# Statistical Inference of the Beta Binomial Exponential 2 Distribution with Application to Environmental Data

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## Abstract

**:**

## 1. Introduction

- (i)
- Provide a generalization of the $BE2$ distribution by including two additional shape parameters that allow for larger adaptability in the form of the beta binomial exponential 2 ($BBE2$) distribution and, as a result, in modeling observed positive data.
- (ii)
- The pdf of the $BBE2$ distribution can take different shapes, such as decreasing, unimodal and right skewness, and the shapes of the hazard rate function (hrf) can be decreasing, increasing and constant.
- (iii)
- Some statistical and mathematical features of the $BBE2$ distribution are computed and investigated.
- (iv)
- Develop an acceptance sampling plan (ASP), derive its operating characteristic function and give the corresponding decision rule by using the $BBE2$ distribution.
- (v)
- Study two classical approaches of estimation; maximum likelihood ($ML$) and maximum product of spacing ($MPSP$). Further, the Bayesian approach of estimation is utilized to estimate the model parameters.
- (vi)
- The significance of the $BBE2$ model is demonstrated through a study of real-world data applications, which demonstrates the flexibility and potential of the $BBE2$ model in comparison to other well-known competitive models.

## 2. Beta Binomial Exponential 2 Distribution

#### A Useful Representation

## 3. Statistical Features of the $\mathit{BBE}\mathbf{2}$ Distribution

#### 3.1. Quantile Function

#### 3.2. Moments

**Theorem**

**1.**

**Proof.**

#### 3.3. Moment-Generating Function

**Theorem**

**2.**

**Proof.**

## 4. Acceptance Sampling Plans

- Take n units at random from the suggested lot as a sample.
- Run the following test for ${t}_{0}$ units of time:If c or fewer units (acceptance number) fail during the test, accept the entire lot; otherwise, the lot is rejected.

- $\alpha =0.25,0.75,0.95$,
- $c=0,1,5,10,20$,
- $k=0.1,0.2,0.4,0.8,1$ (note that when $k=1,{t}_{0}={m}_{0}=0.5\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\lambda ,\theta ,a,b$)
- The parameters of the BBE2 distribution $(\lambda ,\theta )$ are assumed to be:
- Case 1: $(\lambda =0.50,\theta =0.25)$
- Case 2: $(\lambda =0.50,\theta =0.75)$

- Parameter $(a,b)$ of the BBE2 distribution is assumed to be $(0.5,1.5)$ and $(1.5,0.5)$.

- For the parameters of ASP: When $\alpha $ and c are increasing, the required sample size n is increasing, but $L\left({p}_{0}\right)$ is decreasing. While k is increasing, the required n is decreasing, but $L\left({p}_{0}\right)$ is increasing.
- For the parameters of the $BBE2$ distribution: With increases in any parameters of $\lambda ,\theta ,a$ and b where the other parameters are fixed, the required n is increasing, but $L\left({p}_{0}\right)$ is decreasing.

## 5. Non-Bayesian Estimation Methods

#### 5.1. Maximum Likelihood Estimation

#### 5.2. Maximum Product of Spacing Estimation

## 6. Bayesian Estimation

#### 6.1. Markov Chain Monte Carlo

- Starting with an initial guess: just one value that could be gathered from the distribution.
- Creating a series of new samples based on this first guess. Each new sample is created in two steps:
- Proposal: A new sample proposal is produced by adding a small random disturbance to the most recent sample.
- Acceptance: The suggestion is either accepted as the new sample or rejected (in which case the old sample is retained).

#### 6.2. Metropolis–Hasting Algorithm

- Step 1.
- Set initial value of $\xi $ as ${\xi}^{\left(0\right)}=\left({\widehat{\lambda}}_{MLE},{\widehat{\theta}}_{MLE},{\widehat{a}}_{MLE},{\widehat{b}}_{MLE}\right)$.
- Step 2.
- For $i=1,2,\dots ,M$ repeat the next stages:
- 2.1:
- Set $\xi ={\xi}^{(i-1)}$.
- 2.2:
- Generate a new candidate parameter value $\delta $ from ${N}_{4}\left(log\xi ,{S}_{\xi}\right)$.
- 2.3:
- Set ${\theta}^{{}^{\prime}}=exp\left(\delta \right)$.
- 2.4:
- Calculate $\beta =\frac{\pi \left({\xi}^{{}^{\prime}}\right|x)}{\pi \left({\xi}^{|}x\right)}$, where $\pi (\xb7)$ is the posterior density in (23).
- 2.5:
- Generate a sample u from the uniform $U(0,1)$ distribution.
- 2.6:
- Accept or reject the new candidate ${\theta}^{{}^{\prime}}$$$\left\{\begin{array}{cc}\mathrm{If}\phantom{\rule{1.em}{0ex}}u\le \beta \hfill & \mathrm{set}\phantom{\rule{1.em}{0ex}}{\xi}^{\left(i\right)}={\xi}^{{}^{\prime}}\hfill \\ \mathrm{otherwise}\hfill & \mathrm{set}\phantom{\rule{1.em}{0ex}}{\xi}^{\left(i\right)}=\xi .\hfill \end{array}\right.$$

#### 6.3. Highest Posterior Density Intervals

## 7. Simulation Study and Data Analysis

#### Simulation Study

- Sample size generated from the $BBE2$ distribution is supposed to be $n=100,200$.
- For the parameters $(a,b)$ of the beta distribution, we assumed that: $a=0.50,0.75$ and $b=0.50,0.75$.
- For parameter $\left(\lambda \right)$ of the exponential distribution, we assumed that: $\lambda =0.50,1.50$.
- For parameter $\left(\theta \right)$ of the binomial distribution, we assumed that: $\theta =0.25,0.50$.

- In general, the increasing n, MSEs and AILs are decreasing for all methods of Non-BEs and BEs. Further, two Non-BEs methods (MLE and MPSP) are competing well for estimating the parameters of the $BBE2$ distribution. For BE methods, the loss function GE estimates are better than BEs under other loss functions (LINEX and SE).
- For fixed $(\lambda ,\theta ,a)$ and as the value of b increases, the MSEs of $\lambda $ and b estimates are increasing, but the MSEs of $\theta $ and a are decreasing.
- For fixed $(\lambda ,\theta ,b)$ and as the value of a increases, the MSEs of $\lambda $ and $\theta $ estimates are decreasing, but the MSEs of a and b are increasing.
- For fixed $(\lambda ,a,b)$ and as the value of $\theta $ increases, the MSEs of $\lambda $ estimates is decreasing, but the MSEs of a and b are increasing.
- For fixed $(\theta ,a,b)$ and as the value of $\lambda $ increases, the MSEs of all parameters are increasing.

## 8. Applications: Kevlar Data

## 9. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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$\mathit{\lambda}$ | $\mathit{\theta}$ | a | b | Distribution | Authors |
---|---|---|---|---|---|

− | − | − | 1 | generalized $BE2$ | [32] |

− | − | 1 | 1 | $BE2$ | [3] |

− | 0 | − | − | B exponential | [33] |

− | 2 | − | − | B gamma | [34] |

− | 0 | 1 | 1 | exponential | |

− | 2 | 1 | 1 | gamma |

Moments (${\mathit{\mu}}_{\mathit{r}}^{{}^{\prime}}$) | $(\mathit{a},\mathit{b},\mathit{\theta},\mathit{\lambda})$ | |||
---|---|---|---|---|

(0.5, 0.7, 0.30, 3.0) | (2.5, 2.7, 1.0, 3.0) | (1.5, 2.7, 0.70, 3.0) | (1.5, 2.7, 1.00, 3.0) | |

${\mu}_{1}^{{}^{\prime}}$ | 0.362942 | 0.571999 | 2.92529 | 0.435659 |

${\mu}_{2}^{{}^{\prime}}$ | 0.358767 | 0.388811 | 9.50626 | 0.245108 |

${\mu}_{3}^{{}^{\prime}}$ | 0.554513 | 0.30641 | 76.2397 | 0.168439 |

${\mu}_{4}^{{}^{\prime}}$ | 1.14747 | 0.274943 | 550.099 | 0.136526 |

${\mu}_{5}^{{}^{\prime}}$ | 2.95521 | 0.277046 | 4696.42 | 0.127411 |

${\mu}_{6}^{{}^{\prime}}$ | 9.07734 | 0.310069 | 46206.3 | 0.1345 |

Variance | 0.22704 | 0.061628 | 0.948911 | 0.0553086 |

Skewness | 2.39872 | 0.882932 | 46.3885 | 1.03515 |

Kurtosis | 164.972 | 13.4627 | 50623.0 | 3.9319 |

**Table 3.**ASPs for the $BBE2$ distribution with parameters: $\lambda =0.5,\theta =0.25$ and different values for a and b.

