. [Google Scholar] [CrossRef] - Wu, S.F. Bayesian interval estimation for the two-parameter exponential distribution based on the right type II censored sample. Symmetry
**2022**, 14, 352. [Google Scholar] [CrossRef] - Wu, S.F. Interval estimation for the two-parameter exponential distribution under progressive type II censoring on the Bayesian approach. Symmetry
**2022**, 14, 808. [Google Scholar] [CrossRef] - Arnold, B.C.; Balakrishnan, N.; Nagaraja, H.N. Records; John Wiley & Sons, Inc.: New York, NY, USA, 1998. [Google Scholar]
- Proschan, F. Theoretical explanation of observed decreasing failure rate. Technometrics
**1963**, 3, 375–383. [Google Scholar] [CrossRef]

**Table 1.**The average length and area for the interval estimation of the exponential distribution with ($\mu $,$\theta $ ) = (0,1) under 1-α = 0.90 and 0.95.

1-α = 0.90 | 1-α = 0.95 | |||||
---|---|---|---|---|---|---|

n | Length | Method 1 | Method 2 | Length | Method 1 | Method 2 |

1 | 18.9442 | 5170.463 | 5672.394 | 40.1069 | 28621.48 | 38499.85 |

2 | 5.2178 | 181.8382 | 167.7742 | 7.8828 | 471.5438 | 474.2633 |

3 | 3.2142 | 55.4781 | 50.0454 | 4.4151 | 113.3213 | 106.0673 |

4 | 2.4160 | 29.6365 | 26.8569 | 3.1991 | 54.3921 | 50.1530 |

5 | 1.9952 | 19.8614 | 18.1598 | 2.6003 | 35.3613 | 32.5916 |

6 | 1.7454 | 15.3585 | 14.1663 | 2.2028 | 24.9928 | 23.1242 |

7 | 1.5387 | 12.0971 | 11.2444 | 1.9497 | 19.7239 | 18.3398 |

8 | 1.4171 | 10.4444 | 9.7725 | 1.7536 | 16.2681 | 15.2019 |

9 | 1.2945 | 8.9408 | 8.4128 | 1.5989 | 13.7533 | 12.9117 |

10 | 1.2020 | 7.8545 | 7.4262 | 1.4980 | 12.3047 | 11.6006 |

15 | 0.9339 | 5.3172 | 5.109 | 1.1449 | 7.9318 | 7.5936 |

20 | 0.7942 | 4.2413 | 4.1122 | 0.9625 | 6.153 | 5.9454 |

30 | 0.6299 | 3.1236 | 3.0579 | 0.7617 | 4.4811 | 4.3750 |

60 | 0.4365 | 2.0251 | 2.0031 | 0.5219 | 2.8227 | 2.7876 |

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

© 2022 by the author. 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wu, S.-F.
Interval Estimation for the Two-Parameter Exponential Distribution Based on the Upper Record Values. *Symmetry* **2022**, *14*, 1906.
https://doi.org/10.3390/sym14091906

**AMA Style**

Wu S-F.
Interval Estimation for the Two-Parameter Exponential Distribution Based on the Upper Record Values. *Symmetry*. 2022; 14(9):1906.
https://doi.org/10.3390/sym14091906

**Chicago/Turabian Style**

Wu, Shu-Fei.
2022. "Interval Estimation for the Two-Parameter Exponential Distribution Based on the Upper Record Values" *Symmetry* 14, no. 9: 1906.
https://doi.org/10.3390/sym14091906