# A New Generalization of the Generalized Inverse Rayleigh Distribution with Applications

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Beta Generalized Inverse Rayleigh Distribution

#### 2.1. Probability Density Function of BGIRD

#### 2.2. Cumulative Distribution Function of BGIRD

#### 2.3. Mixture Representation

#### 2.4. The Reliability Function

#### 2.5. The Hazard Function

#### 2.6. Special Sup-Models

- In particular, BGIRD becomes GIRD ($\eta $, $\rho $) when $\nu $ and $\tau =1$.
- The beta inverse Rayleigh distribution BIRD ($\nu ,\phantom{\rule{4pt}{0ex}}\tau ,\phantom{\rule{4pt}{0ex}}\rho $) is clearly a special case of BGIRD when $\eta $ = 1.
- IRD ($\rho $) can be obtained from (8) by making $\nu ,\phantom{\rule{4pt}{0ex}}\tau ,\phantom{\rule{4pt}{0ex}}\eta $ = 1.
- In addition, the exponentiated Rayleigh distribution ERD ($\eta $, $\rho $) is a special case of BGIRD when $\nu $ = $\tau $ = 1 and the random variable Y = 1/X.
- If $\nu $ = $\tau $ = 1 and $\eta $ = 1 in Equation (8), the random variable Y = 1/X has the Rayleigh distribution ($\rho $).

## 3. Statistical Properties

#### 3.1. Quantile Function

#### 3.2. Median

#### 3.3. Mode

#### 3.4. Moments

#### 3.5. Harmonic Mean

#### 3.6. Skewness and Kurtosis

#### 3.7. Mean Deviations

#### 3.7.1. The Mean Deviation about the Mean

**Theorem**

**1.**

**Proof.**

#### 3.7.2. The Mean Deviation about the Median

**Theorem**

**2.**

**Proof.**

#### 3.8. Rényi and Shannon Entropies

#### 3.8.1. The Rényi Entropy for the BGIRD

**Theorem**

**3.**

**Proof.**

#### 3.8.2. The Shannon Entropy for the BGIRD

**Theorem**

**4.**

## 4. Order Statistics

## 5. Maximum Likelihood Estimation Method

## 6. Simulation Study

- From Table 2, we note that the MSEs of the ML estimates for BGIR($\nu $, $\tau $, $\eta $, $\rho $), $R\left({t}_{0}\right)$, and $h\left({t}_{0}\right)$ decrease as the sample size increases which show consistency of the estimated parameters.
- According to the simulation results given in Table 2, as the sample size n increases, the ARBias is close to zero, the mean estimates tend to be closer to the true parameter values.

## 7. Application

**Data Set 1:**This data represents the survival times of a group of patients suffering from Head and Neck cancer disease and treated using a combination of radiotherapy and chemotherapy. These data are: 12.2, 23.56, 23.74, 25.87, 31.98, 37, 41.35, 47.38, 55.46, 58.36, 63.47, 68.46, 78.26, 74.47, 81.43, 84, 92, 94, 110, 112, 119, 127, 130, 133, 140, 146, 155, 159, 173, 179, 194, 195, 209, 249, 281, 319, 339, 432, 469, 519, 633, 725, 817, 1776.

**Data Set 2:**The data set represents the failure times of the air conditioning system of an airplane. It has 30 observations as follows: 23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95.

**Data Set 3:**The data set represents the lifetime data relating to relief times (in minutes) of patients receiving an analgesic. The data set consists of 20 observations and it is as follows: 1.1, 1.4, 1.3, 1.7, 1.9, 1.8, 1.6, 2.2, 1.7, 2.7, 4.1, 1.8, 1.5, 1.2, 1.4, 3.0, 1.7, 2.3, 1.6, 2.0.

**Data Set 4:**These data are the strengths of 1.5 cm glass fibers, measured at the National Physical Laboratory, England. The data set is: 0.55, 0.93, 1.25, 1.36, 1.49, 1.52, 1.58, 1.61, 1.64, 1.68, 1.73, 1.81, 2, 0.74, 1.04, 1.27, 1.39, 1.49, 1.53, 1.59, 1.61, 1.66, 1.68, 1.76, 1.82, 2.01, 0.77, 1.11, 1.28, 1.42, 1.5, 1.54, 1.6, 1.62, 1.66, 1.69, 1.76, 1.84, 2.24, 0.81, 1.13, 1.29, 1.48, 1.5, 1.55, 1.61, 1.62, 1.66, 1.7, 1.77, 1.84, 0.84, 1.24, 1.3, 1.48, 1.51, 1.55, 1.61, 1.63, 1.67, 1.7, 1.78, 1.89.