$\mathit{\alpha}$ | c | $\mathit{k}\to $ | $0.10$ | $0.20$ | $0.4$ | $0.80$ | 1 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{n}$ | $\mathit{L}\left({\mathit{p}}_{\mathbf{0}}\right)$ | $\mathit{n}$ | $\mathit{L}\left({\mathit{p}}_{\mathbf{0}}\right)$ | $\mathit{n}$ | $\mathit{L}\left({\mathit{p}}_{\mathbf{0}}\right)$ | $\mathit{n}$ | $\mathit{L}\left({\mathit{p}}_{\mathbf{0}}\right)$ | $\mathit{n}$ | $\mathit{L}\left({\mathit{p}}_{\mathbf{0}}\right)$ | |||

$a=0.5$ and $b=1.5$ | ||||||||||||

0.25 | 0 | 2 | 0.8315 | 2 | 0.7634 | 1 | 1.0000 | 1 | 1.0000 | 1 | 1.0000 | |

1 | 6 | 0.8002 | 5 | 0.7607 | 3 | 0.8912 | 3 | 0.7944 | 3 | 0.7500 | ||

5 | 26 | 0.7640 | 19 | 0.7623 | 14 | 0.7673 | 10 | 0.8288 | 9 | 0.8555 | ||

10 | 52 | 0.7683 | 38 | 0.7560 | 27 | 0.7912 | 20 | 0.8076 | 19 | 0.7597 | ||

20 | 107 | 0.7579 | 77 | 0.7556 | 56 | 0.7533 | 41 | 0.7738 | 37 | 0.7975 | ||

0.75 | 0 | 8 | 0.2748 | 6 | 0.2593 | 4 | 0.3010 | 3 | 0.2987 | 3 | 0.2500 | |

1 | 16 | 0.2536 | 11 | 0.2756 | 8 | 0.2699 | 6 | 0.2510 | 5 | 0.3125 | ||

5 | 43 | 0.2669 | 31 | 0.2531 | 22 | 0.2595 | 16 | 0.2520 | 14 | 0.2905 | ||

10 | 76 | 0.2612 | 54 | 0.2604 | 38 | 0.2804 | 28 | 0.2516 | 25 | 0.2706 | ||

20 | 140 | 0.2585 | 99 | 0.2657 | 71 | 0.2581 | 51 | 0.2696 | 46 | 0.2757 | ||

0.95 | 0 | 17 | 0.0522 | 12 | 0.0513 | 8 | 0.0607 | 5 | 0.0892 | 5 | 0.0625 | |

1 | 27 | 0.0517 | 19 | 0.0510 | 13 | 0.0567 | 9 | 0.0608 | 8 | 0.0625 | ||

5 | 60 | 0.0529 | 42 | 0.0544 | 29 | 0.0616 | 21 | 0.0520 | 18 | 0.0717 | ||

10 | 98 | 0.0502 | 69 | 0.0501 | 48 | 0.0566 | 34 | 0.0576 | 30 | 0.0680 | ||

20 | 168 | 0.0526 | 119 | 0.0502 | 84 | 0.0515 | 59 | 0.0619 | 53 | 0.0632 | ||

$a=1.5$ and $b=0.5$ | ||||||||||||

0.25 | 0 | 10 | 0.7641 | 4 | 0.7855 | 2 | 0.8107 | 1 | 1.0000 | 1 | 1.0000 | |

1 | 33 | 0.7572 | 13 | 0.7635 | 6 | 0.7591 | 3 | 0.8343 | 3 | 0.7500 | ||

5 | 144 | 0.7531 | 55 | 0.7626 | 23 | 0.7747 | 11 | 0.8217 | 9 | 0.8555 | ||

10 | 294 | 0.7512 | 113 | 0.7515 | 47 | 0.7569 | 23 | 0.7506 | 19 | 0.7597 | ||

20 | 604 | 0.7521 | 231 | 0.7545 | 96 | 0.7502 | 45 | 0.7874 | 37 | 0.7975 | ||

0.75 | 0 | 47 | 0.2528 | 18 | 0.2545 | 7 | 0.2840 | 3 | 0.3516 | 3 | 0.2500 | |

1 | 91 | 0.2531 | 34 | 0.2644 | 14 | 0.2638 | 6 | 0.3249 | 5 | 0.3125 | ||

5 | 251 | 0.2526 | 95 | 0.2568 | 38 | 0.2734 | 17 | 0.3081 | 14 | 0.2905 | ||

10 | 441 | 0.2513 | 167 | 0.2556 | 68 | 0.2540 | 31 | 0.2650 | 25 | 0.2706 | ||

20 | 809 | 0.2517 | 307 | 0.2543 | 125 | 0.2522 | 57 | 0.2681 | 46 | 0.2757 | ||

0.95 | 0 | 101 | 0.0503 | 38 | 0.0509 | 15 | 0.0530 | 6 | 0.0733 | 5 | 0.0625 | |

1 | 160 | 0.0502 | 60 | 0.0515 | 24 | 0.0511 | 10 | 0.0650 | 8 | 0.0625 | ||

5 | 355 | 0.0501 | 134 | 0.0503 | 53 | 0.0543 | 23 | 0.0635 | 18 | 0.0717 | ||

10 | 573 | 0.0503 | 216 | 0.0515 | 86 | 0.0551 | 38 | 0.0610 | 30 | 0.0680 | ||

20 | 983 | 0.0501 | 372 | 0.0502 | 149 | 0.0532 | 67 | 0.0536 | 53 | 0.0632 |

**Table 4.**ASPs for the $BBE2$ distribution with parameters: $\lambda =0.5,\theta =0.75$ and different values for a and b.

$\mathit{\alpha}$ | c | $\mathit{k}\to $ | $0.10$ | $0.20$ | $0.4$ | $0.80$ | 1 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{n}$ | $\mathit{L}\left({\mathit{p}}_{\mathbf{0}}\right)$ | $\mathit{n}$ | $\mathit{L}\left({\mathit{p}}_{\mathbf{0}}\right)$ | $\mathit{n}$ | $\mathit{L}\left({\mathit{p}}_{\mathbf{0}}\right)$ | $\mathit{n}$ | $\mathit{L}\left({\mathit{p}}_{\mathbf{0}}\right)$ | $\mathit{n}$ | $\mathit{L}\left({\mathit{p}}_{\mathbf{0}}\right)$ | |||

$a=0.5$ and $b=1.5$ | ||||||||||||

0.25 | 0 | 2 | 0.8413 | 2 | 0.7752 | 1 | 1.0000 | 1 | 1.0000 | 1 | 1.0000 | |

1 | 7 | 0.7559 | 5 | 0.7801 | 4 | 0.7606 | 3 | 0.7986 | 3 | 0.7500 | ||

5 | 28 | 0.7503 | 20 | 0.7589 | 14 | 0.7948 | 10 | 0.8360 | 9 | 0.8555 | ||

10 | 56 | 0.7510 | 40 | 0.7532 | 28 | 0.7873 | 21 | 0.7541 | 19 | 0.7597 | ||

20 | 114 | 0.7508 | 81 | 0.7542 | 58 | 0.7518 | 41 | 0.7913 | 37 | 0.7975 | ||

0.75 | 0 | 9 | 0.2509 | 6 | 0.2800 | 4 | 0.3169 | 3 | 0.3038 | 3 | 0.2500 | |

1 | 17 | 0.2530 | 12 | 0.2546 | 8 | 0.2921 | 6 | 0.2580 | 5 | 0.3125 | ||

5 | 46 | 0.2602 | 32 | 0.2724 | 23 | 0.2513 | 16 | 0.2638 | 14 | 0.2905 | ||

10 | 81 | 0.2571 | 57 | 0.2576 | 40 | 0.2597 | 28 | 0.2674 | 25 | 0.2706 | ||

20 | 149 | 0.2552 | 105 | 0.2536 | 73 | 0.2741 | 52 | 0.2516 | 46 | 0.2757 | ||

0.95 | 0 | 18 | 0.0530 | 12 | 0.0608 | 8 | 0.0684 | 6 | 0.0509 | 5 | 0.0625 | |

1 | 28 | 0.0573 | 20 | 0.0516 | 13 | 0.0665 | 9 | 0.0640 | 8 | 0.0625 | ||

5 | 64 | 0.0520 | 44 | 0.0569 | 31 | 0.0508 | 21 | 0.0565 | 18 | 0.0717 | ||

10 | 104 | 0.0511 | 72 | 0.0550 | 50 | 0.0551 | 34 | 0.0640 | 30 | 0.0680 | ||

20 | 179 | 0.0513 | 125 | 0.0525 | 87 | 0.0530 | 60 | 0.0576 | 53 | 0.0632 | ||

$a=1.5$ and $b=0.5$ | ||||||||||||

0.25 | 0 | 17 | 0.7519 | 6 | 0.7622 | 2 | 0.8462 | 1 | 1.0000 | 1 | 1.0000 | |

1 | 55 | 0.7530 | 19 | 0.7541 | 7 | 0.7674 | 3 | 0.8469 | 3 | 0.7500 | ||

5 | 240 | 0.7509 | 81 | 0.7518 | 28 | 0.7730 | 12 | 0.7725 | 9 | 0.8555 | ||

10 | 489 | 0.7515 | 164 | 0.7545 | 57 | 0.7643 | 23 | 0.7971 | 19 | 0.7597 | ||

20 | 1007 | 0.7506 | 337 | 0.7540 | 117 | 0.7577 | 47 | 0.7766 | 37 | 0.7975 | ||

0.75 | 0 | 78 | 0.2536 | 26 | 0.2572 | 9 | 0.2628 | 3 | 0.3706 | 3 | 0.2500 | |

1 | 152 | 0.2520 | 51 | 0.2507 | 17 | 0.2700 | 6 | 0.3523 | 5 | 0.3125 | ||

5 | 420 | 0.2502 | 140 | 0.2510 | 48 | 0.2501 | 18 | 0.2888 | 14 | 0.2905 | ||

10 | 736 | 0.2511 | 245 | 0.2534 | 84 | 0.2511 | 32 | 0.2779 | 25 | 0.2706 | ||

20 | 1351 | 0.2504 | 450 | 0.2526 | 154 | 0.2526 | 59 | 0.2801 | 46 | 0.2757 | ||

0.95 | 0 | 169 | 0.0501 | 56 | 0.0504 | 18 | 0.0584 | 7 | 0.0509 | 5 | 0.0625 | |

1 | 267 | 0.0505 | 88 | 0.0519 | 29 | 0.0567 | 11 | 0.0519 | 8 | 0.0625 | ||

5 | 593 | 0.0503 | 197 | 0.0501 | 66 | 0.0525 | 24 | 0.0638 | 18 | 0.0717 | ||

10 | 957 | 0.0504 | 318 | 0.0504 | 107 | 0.0526 | 40 | 0.0565 | 30 | 0.0680 | ||

20 | 1642 | 0.0500 | 546 | 0.0501 | 185 | 0.0507 | 70 | 0.0526 | 53 | 0.0632 |

**Table 5.**Avg. estimated values, MSEs, Asy-CI, AILs and CP (in %) of Non-BE (MLE and MPSP) for the $BBE2$ distribution at different sample sizes n and different values of $(a,b)$ when $\lambda =0.50,\theta =0.25$.

n | MLE | MPSP | Asy-CI | ||||||
---|---|---|---|---|---|---|---|---|---|

Avg. | MSE | Avg. | MSE | Lower | Upper | AIL | CP (%) | ||

$a=0.50$ and $b=0.50$ | |||||||||

100 | $\lambda $ | 0.6629 | 0.1951 | 0.6599 | 0.2084 | 0.0000 | 1.4682 | 1.4682 | 95.80 |

$\theta $ | 0.4114 | 0.0382 | 0.3850 | 0.0399 | 0.1952 | 0.6275 | 0.4323 | 96.70 | |

a | 0.5658 | 0.0083 | 0.5382 | 0.0052 | 0.4425 | 0.6891 | 0.2466 | 97.00 | |

b | 0.5980 | 0.0918 | 0.6370 | 0.3456 | 0.0358 | 1.1602 | 1.1244 | 96.60 | |

200 | $\lambda $ | 0.5922 | 0.0733 | 0.5939 | 0.0931 | 0.0930 | 1.0915 | 0.9985 | 94.80 |

$\theta $ | 0.4029 | 0.0300 | 0.3880 | 0.0278 | 0.2436 | 0.5623 | 0.3187 | 96.90 | |

a | 0.5760 | 0.0082 | 0.5618 | 0.0061 | 0.4804 | 0.6717 | 0.1913 | 97.50 | |

b | 0.6139 | 0.0599 | 0.6115 | 0.0705 | 0.1891 | 1.0388 | 0.8497 | 95.50 | |

$a=0.50$ and $b=0.75$ | |||||||||

100 | $\lambda $ | 0.7995 | 0.2320 | 0.7655 | 0.2944 | 0.0597 | 1.5393 | 1.4796 | 95.40 |

$\theta $ | 0.3798 | 0.0301 | 0.3414 | 0.0331 | 0.1540 | 0.6056 | 0.4515 | 95.20 | |

a | 0.5613 | 0.0074 | 0.5365 | 0.0046 | 0.4433 | 0.6793 | 0.2360 | 97.20 | |

b | 0.7402 | 0.1308 | 0.8397 | 0.5737 | 0.0311 | 1.4492 | 1.4181 | 97.30 | |

200 | $\lambda $ | 0.7841 | 0.1755 | 0.7697 | 0.2010 | 0.1803 | 1.3880 | 1.2077 | 96.10 |

$\theta $ | 0.3766 | 0.0238 | 0.3529 | 0.0224 | 0.2036 | 0.5496 | 0.3460 | 96.30 | |

a | 0.5658 | 0.0064 | 0.5529 | 0.0048 | 0.4753 | 0.6562 | 0.1808 | 95.70 | |

b | 0.7135 | 0.0713 | 0.7383 | 0.1529 | 0.1947 | 1.2323 | 1.0376 | 96.00 | |

$a=0.75$ and $b=0.50$ | |||||||||

100 | $\lambda $ | 0.4720 | 0.0605 | 0.4387 | 0.2247 | 0.0000 | 0.9526 | 0.9526 | 96.20 |

$\theta $ | 0.3255 | 0.0336 | 0.3335 | 0.0785 | 0.0000 | 0.6540 | 0.6540 | 96.80 | |

a | 0.8641 | 0.0276 | 0.8136 | 0.0160 | 0.6271 | 1.1012 | 0.4741 | 98.70 | |

b | 0.8028 | 0.2435 | 1.0501 | 2.3357 | 0.0367 | 1.5689 | 1.5322 | 96.20 | |

200 | $\lambda $ | 0.4495 | 0.0474 | 0.3067 | 0.0991 | 0.0333 | 0.8657 | 0.8325 | 97.10 |

$\theta $ | 0.3199 | 0.0267 | 0.3942 | 0.0921 | 0.0294 | 0.6104 | 0.5810 | 96.60 | |

a | 0.8921 | 0.0277 | 0.8650 | 0.0200 | 0.7220 | 1.0621 | 0.3401 | 98.90 | |

b | 0.8125 | 0.2128 | 1.2654 | 2.6261 | 0.1456 | 1.4795 | 1.3339 | 95.40 | |

$a=0.75$ and $b=0.75$ | |||||||||

100 | $\lambda $ | 0.5471 | 0.0587 | 0.4611 | 0.1206 | 0.0802 | 1.0141 | 0.9339 | 98.60 |

$\theta $ | 0.3169 | 0.0315 | 0.3026 | 0.0696 | 0.0000 | 0.6399 | 0.6399 | 95.40 | |

a | 0.8618 | 0.0265 | 0.8149 | 0.0162 | 0.6289 | 1.0947 | 0.4658 | 96.80 | |

b | 0.9678 | 0.2120 | 1.3863 | 4.5384 | 0.1709 | 1.7647 | 1.5937 | 97.20 | |

200 | $\lambda $ | 0.5292 | 0.0313 | 0.3797 | 0.0551 | 0.1862 | 0.8721 | 0.6859 | 95.50 |

$\theta $ | 0.2946 | 0.0252 | 0.3386 | 0.0752 | 0.0000 | 0.5937 | 0.5937 | 95.00 | |

a | 0.8920 | 0.0275 | 0.8661 | 0.0203 | 0.7236 | 1.0603 | 0.3367 | 98.60 | |

b | 0.9877 | 0.2839 | 1.9519 | 9.0478 | 0.0510 | 1.9245 | 1.8734 | 98.20 |

**Table 6.**Avg. estimated values, MSEs, Asy-CI, AILs and CP (in %) of Non-BE (MLE) for the MPSPr $BBE2$ distribution at different sample sizes n and different values of $(a,b)$ when $\lambda =0.50,\theta =0.50$.

n | MLE | MPSP | Asy-CI | ||||||
---|---|---|---|---|---|---|---|---|---|

Avg. | MSE | Avg. | MSE | Lower | Upper | AIL | CP (%) | ||

$a=0.50$ and $b=0.50$ | |||||||||

100 | $\lambda $ | 0.6544 | 0.1423 | 0.6864 | 0.2201 | 0.0000 | 1.3295 | 1.3295 | 95.10 |

$\theta $ | 0.4545 | 0.0162 | 0.4375 | 0.0270 | 0.2217 | 0.6872 | 0.4655 | 95.20 | |

a | 0.5817 | 0.0109 | 0.5499 | 0.0066 | 0.4537 | 0.7097 | 0.2560 | 97.50 | |

b | 0.5521 | 0.0771 | 0.5499 | 0.1646 | 0.0174 | 1.0869 | 1.0695 | 97.00 | |

200 | $\lambda $ | 0.6190 | 0.0790 | 0.6332 | 0.1054 | 0.1195 | 1.1184 | 0.9989 | 95.20 |

$\theta $ | 0.4509 | 0.0107 | 0.4406 | 0.0146 | 0.2720 | 0.6297 | 0.3577 | 96.70 | |

a | 0.5867 | 0.0099 | 0.5708 | 0.0073 | 0.4915 | 0.6820 | 0.1905 | 96.90 | |

b | 0.5440 | 0.0441 | 0.5347 | 0.0664 | 0.1415 | 0.9464 | 0.8049 | 95.70 | |

$a=0.50$ and $b=0.75$ | |||||||||

100 | $\lambda $ | 0.7933 | 0.2350 | 0.7753 | 0.2725 | 0.0364 | 1.5502 | 1.5139 | 95.90 |