**Data Set 5:**These data arose in testing on the cycle at which the Yarn failed. The data are the number of cycles until failure of the yarn and they are: 86, 146, 251, 653, 98, 249, 400, 292, 131, 169, 175, 176, 76, 264, 15, 364, 195, 262, 88, 264, 157, 220, 42, 321, 180, 198, 38, 20, 61, 121, 282, 224, 149, 180, 325, 250, 196, 90, 229, 166, 38, 337, 65, 151, 341, 40, 40, 135, 597, 246, 211, 180, 93, 315, 353, 571, 124, 279, 81, 186, 497, 182, 423, 185, 229, 400, 338, 290, 398, 71, 246, 185, 188, 568, 55, 55, 61, 244,20, 284, 393, 396, 203, 829, 239, 236, 286, 194, 277, 143, 198, 264, 105, 203, 124, 137, 135, 350, 193, 188.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Conflicts of Interest

## Appendix A

**Theorem**

**A1.**

**Proof.**

## References

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$\mathit{\nu}$ | $\mathit{\tau}$ | $\mathit{\eta}$ | $\mathit{\rho}$ | Skewness | Kurtosis | Mean | Median | Mode | Harmonic Mean |
---|---|---|---|---|---|---|---|---|---|

2 | 2 | 2 | 1.5 | 0.13731 | 1.29818 | 0.637261 | 0.601615 | 0.545538 | 0.592602 |

2 | 2 | 2 | 2 | 0.13731 | 1.29818 | 0.477946 | 0.451212 | 0.409154 | 0.444451 |

2 | 2 | 0.5 | 2 | 0.31697 | 1.58531 | 1.42599 | 0.93221 | 0.60726 | 0.862293 |

2 | 2 | 1.5 | 2 | 0.162677 | 1.32328 | 0.543969 | 0.50147 | 0.439831 | 0.492135 |

2 | 2 | 5 | 2 | 0.0792963 | 1.25709 | 0.357224 | 0.349688 | 0.336093 | 0.346181 |

1 | 2 | 2 | 2 | 0.136143 | 1.29595 | 0.392529 | 0.368785 | 0.331165 | 0.360099 |

3 | 2 | 2 | 2 | 0.137703 | 1.29892 | 0.536444 | 0.507568 | 0.462275 | 0.501744 |

2 | 0.75 | 2 | 2 | 0.240794 | 1.4393 | 0.770942 | 0.63456 | 0.498165 | 0.616217 |

2 | 1 | 2 | 2 | 0.203663 | 1.37855 | 0.64575 | 0.566454 | 0.470633 | 0.553656 |

2 | 4 | 2 | 2 | 0.0921975 | 1.26287 | 0.387276 | 0.376634 | 0.35771 | 0.372261 |

1 | 1 | 1.5 | 0.5 | 0.24225 | 1.43601 | 2.46538 | 2.00588 | 1.53797 | 1.92415 |

1 | 1 | 1 | 1.5 | 0.30686 | 1.57048 | 3.54491 | 0.800748 | 0.54433 | 0.75225 |

**Table 2.**Maximum likelihood estimates, ARBias, and MSE of the parameters $\underline{\theta}=(\nu ,\tau ,\eta ,\rho )$ = (2, 2, 0.5, 2) and R(${t}_{0}$) = 0.1519, h(${t}_{0}$) = 0.8740 at t${}_{0}$ = 2.

n | $\widehat{\mathit{\nu}}$ | $\widehat{\mathit{\tau}}$ | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{\rho}}$ | $\widehat{\mathit{R}}\left({\mathit{t}}_{0}\right)$ | $\widehat{\mathit{h}}\left({\mathit{t}}_{0}\right)$ | |
---|---|---|---|---|---|---|---|

MLEs | 1.8771 | 2.1164 | 0.5575 | 1.9080 | 0.1219 | 1.0487 | |

10 | ARBias | 0.0614 | 0.0582 | 0.1149 | 0.0460 | 0.1977 | 0.1999 |

MSE | 0.1384 | 0.1120 | 0.0204 | 0.0963 | 0.0056 | 0.1160 | |

MLEs | 1.9092 | 2.0567 | 0.5353 | 1.9394 | 0.1348 | 0.9684 | |

20 | ARBias | 0.0454 | 0.0283 | 0.0706 | 0.0303 | 0.1130 | 0.1080 |

MSE | 0.1217 | 0.0846 | 0.0133 | 0.0812 | 0.0032 | 0.0576 | |

MLEs | 1.9511 | 2.0456 | 0.5190 | 1.9713 | 0.1412 | 0.9281 | |

30 | ARBias | 0.0244 | 0.0228 | 0.0381 | 0.0143 | 0.0708 | 0.0619 |

MSE | 0.1191 | 0.0830 | 0.0064 | 0.0761 | 0.0019 | 0.0283 | |

MLEs | 1.9430 | 2.0373 | 0.5131 | 1.9681 | 0.1447 | 0.9104 | |

50 | ARBias | 0.0285 | 0.0187 | 0.0261 | 0.0159 | 0.0477 | 0.0417 |

MSE | 0.1160 | 0.0771 | 0.0057 | 0.0730 | 0.0014 | 0.0182 | |

MLEs | 1.9714 | 2.0082 | 0.5049 | 1.9879 | 0.1490 | 0.8857 | |

100 | ARBias | 0.0143 | 0.0041 | 0.0098 | 0.0060 | 0.0191 | 0.0135 |

MSE | 0.0373 | 0.0245 | 0.0019 | 0.0247 | 0.0005 | 0.0052 |

**Table 3.**Parameter estimates, goodness-of-fit measures of the fitted distributions of Data Sets 1–4.

Data | Model | MLEs | Statistics | ||||||
---|---|---|---|---|---|---|---|---|---|