$\theta $ | 0.4253 | 0.0198 | 0.4041 | 0.0357 | 0.1911 | 0.6594 | 0.4683 | 94.60 | |

a | 0.5774 | 0.0101 | 0.5489 | 0.0061 | 0.4521 | 0.7026 | 0.2505 | 96.20 | |

b | 0.6679 | 0.0789 | 0.7489 | 0.6881 | 0.1410 | 1.1948 | 1.0538 | 95.40 | |

200 | $\lambda $ | 0.7735 | 0.1568 | 0.7669 | 0.1753 | 0.2121 | 1.3350 | 1.1229 | 95.50 |

$\theta $ | 0.4165 | 0.0154 | 0.4043 | 0.0216 | 0.2369 | 0.5960 | 0.3591 | 95.60 | |

a | 0.5826 | 0.0093 | 0.5682 | 0.0070 | 0.4854 | 0.6798 | 0.1944 | 96.20 | |

b | 0.6460 | 0.0628 | 0.7268 | 1.3109 | 0.1987 | 1.0934 | 0.8946 | 95.60 | |

$a=0.75$ and $b=0.50$ | |||||||||

100 | $\lambda $ | 0.4447 | 0.0640 | 0.4102 | 0.2185 | 0.0000 | 0.9304 | 0.9304 | 94.70 |

$\theta $ | 0.3149 | 0.0638 | 0.3539 | 0.1009 | 0.0000 | 0.6532 | 0.6532 | 96.20 | |

a | 0.9530 | 0.0616 | 0.8937 | 0.0374 | 0.6723 | 1.2338 | 0.5615 | 95.50 | |

b | 0.7806 | 0.1991 | 0.9517 | 0.8384 | 0.0980 | 1.4632 | 1.3652 | 96.20 | |

200 | $\lambda $ | 0.3823 | 0.0439 | 0.3623 | 0.2497 | 0.0407 | 0.7240 | 0.6833 | 95.20 |

$\theta $ | 0.3579 | 0.0514 | 0.4617 | 0.0898 | 0.0102 | 0.7056 | 0.6953 | 97.10 | |

a | 0.9604 | 0.0540 | 0.9299 | 0.0431 | 0.7657 | 1.1551 | 0.3894 | 97.10 | |

b | 0.9183 | 0.3579 | 1.5568 | 6.7943 | 0.0759 | 1.7607 | 1.6848 | 97.10 | |

$a=0.75$ and $b=0.75$ | |||||||||

100 | $\lambda $ | 0.5287 | 0.0549 | 0.4691 | 0.1171 | 0.0719 | 0.9855 | 0.9136 | 95.80 |

$\theta $ | 0.3188 | 0.0304 | 0.2878 | 0.0581 | 0.0038 | 0.6338 | 0.6301 | 96.20 | |

a | 0.8490 | 0.0205 | 0.8065 | 0.0132 | 0.6457 | 1.0522 | 0.4065 | 97.20 | |

b | 0.9937 | 0.1904 | 1.2301 | 2.4442 | 0.2825 | 1.7048 | 1.4223 | 97.20 | |

200 | $\lambda $ | 0.5257 | 0.0405 | 0.3588 | 0.0671 | 0.1333 | 0.9181 | 0.7848 | 95.10 |

$\theta $ | 0.3012 | 0.0243 | 0.3547 | 0.0845 | 0.0117 | 0.5907 | 0.5790 | 96.20 | |

a | 0.8875 | 0.0260 | 0.8625 | 0.0194 | 0.7216 | 1.0533 | 0.3317 | 97.30 | |

b | 1.0043 | 0.2560 | 1.7036 | 6.4666 | 0.1448 | 1.8639 | 1.7192 | 98.40 |

**Table 7.**Avg. estimated values, MSEs, Asy-CI, AILs and CP (in %) of Non-BE (MLE and MPSP) for the $BBE2$ distribution at different sample sizes n and different values of $(a,b)$ when $\lambda =1.50,\theta =0.25$.

n | MLE | MPSP | Asy-CI | ||||||
---|---|---|---|---|---|---|---|---|---|

Avg. | MSE | Avg. | MSE | Lower | Upper | AIL | CP (%) | ||

$a=0.50$ and $b=0.50$ | |||||||||

100 | $\lambda $ | 1.7799 | 0.7489 | 1.7769 | 1.0536 | 0.1736 | 3.3861 | 3.2125 | 95.00 |

$\theta $ | 0.4144 | 0.0392 | 0.3839 | 0.0388 | 0.1985 | 0.6304 | 0.4319 | 95.60 | |

a | 0.5694 | 0.0080 | 0.5426 | 0.0048 | 0.4587 | 0.6801 | 0.2214 | 96.20 | |

b | 0.6304 | 0.0884 | 0.6892 | 0.6447 | 0.1064 | 1.1545 | 1.0481 | 96.50 | |

200 | $\lambda $ | 1.6477 | 0.3536 | 1.6560 | 0.4899 | 0.5180 | 2.7774 | 2.2593 | 95.80 |

$\theta $ | 0.4057 | 0.0315 | 0.3886 | 0.0287 | 0.2387 | 0.5726 | 0.3338 | 96.50 | |

a | 0.5816 | 0.0088 | 0.5676 | 0.0066 | 0.4911 | 0.6720 | 0.1809 | 97.00 | |

b | 0.6432 | 0.0675 | 0.6343 | 0.0764 | 0.2182 | 1.0683 | 0.8501 | 95.50 | |

$a=0.50$ and $b=0.75$ | |||||||||

100 | $\lambda $ | 1.8993 | 0.8923 | 1.7612 | 1.1563 | 0.2198 | 3.5787 | 3.3589 | 94.00 |

$\theta $ | 0.3611 | 0.0270 | 0.3165 | 0.0340 | 0.1231 | 0.5990 | 0.4758 | 93.10 | |

a | 0.5877 | 0.0107 | 0.5627 | 0.0067 | 0.4794 | 0.6960 | 0.2167 | 96.50 | |

b | 0.8848 | 0.1525 | 1.0652 | 1.7052 | 0.1658 | 1.6037 | 1.4379 | 96.70 | |

200 | $\lambda $ | 1.8545 | 0.5077 | 1.7559 | 0.6021 | 0.6422 | 3.0668 | 2.4247 | 96.30 |

$\theta $ | 0.3525 | 0.0202 | 0.3327 | 0.0254 | 0.1597 | 0.5453 | 0.3856 | 94.70 | |

a | 0.5999 | 0.0117 | 0.5867 | 0.0092 | 0.5182 | 0.6815 | 0.1633 | 96.80 | |

b | 0.8671 | 0.0944 | 0.8991 | 0.2018 | 0.3099 | 1.4243 | 1.1144 | 95.60 | |

$a=0.75$ and $b=0.50$ | |||||||||

100 | $\lambda $ | 1.2553 | 0.2930 | 1.2813 | 2.8457 | 0.3053 | 2.2054 | 1.9001 | 96.10 |

$\theta $ | 0.3443 | 0.0746 | 0.3157 | 0.0662 | 0.0000 | 0.8488 | 0.8488 | 99.20 | |

a | 0.8580 | 0.0259 | 0.8080 | 0.0155 | 0.6232 | 1.0927 | 0.4695 | 98.40 | |

b | 0.8357 | 0.2413 | 0.9190 | 0.7613 | 0.1300 | 1.5413 | 1.4113 | 96.10 | |

200 | $\lambda $ | 1.2239 | 0.2137 | 0.9158 | 0.9213 | 0.4938 | 1.9539 | 1.4601 | 97.30 |

$\theta $ | 0.3127 | 0.0276 | 0.3944 | 0.1111 | 0.0096 | 0.6159 | 0.6063 | 98.20 | |

a | 0.9045 | 0.0319 | 0.8972 | 0.0962 | 0.7280 | 1.0811 | 0.3531 | 99.10 | |

b | 0.8640 | 0.3663 | 1.1613 | 2.6713 | 0.0000 | 1.8160 | 1.8160 | 97.30 | |

$a=0.75$ and $b=0.75$ | |||||||||

100 | $\lambda $ | 1.4147 | 0.1615 | 1.2993 | 0.4753 | 0.6421 | 2.1872 | 1.5451 | 97.80 |

$\theta $ | 0.3186 | 0.0276 | 0.2755 | 0.0480 | 0.0213 | 0.6160 | 0.5946 | 94.80 | |

a | 0.8478 | 0.0226 | 0.8056 | 0.0146 | 0.6233 | 1.0723 | 0.4490 | 97.80 | |

b | 1.0633 | 0.2446 | 1.0911 | 0.3475 | 0.3104 | 1.8162 | 1.5058 | 94.80 | |

200 | $\lambda $ | 1.3527 | 0.1320 | 1.1632 | 0.8639 | 0.6995 | 2.0059 | 1.3064 | 96.60 |

$\theta $ | 0.2874 | 0.0204 | 0.3059 | 0.0629 | 0.0161 | 0.5586 | 0.5425 | 97.30 | |

a | 0.8744 | 0.0220 | 0.8507 | 0.0167 | 0.7158 | 1.0329 | 0.3171 | 97.90 | |

b | 1.0731 | 0.1735 | 1.4224 | 4.6088 | 0.5563 | 1.5900 | 1.0337 | 95.90 |

**Table 8.**Avg. estimated values, MSEs, Asy-CI, AILs and CP (in %) of Non-BE (MLE and MPSP) for the $BBE2$ distribution at different sample sizes n and different values of $(a,b)$ when $\lambda =1.5,\theta =0.50$.

n | MLE | MPSP | Asy-CI | ||||||
---|---|---|---|---|---|---|---|---|---|

Avg. | MSE | Avg. | MSE | Lower | Upper | AIL | CP (%) | ||

$a=0.50$ and $b=0.50$ | |||||||||

100 | $\lambda $ | 1.7500 | 0.5951 | 1.8153 | 0.9300 | 0.3187 | 3.1813 | 2.8626 | 94.80 |

$\theta $ | 0.4586 | 0.0177 | 0.4411 | 0.0305 | 0.2107 | 0.7064 | 0.4956 | 94.90 | |

a | 0.5835 | 0.0109 | 0.5516 | 0.0063 | 0.4603 | 0.7067 | 0.2464 | 97.20 | |

b | 0.5822 | 0.0622 | 0.5949 | 0.4472 | 0.1204 | 1.0440 | 0.9236 | 95.60 | |

200 | $\lambda $ | 1.7314 | 0.4019 | 1.8013 | 0.5867 | 0.5739 | 2.8889 | 2.3150 | 95.40 |

$\theta $ | 0.4491 | 0.0102 | 0.4349 | 0.0126 | 0.2779 | 0.6203 | 0.3424 | 96.10 | |

a | 0.5906 | 0.0107 | 0.5746 | 0.0079 | 0.4924 | 0.6889 | 0.1965 | 97.20 | |

b | 0.5643 | 0.0416 | 0.5403 | 0.0452 | 0.1846 | 0.9440 | 0.7594 | 95.60 | |

$a=0.50$ and $b=0.75$ | |||||||||

100 | $\lambda $ | 1.8728 | 0.6567 | 1.7872 | 1.0015 | 0.4613 | 3.2843 | 2.8230 | 95.00 |

$\theta $ | 0.3899 | 0.0256 | 0.3529 | 0.0496 | 0.1626 | 0.6171 | 0.4546 | 94.20 | |

a | 0.6050 | 0.0155 | 0.5777 | 0.0100 | 0.4730 | 0.7369 | 0.2639 | 96.90 | |

b | 0.8055 | 0.1052 | 0.8585 | 0.2975 | 0.1787 | 1.4322 | 1.2536 | 95.50 | |

200 | $\lambda $ | 1.8906 | 0.5142 | 1.8276 | 0.6019 | 0.7111 | 3.0700 | 2.3590 | 96.70 |

$\theta $ | 0.3899 | 0.0217 | 0.3766 | 0.0320 | 0.1979 | 0.5819 | 0.3840 | 94.30 | |

a | 0.6078 | 0.0138 | 0.5935 | 0.0108 | 0.5169 | 0.6986 | 0.1817 | 96.80 | |

b | 0.7688 | 0.0618 | 0.8135 | 0.6507 | 0.2827 | 1.2550 | 0.9724 | 95.30 | |

$a=0.75$ and $b=0.50$ | |||||||||

100 | $\lambda $ | 1.3773 | 0.2912 | 1.3769 | 1.4309 | 0.3428 | 2.4119 | 2.0690 | 96.50 |

$\theta $ | 0.3389 | 0.0733 | 0.3603 | 0.1203 | 0.0000 | 0.7673 | 0.7673 | 93.90 | |

a | 0.9531 | 0.0653 | 0.9171 | 0.1759 | 0.6474 | 1.2587 | 0.6113 | 97.40 | |

b | 0.8073 | 0.4344 | 0.8034 | 2.2071 | 0.0000 | 1.9551 | 1.9551 | 94.70 | |

200 | $\lambda $ | 1.2145 | 0.2265 | 0.8009 | 0.8269 | 0.4630 | 1.9659 | 1.5029 | 97.60 |

$\theta $ | 0.3300 | 0.0692 | 0.4241 | 0.1110 | 0.0000 | 0.7264 | 0.7264 | 97.30 | |

a | 0.9534 | 0.0508 | 0.9202 | 0.0383 | 0.7622 | 1.1447 | 0.3825 | 97.30 | |

b | 0.8614 | 0.4029 | 1.2763 | 4.4615 | 0.0000 | 1.8912 | 1.8912 | 94.60 | |

$a=0.75$ and $b=0.75$ | |||||||||

100 | $\lambda $ | 1.3441 | 0.1501 | 1.4643 | 3.0728 | 0.6448 | 2.0434 | 1.3986 | 96.50 |

$\theta $ | 0.3433 | 0.0897 | 0.3196 | 0.1223 | 0.0000 | 0.8464 | 0.8464 | 97.70 | |

a | 0.9316 | 0.0510 | 0.8796 | 0.0350 | 0.6666 | 1.1966 | 0.5300 | 96.50 | |

b | 1.0458 | 0.2340 | 1.0464 | 0.3567 | 0.2913 | 1.8004 | 1.5090 | 95.30 | |

200 | $\lambda $ | 1.2832 | 0.1969 | 1.0449 | 0.8917 | 0.5199 | 2.0466 | 1.5267 | 96.50 |

$\theta $ | 0.3119 | 0.0681 | 0.3704 | 0.1266 | 0.0000 | 0.6687 | 0.6687 | 95.30 | |

a | 0.9621 | 0.0529 | 0.9304 | 0.0409 | 0.7866 | 1.1377 | 0.3510 | 97.70 | |

b | 1.1514 | 0.5048 | 1.9029 | 0.1950 | 0.0000 | 2.3071 | 2.3071 | 94.20 |

**Table 9.**Avg. estimated values and MSEs of the BE using MCMC for the $BBE2$ distribution at different sample sizes n and different values of $(a,b)$ when $\lambda =0.50,\theta =0.25$.

n | BE: SEL | BE:LINEX | BE: GE | HPD | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Avg. | MSE | Avg. | MSE | Avg. | MSE | Lower | Upper | AIL | CP (%) | ||

$a=0.50$ and $b=0.50$ | |||||||||||

100 | $\lambda $ | 0.6513 | 0.1480 | 0.6469 | 0.1428 | 0.6356 | 0.1376 | 0.1306 | 1.3829 | 1.2523 | 95.30 |

$\theta $ | 0.4091 | 0.0570 | 0.4069 | 0.0557 | 0.3944 | 0.0511 | 0.0881 | 0.7280 | 0.6399 | 96.40 | |

a | 0.5563 | 0.0091 | 0.5551 | 0.0089 | 0.5503 | 0.0084 | 0.4209 | 0.7235 | 0.3027 | 98.10 | |

b | 0.6258 | 0.1299 | 0.6218 | 0.1252 | 0.6114 | 0.1205 | 0.1866 | 1.3006 | 1.1139 | 96.20 | |

200 | $\lambda $ | 0.6024 | 0.1001 | 0.5991 | 0.0972 | 0.5891 | 0.0937 | 0.1458 | 1.1631 | 1.0173 | 95.00 |

$\theta $ | 0.4108 | 0.0507 | 0.4087 | 0.0496 | 0.3973 | 0.0457 | 0.1547 | 0.7023 | 0.5476 | 97.00 | |

a | 0.5647 | 0.0071 | 0.5640 | 0.0070 | 0.5611 | 0.0067 | 0.4672 | 0.6780 | 0.2109 | 98.50 | |

b | 0.6530 | 0.1095 | 0.6488 | 0.1054 | 0.6381 | 0.0997 | 0.1881 | 1.2569 | 1.0688 | 95.50 | |

$a=0.50$ and $b=0.75$ | |||||||||||

100 | $\lambda $ | 0.8091 | 0.2685 | 0.8024 | 0.2591 | 0.7887 | 0.2494 | 0.1823 | 1.5890 | 1.4068 | 95.70 |

$\theta $ | 0.3810 | 0.0445 | 0.3791 | 0.0435 | 0.3677 | 0.0400 | 0.0772 | 0.6911 | 0.6140 | 96.00 | |

a | 0.5455 | 0.0066 | 0.5445 | 0.0065 | 0.5399 | 0.0062 | 0.4125 | 0.6696 | 0.2571 | 96.40 | |

b | 0.7605 | 0.1464 | 0.7543 | 0.1407 | 0.7416 | 0.1380 | 0.2130 | 1.5316 | 1.3186 | 95.80 | |

200 | $\lambda $ | 0.7649 | 0.1888 | 0.7597 | 0.1828 | 0.7474 | 0.1750 | 0.2591 | 1.4660 | 1.2069 | 95.70 |

$\theta $ | 0.3754 | 0.0370 | 0.3737 | 0.0362 | 0.3628 | 0.0331 | 0.1190 | 0.6343 | 0.5153 | 96.30 | |

a | 0.5617 | 0.0063 | 0.5611 | 0.0062 | 0.5585 | 0.0059 | 0.4738 | 0.6615 | 0.1877 | 97.90 | |

b | 0.7692 | 0.0992 | 0.7644 | 0.0966 | 0.7531 | 0.0950 | 0.2360 | 1.3608 | 1.1249 | 95.40 | |