$\widehat{\mathit{\rho}}$ | $\widehat{\mathit{\nu}}$ | $\widehat{\mathit{\tau}}$ | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{\theta}}$ | AIC | BIC | −2 $\mathit{logL}$ | ||

Data 1 | BGIRD | 14.0184 | 74.0009 | 8.71669 | 0.15366 | 563.039 | 570.175 | 555.039 | |

BIRD | 6899.62 | 4900.34 | 0.0543026 | 715.539 | 720.892 | 709.539 | |||

BGIWD | 4938.59 | 0.001181 | 0.0862678 | 0.750759 | 39.904 | 576.569 | 585.49 | 566.569 | |

Exponential | 0.004475 | 566.02 | 567.80 | 564.02 | |||||

Lindley | 0.00891 | 581.16 | 582.95 | 579.16 | |||||

Akash | 0.013423 | 611.93 | 613.71 | 609.93 | |||||

Data 2 | BGIRD | 5.0833 | 14.4244 | 2.9408 | 0.1921 | 316.199 | 321.80 | 308.199 | |

BIRD | 0.5395 | 9.6731 | 0.2074 | 330.41 | 333.21 | 326.41 | |||

BGIWD | 4227.66 | 0.0013 | 0.0586 | 0.534319 | 40.7341 | 326.17 | 333.18 | 316.17 | |

Exponential | 0.016779 | 307.26 | 308.66 | 305.26 | |||||

Lindley | 0.033021 | 325.2 | 326.6 | 323.2 | |||||

Akash | 0.050293 | 356.88 | 358.2 | 354.8 | |||||

Data 3 | BGIRD | 17.5039 | 58.042 | 6.3644 | 0.3428 | 51.067 | 55.05 | 43.067 | |

BIRD | 7569.7 | 5201.94 | 0.098989 | 133.896 | 136.883 | 127.896 | |||

BGIWD | 2464.11 | 0.0015 | 0.0396 | 0.4256 | 29.545 | 109.587 | 114.566 | 99.5871 | |

Exponential | 0.526316 | 67.67 | 68.67 | 65.67 | |||||

Lindley | 0.816118 | 60.50 | 62.50 | 63.49 | |||||

Akash | 1.156923 | 61.52 | 62.51 | 59.52 | |||||

Data 4 | BGIRD | 21.0277 | 69.3754 | 9.45318 | 0.3144 | 102.438 | 111.01 | 94.4377 | |

BIRD | 0.65579 | 0.808613 | 1.00416 | 115.251 | 121.681 | 109.251 | |||

BGIWD | 37.7781 | 0.0052 | 0.174197 | 1.28585 | 2.63091 | 164.39 | 175.106 | 154.39 | |

Exponential | 0.663647 | 179.66 | 181.80 | 177.66 | |||||

Lindley | 0.996116 | 164.56 | 166.70 | 162.56 | |||||

Akash | 1.355445 | 165.73 | 169.93 | 163.73 | |||||

Data 5 | BGIRD | 12.0129 | 155.906 | 7.49802 | 0.205185 | 1285.79 | 1296.21 | 1277.79 | |

BIRD | 19758 | 33641.7 | 0.0520308 | 1690.43 | 1698.24 | 1684.43 | |||

BGIWD | 5.54157 | 1.93615 | 3.77834 | 0.629758 | 9.71891 | 1289.55 | 1302.57 | 1279.55 | |

Exponential | 0.004505 | 1282.52 | 1285.12 | 1280.52 | |||||

Lindley | 0.00897 | 1253.34 | 1255.95 | 1251.34 | |||||

Akash | 0.013514 | 1257.83 | 1260.43 | 1255.83 |

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**MDPI and ACS Style**

Bakoban, R.A.; Al-Shehri, A.M.
A New Generalization of the Generalized Inverse Rayleigh Distribution with Applications. *Symmetry* **2021**, *13*, 711.
https://doi.org/10.3390/sym13040711

**AMA Style**

Bakoban RA, Al-Shehri AM.
A New Generalization of the Generalized Inverse Rayleigh Distribution with Applications. *Symmetry*. 2021; 13(4):711.
https://doi.org/10.3390/sym13040711

**Chicago/Turabian Style**

Bakoban, Rana Ali, and Ashwaq Mohammad Al-Shehri.
2021. "A New Generalization of the Generalized Inverse Rayleigh Distribution with Applications" *Symmetry* 13, no. 4: 711.
https://doi.org/10.3390/sym13040711