$a=0.75$ and $b=0.50$ | |||||||||||

100 | $\lambda $ | 0.4809 | 0.0563 | 0.4787 | 0.0553 | 0.4702 | 0.0543 | 0.1399 | 1.0081 | 0.8682 | 96.00 |

$\theta $ | 0.3096 | 0.0443 | 0.3082 | 0.0436 | 0.2989 | 0.0413 | 0.0009 | 0.6334 | 0.6325 | 95.50 | |

a | 0.8275 | 0.0244 | 0.8249 | 0.0239 | 0.8182 | 0.0229 | 0.5893 | 1.0826 | 0.4934 | 97.20 | |

b | 0.7806 | 0.2892 | 0.7751 | 0.2795 | 0.7640 | 0.2732 | 0.2631 | 1.4980 | 1.2349 | 96.60 | |

200 | $\lambda $ | 0.4979 | 0.0765 | 0.4954 | 0.0736 | 0.4877 | 0.0713 | 0.1726 | 1.0339 | 0.8613 | 96.30 |

$\theta $ | 0.3123 | 0.0361 | 0.3110 | 0.0355 | 0.3019 | 0.0336 | 0.0695 | 0.6814 | 0.6119 | 97.80 | |

a | 0.8837 | 0.0270 | 0.8818 | 0.0265 | 0.8775 | 0.0256 | 0.7195 | 1.0529 | 0.3333 | 97.80 | |

b | 0.7820 | 0.2022 | 0.7772 | 0.1966 | 0.7668 | 0.1894 | 0.2254 | 1.3674 | 1.1419 | 96.30 | |

$a=0.75$ and $b=0.75$ | |||||||||||

100 | $\lambda $ | 0.5980 | 0.1115 | 0.5944 | 0.1078 | 0.5832 | 0.1042 | 0.2021 | 1.2069 | 1.0048 | 96.70 |

$\theta $ | 0.3263 | 0.0389 | 0.3245 | 0.0380 | 0.3132 | 0.0348 | 0.0150 | 0.6615 | 0.6464 | 96.70 | |

a | 0.8225 | 0.0163 | 0.8201 | 0.0158 | 0.8140 | 0.0149 | 0.6520 | 1.0320 | 0.3800 | 97.20 | |

b | 0.9579 | 0.2684 | 0.9493 | 0.2557 | 0.9355 | 0.2512 | 0.2655 | 1.8190 | 1.5534 | 96.10 | |

200 | $\lambda $ | 0.5567 | 0.0635 | 0.5540 | 0.0618 | 0.5450 | 0.0594 | 0.2204 | 1.1121 | 0.8916 | 95.90 |

$\theta $ | 0.3086 | 0.0313 | 0.3071 | 0.0306 | 0.2969 | 0.0283 | 0.0629 | 0.6873 | 0.6244 | 98.50 | |

a | 0.8658 | 0.0220 | 0.8642 | 0.0216 | 0.8602 | 0.0207 | 0.7201 | 1.0532 | 0.3332 | 99.50 | |

b | 0.9864 | 0.2140 | 0.9785 | 0.2045 | 0.9663 | 0.1975 | 0.3605 | 1.8260 | 1.4655 | 96.40 |

**Table 10.**Avg. estimated values and MSEs of the BE using MCMC for the $BBE2$ distribution at different sample sizes n and different values of $(a,b)$ when $\lambda =0.50,\theta =0.50$.

n | BE: SEL | BE:LINEX | BE: GE | HPD | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Avg. | MSE | Avg. | MSE | Avg. | MSE | Lower | Upper | AIL | CP (%) | ||

$a=0.50$ and $b=0.50$ | |||||||||||

100 | $\lambda $ | 0.6658 | 0.1536 | 0.6613 | 0.1484 | 0.6497 | 0.1426 | 0.1455 | 1.3650 | 1.2195 | 95.20 |

$\theta $ | 0.4548 | 0.0393 | 0.4523 | 0.0388 | 0.4397 | 0.0395 | 0.1388 | 0.7979 | 0.6591 | 95.80 | |

a | 0.5672 | 0.0108 | 0.5660 | 0.0106 | 0.5607 | 0.0099 | 0.4224 | 0.7208 | 0.2983 | 97.30 | |

b | 0.5675 | 0.1006 | 0.5641 | 0.0976 | 0.5539 | 0.0940 | 0.1314 | 1.1523 | 1.0209 | 95.20 | |

200 | $\lambda $ | 0.6356 | 0.0959 | 0.6320 | 0.0927 | 0.6214 | 0.0882 | 0.1976 | 1.1913 | 0.9937 | 95.60 |

$\theta $ | 0.4348 | 0.0303 | 0.4324 | 0.0298 | 0.4200 | 0.0308 | 0.1497 | 0.6999 | 0.5503 | 95.70 | |

a | 0.5822 | 0.0102 | 0.5815 | 0.0101 | 0.5784 | 0.0096 | 0.4830 | 0.7011 | 0.2181 | 97.50 | |

b | 0.5522 | 0.0614 | 0.5497 | 0.0600 | 0.5408 | 0.0581 | 0.1729 | 1.0280 | 0.8551 | 95.40 | |

$a=0.50$ and $b=0.75$ | |||||||||||

100 | $\lambda $ | 0.7804 | 0.2526 | 0.7745 | 0.2439 | 0.7618 | 0.2360 | 0.1986 | 1.5670 | 1.3684 | 96.00 |

$\theta $ | 0.4173 | 0.0448 | 0.4150 | 0.0444 | 0.4026 | 0.0459 | 0.1080 | 0.7519 | 0.6438 | 95.50 | |

a | 0.5639 | 0.0106 | 0.5628 | 0.0105 | 0.5580 | 0.0099 | 0.4138 | 0.6983 | 0.2845 | 96.70 | |

b | 0.7286 | 0.1998 | 0.7224 | 0.1833 | 0.7112 | 0.1881 | 0.1863 | 1.3979 | 1.2116 | 95.10 | |

200 | $\lambda $ | 0.7554 | 0.1803 | 0.7502 | 0.1744 | 0.7381 | 0.1670 | 0.2744 | 1.4806 | 1.2063 | 96.30 |

$\theta $ | 0.4145 | 0.0317 | 0.4126 | 0.0317 | 0.4015 | 0.0335 | 0.1361 | 0.6977 | 0.5617 | 96.20 | |

a | 0.5769 | 0.0089 | 0.5763 | 0.0088 | 0.5735 | 0.0084 | 0.4798 | 0.6892 | 0.2094 | 98.20 | |

b | 0.7129 | 0.0970 | 0.7082 | 0.0944 | 0.6966 | 0.0935 | 0.2340 | 1.3315 | 1.0975 | 95.60 | |

$a=0.75$ and $b=0.50$ | |||||||||||

100 | $\lambda $ | 0.5024 | 0.0801 | 0.4995 | 0.0779 | 0.4901 | 0.0750 | 0.1226 | 1.2241 | 1.1016 | 95.80 |

$\theta $ | 0.3734 | 0.0535 | 0.3715 | 0.0534 | 0.3602 | 0.0552 | 0.0416 | 0.7058 | 0.6642 | 95.80 | |

a | 0.9352 | 0.0625 | 0.9315 | 0.0605 | 0.9238 | 0.0574 | 0.6001 | 1.2427 | 0.6427 | 96.70 | |

b | 0.7618 | 0.2500 | 0.7562 | 0.2419 | 0.7442 | 0.2347 | 0.1930 | 1.8617 | 1.6687 | 95.80 | |

200 | $\lambda $ | 0.4132 | 0.0494 | 0.4113 | 0.0487 | 0.4033 | 0.0483 | 0.0947 | 0.8132 | 0.7185 | 96.00 |

$\theta $ | 0.3463 | 0.0653 | 0.3444 | 0.0650 | 0.3341 | 0.0665 | 0.0498 | 0.7401 | 0.6903 | 96.00 | |

a | 0.9399 | 0.0505 | 0.9378 | 0.0497 | 0.9330 | 0.0482 | 0.7321 | 1.1718 | 0.4397 | 99.00 | |

b | 0.8996 | 0.3778 | 0.8915 | 0.3640 | 0.8777 | 0.3520 | 0.2745 | 1.7283 | 1.4538 | 96.00 | |

$a=0.75$ and $b=0.75$ | |||||||||||

100 | $\lambda $ | 0.5441 | 0.0594 | 0.5412 | 0.0575 | 0.5311 | 0.0549 | 0.1428 | 0.9901 | 0.8473 | 95.40 |

$\theta $ | 0.3335 | 0.0440 | 0.3315 | 0.0426 | 0.3207 | 0.0393 | 0.0000 | 0.6359 | 0.6359 | 95.40 | |

a | 0.8283 | 0.0204 | 0.8259 | 0.0198 | 0.8200 | 0.0188 | 0.6028 | 1.0679 | 0.4650 | 97.40 | |

b | 1.0214 | 0.2856 | 1.0115 | 0.2702 | 0.9981 | 0.2618 | 0.3154 | 1.9135 | 1.5981 | 95.40 | |

200 | $\lambda $ | 0.5788 | 0.0725 | 0.5761 | 0.0708 | 0.5671 | 0.0682 | 0.2098 | 1.0824 | 0.8726 | 95.30 |

$\theta $ | 0.3227 | 0.0329 | 0.3210 | 0.0322 | 0.3096 | 0.0295 | 0.0157 | 0.6086 | 0.5929 | 96.30 | |

a | 0.8720 | 0.0242 | 0.8704 | 0.0237 | 0.8664 | 0.0229 | 0.7249 | 1.0765 | 0.3516 | 99.50 | |

b | 0.9800 | 0.2226 | 0.9724 | 0.2144 | 0.9599 | 0.2078 | 0.3522 | 1.8339 | 1.4817 | 96.70 |

**Table 11.**Avg. estimated values and MSEs of the BE using MCMC for the $BBE2$ distribution at different sample sizes n and different values of $(a,b)$ when $\lambda =1.50,\theta =0.25$.

n | BE: SEL | BE:LINEX | BE: GE | HPD | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Avg. | MSE | Avg. | MSE | Avg. | MSE | Lower | Upper | AIL | CP (%) | ||

$a=0.50$ and $b=0.50$ | |||||||||||

100 | $\lambda $ | 1.5865 | 0.5295 | 1.5627 | 0.4923 | 1.5485 | 0.4999 | 0.4954 | 3.0986 | 2.6033 | 95.50 |

$\theta $ | 0.4091 | 0.0563 | 0.4070 | 0.0551 | 0.3949 | 0.0509 | 0.1118 | 0.7391 | 0.6273 | 96.30 | |

a | 0.5625 | 0.0088 | 0.5614 | 0.0086 | 0.5566 | 0.0081 | 0.4457 | 0.7099 | 0.2642 | 98.50 | |

b | 0.7110 | 0.1426 | 0.7060 | 0.1372 | 0.6944 | 0.1302 | 0.2133 | 1.2958 | 1.0825 | 95.50 | |

200 | $\lambda $ | 1.5255 | 0.3875 | 1.5053 | 0.3671 | 1.4903 | 0.3725 | 0.4860 | 2.6242 | 2.1381 | 95.20 |

$\theta $ | 0.4014 | 0.0446 | 0.3994 | 0.0437 | 0.3877 | 0.0399 | 0.1672 | 0.6924 | 0.5252 | 97.20 | |

a | 0.5743 | 0.0081 | 0.5736 | 0.0080 | 0.5708 | 0.0076 | 0.4806 | 0.6745 | 0.1938 | 97.70 | |

b | 0.7281 | 0.1339 | 0.7233 | 0.1291 | 0.7113 | 0.1219 | 0.2825 | 1.3011 | 1.0186 | 96.10 | |

$a=0.50$ and $b=0.75$ | |||||||||||

100 | $\lambda $ | 1.6533 | 0.5052 | 1.6292 | 0.4702 | 1.6155 | 0.4742 | 0.5608 | 3.0550 | 2.4942 | 95.40 |

$\theta $ | 0.3718 | 0.0430 | 0.3700 | 0.0421 | 0.3594 | 0.0389 | 0.0806 | 0.7218 | 0.6412 | 96.40 | |

a | 0.5750 | 0.0096 | 0.5739 | 0.0094 | 0.5694 | 0.0088 | 0.4659 | 0.7065 | 0.2406 | 97.20 | |

b | 1.0035 | 0.2142 | 0.9943 | 0.2033 | 0.9803 | 0.1944 | 0.3801 | 1.7651 | 1.3851 | 95.60 | |

200 | $\lambda $ | 1.7344 | 0.4795 | 1.7104 | 0.4493 | 1.6962 | 0.4504 | 0.7395 | 3.1014 | 2.3618 | 96.40 |

$\theta $ | 0.3463 | 0.0277 | 0.3448 | 0.0272 | 0.3342 | 0.0249 | 0.1245 | 0.5995 | 0.4750 | 96.70 | |

a | 0.5889 | 0.0101 | 0.5882 | 0.0099 | 0.5857 | 0.0095 | 0.4970 | 0.6762 | 0.1792 | 96.30 | |

b | 0.9472 | 0.1602 | 0.9395 | 0.1523 | 0.9264 | 0.1455 | 0.3824 | 1.5918 | 1.2094 | 95.70 | |

$a=0.75$ and $b=0.50$ | |||||||||||

100 | $\lambda $ | 1.2376 | 0.2899 | 1.2244 | 0.2897 | 1.2082 | 0.3005 | 0.4985 | 2.2758 | 1.7773 | 97.40 |

$\theta $ | 0.3196 | 0.0475 | 0.3179 | 0.0468 | 0.3071 | 0.0444 | 0.0025 | 0.7014 | 0.6989 | 95.70 | |

a | 0.8419 | 0.0269 | 0.8391 | 0.0262 | 0.8323 | 0.0249 | 0.6153 | 1.0881 | 0.4728 | 97.40 | |

b | 0.8761 | 0.3155 | 0.8659 | 0.2947 | 0.8507 | 0.2790 | 0.3109 | 1.5729 | 1.2620 | 95.70 | |

200 | $\lambda $ | 1.2803 | 0.3823 | 1.2648 | 0.3662 | 1.2518 | 0.3740 | 0.4053 | 2.2012 | 1.7959 | 95.20 |

$\theta $ | 0.3266 | 0.0370 | 0.3252 | 0.0364 | 0.3152 | 0.0345 | 0.0627 | 0.6774 | 0.6146 | 96.40 | |

a | 0.8934 | 0.0295 | 0.8916 | 0.0289 | 0.8876 | 0.0278 | 0.7496 | 1.0709 | 0.3214 | 97.60 | |

b | 0.8990 | 0.3388 | 0.8913 | 0.3264 | 0.8780 | 0.3145 | 0.3484 | 1.8689 | 1.5206 | 97.60 | |

$a=0.75$ and $b=0.75$ | |||||||||||

100 | $\lambda $ | 1.3833 | 0.2552 | 1.3668 | 0.2485 | 1.3504 | 0.2554 | 0.5322 | 2.2118 | 1.6796 | 96.70 |

$\theta $ | 0.2915 | 0.0361 | 0.2903 | 0.0357 | 0.2819 | 0.0337 | 0.0010 | 0.6494 | 0.6484 | 95.40 | |

a | 0.8479 | 0.0229 | 0.8454 | 0.0222 | 0.8393 | 0.0209 | 0.6459 | 1.0501 | 0.4042 | 96.70 | |

b | 1.1431 | 0.3846 | 1.1285 | 0.3538 | 1.1130 | 0.3411 | 0.4450 | 2.1598 | 1.7147 | 96.10 | |

200 | $\lambda $ | 1.3792 | 0.3270 | 1.3640 | 0.3136 | 1.3512 | 0.3200 | 0.5131 | 2.3935 | 1.8804 | 95.30 |

$\theta $ | 0.3530 | 0.0481 | 0.3513 | 0.0473 | 0.3406 | 0.0444 | 0.0075 | 0.6698 | 0.6623 | 95.30 | |

a | 0.8574 | 0.0223 | 0.8559 | 0.0220 | 0.8518 | 0.0215 | 0.6812 | 1.0437 | 0.3625 | 97.70 | |

b | 1.2064 | 0.4315 | 1.1932 | 0.4067 | 1.1793 | 0.3924 | 0.4454 | 2.1931 | 1.7477 | 96.10 |

**Table 12.**Avg. estimated values and MSEs of the BE using MCMC for the $BBE2$ distribution at different sample sizes n and different values of $(a,b)$ when $\lambda =1.50,\theta =0.50$.

n | BE: SEL | BE:LINEX | BE: GE | HPD | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Avg. | MSE | Avg. | MSE | Avg. | MSE | Lower | Upper | AIL | CP (%) | ||

$a=0.50$ and $b=0.50$ | |||||||||||

100 | $\lambda $ | 1.5949 | 0.4241 | 1.5707 | 0.3963 | 1.5552 | 0.3990 | 0.4817 | 2.8664 | 2.3847 | 95.60 |

$\theta $ | 0.4526 | 0.0418 | 0.4500 | 0.0412 | 0.4370 | 0.0416 | 0.1209 | 0.7990 | 0.6781 | 95.20 | |

a | 0.5628 | 0.0096 | 0.5616 | 0.0094 | 0.5566 | 0.0089 | 0.4147 | 0.7011 | 0.2864 | 96.40 | |

b | 0.6333 | 0.0935 | 0.6293 | 0.0905 | 0.6176 | 0.0857 | 0.1976 | 1.1518 | 0.9541 | 95.20 | |

200 | $\lambda $ | 1.6267 | 0.4213 | 1.6063 | 0.3994 | 1.5925 | 0.4020 | 0.6017 | 2.8506 | 2.2489 | 96.10 |

$\theta $ | 0.4479 | 0.0295 | 0.4456 | 0.0292 | 0.4338 | 0.0302 | 0.1657 | 0.7233 | 0.5576 | 96.00 | |

a | 0.5802 | 0.0100 | 0.5795 | 0.0099 | 0.5765 | 0.0095 | 0.4659 | 0.6950 | 0.2291 | 97.50 | |

b | 0.6299 | 0.0917 | 0.6261 | 0.0882 | 0.6159 | 0.0835 | 0.2437 | 1.1356 | 0.8919 | 96.00 | |

$a=0.50$ and $b=0.75$ | |||||||||||

100 | $\lambda $ | 1.6810 | 0.5096 | 1.6553 | 0.4738 | 1.6399 | 0.4789 | 0.4820 | 3.0446 | 2.5626 | 95.00 |

$\theta $ | 0.3865 | 0.0405 | 0.3844 | 0.0402 | 0.3728 | 0.0419 | 0.1189 | 0.6796 | 0.5608 | 96.20 | |

a | 0.5850 | 0.0118 | 0.5839 | 0.0115 | 0.5793 | 0.0107 | 0.4692 | 0.7145 | 0.2453 | 97.40 | |

b | 0.8896 | 0.1460 | 0.8813 | 0.1381 | 0.8668 | 0.1312 | 0.3052 | 1.6220 | 1.3168 | 95.50 | |

200 | $\lambda $ | 1.7339 | 0.4436 | 1.7104 | 0.4134 | 1.6970 | 0.4136 | 0.7003 | 3.0303 | 2.3299 | 96.20 |

$\theta $ | 0.3927 | 0.0329 | 0.3908 | 0.0331 | 0.3794 | 0.0354 | 0.1379 | 0.6684 | 0.5305 | 96.50 | |

a | 0.5989 | 0.0126 | 0.5982 | 0.0125 | 0.5954 | 0.0119 | 0.4832 | 0.6917 | 0.2084 | 95.60 | |

b | 0.8693 | 0.1271 | 0.8624 | 0.1206 | 0.8498 | 0.1154 | 0.3559 | 1.5547 | 1.1988 | 96.00 | |

$a=0.75$ and $b=0.50$ | |||||||||||

100 | $\lambda $ | 1.2899 | 0.3308 | 1.2755 | 0.3247 | 1.2611 | 0.3314 | 0.5131 | 2.4233 | 1.9102 | 96.90 |

$\theta $ | 0.3458 | 0.0728 | 0.3439 | 0.0726 | 0.3329 | 0.0745 | 0.0192 | 0.7276 | 0.7085 | 96.90 | |

a | 0.9204 | 0.0514 | 0.9167 | 0.0499 | 0.9084 | 0.0474 | 0.5942 | 1.1140 | 0.5198 | 97.90 | |

b | 0.8444 | 0.3324 | 0.8382 | 0.3233 | 0.8258 | 0.3142 | 0.2734 | 1.7783 | 1.5050 | 95.80 | |

200 | $\lambda $ | 1.1301 | 0.3316 | 1.1212 | 0.3342 | 1.1080 | 0.3452 | 0.3093 | 1.9100 | 1.6007 | 95.50 |

$\theta $ | 0.3039 | 0.0734 | 0.3030 | 0.0735 | 0.2957 | 0.0762 | 0.0598 | 0.7096 | 0.6498 | 95.50 | |

a | 0.9675 | 0.0570 | 0.9654 | 0.0561 | 0.9612 | 0.0543 | 0.7966 | 1.1437 | 0.3471 | 98.50 | |

b | 0.9585 | 0.5883 | 0.9451 | 0.5285 | 0.9340 | 0.5299 | 0.3774 | 2.0226 | 1.6452 | 97.00 | |

$a=0.75$ and $b=0.75$ | |||||||||||

100 | $\lambda $ | 1.3484 | 0.2612 | 1.3342 | 0.2551 | 1.3196 | 0.2616 | 0.4972 | 2.0795 | 1.5823 | 95.10 |

$\theta $ | 0.3393 | 0.1481 | 0.3378 | 0.1469 | 0.3292 | 0.1488 | 0.0014 | 0.7717 | 0.7703 | 95.10 | |

a | 0.9262 | 0.0872 | 0.9228 | 0.0852 | 0.9157 | 0.0831 | 0.6704 | 1.1831 | 0.5127 | 98.10 | |

b | 1.0871 | 0.4269 | 1.0753 | 0.3921 | 1.0629 | 0.3931 | 0.3857 | 1.8090 | 1.4234 | 95.10 | |

200 | $\lambda $ | 1.1800 | 0.2372 | 1.1686 | 0.2372 | 1.1548 | 0.2447 | 0.5466 | 1.9582 | 1.4116 | 96.00 |

$\theta $ | 0.2716 | 0.0908 | 0.2704 | 0.0907 | 0.2625 | 0.0930 | 0.0099 | 0.6410 | 0.6311 | 96.00 | |

a | 0.9434 | 0.0462 | 0.9416 | 0.0455 | 0.9377 | 0.0442 | 0.7688 | 1.1019 | 0.3330 | 98.70 | |

b | 1.2299 | 0.4617 | 1.2179 | 0.4424 | 1.2035 | 0.4292 | 0.5190 | 2.2479 | 1.7289 | 96.00 |

**Table 13.**Failure time data of Stress-Rupture Life of Kevlar 49/Epoxy Strands with pressure at $90\%$ Data.

0.01 | 0.01 | 0.02 | 0.02 | 0.02 | 0.03 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.07 | 0.08 | 0.09 | 0.09 |

0.10 | 0.10 | 0.11 | 0.11 | 0.12 | 0.13 | 0.18 | 0.19 | 0.20 | 0.23 | 0.24 | 0.24 | 0.29 | 0.34 | 0.35 |

0.36 | 0.38 | 0.40 | 0.42 | 0.43 | 0.52 | 0.54 | 0.56 | 0.60 | 0.60 | 0.63 | 0.65 | 0.67 | 0.68 | 0.72 |

0.72 | 0.72 | 0.73 | 0.79 | 0.79 | 0.80 | 0.80 | 0.83 | 0.85 | 0.90 | 0.92 | 0.95 | 0.99 | 1.00 | 1.01 |

1.02 | 1.03 | 1.05 | 1.10 | 1.10 | 1.11 | 1.15 | 1.18 | 1.20 | 1.29 | 1.31 | 1.33 | 1.34 | 1.40 | 1.43 |

1.45 | 1.50 | 1.51 | 1.52 | 1.53 | 1.54 | 1.54 | 1.55 | 1.58 | 1.60 | 1.63 | 1.64 | 1.8 | 1.8 | 1.81 |

2.02 | 2.05 | 2.14 | 2.17 | 2.33 | 3.03 | 3.03 | 3.34 | 4.20 | 4.69 | 7.89 |

${\mathit{Q}}_{1}$ | Median | ${\mathit{Q}}_{3}$ | Mean | Variance | Kurtosis | Skewness |
---|---|---|---|---|---|---|

0.2400 | 0.800 | 1.4500 | 1.0249 | 1.2530 | 14.4745 | 3.0472 |

**Table 15.**Summary of fitted distributions corresponding to Stress-Rupture life of Kevlar 49/Epoxy Strands data.

Distribution | Estimates | Measures | |||||||
---|---|---|---|---|---|---|---|---|---|

$-\mathbf{2}\mathit{logL}$ | $\mathit{AIC}$ | $\mathit{CAIC}$ | $\mathit{BIC}$ | ||||||

$KLLoGR(a,b,\alpha ,\beta )$ | 0.2734 | 0.5547 | 2.7501 | 0.06908 | 2.0 | 205.9 | 213.9 | 214.3 | 224.3 |

$ELLoGW(a,b,\alpha ,\beta )$ | 1.7650 | 1.0 | 0.4598 | 0.7387 | 1.0910 | 204.9 | 212.9 | 213.3 | 223.3 |

$ELLoGE(a,b,\alpha ,\beta )$ | 1.9878 | 1.0 | 0.4051 | 0.8586 | 1.0 | 205.0 | 211.0 | 211.2 | 218.8 |

$ELLoGR(a,b,\alpha ,\beta )$ | 0.6750 | 1.0 | 1.3884 | 0.06580 | 2.0 | 210.1 | 216.1 | 216.4 | 224.0 |

$LLoGW(a,b,\alpha ,\beta )$ | 1.0 | 1.0 | 2.1998 | 0.4113 | 0.5413 | 207.5 | 213.5 | 213.7 | 221.3 |

$LLoGR(a,b,\alpha ,\beta )$ | 1.0 | 1.0 | 0.9114 | 0.1423 | 2.0 | 213.3 | 217.3 | 217.5 | 226 |

$BetaMW(a,b,\alpha ,\gamma ,\lambda )$ | 108.86 | 25.631 | 1.6632 | 0.0534 | 0.0343 | 207.3 | 217.3 | 217.9 | 230.38 |

$GLLoGW(c,\alpha ,\beta ,\delta ,\theta )$ | 0.2365 | 0.2591 | 0.9648 | 4.3962 | 0.1396 | 204.1 | 214.01 | 214.6 | 227.1 |

$BBE2(a,b,\theta ,\lambda )$ | 0.736 | 0.167 | 0.371 | 6.181 | 204.722 | 212.722 | 213.13 | 212.739 |

**Table 16.**Goodness-of-fit tests corresponding to the Stress-Rupture life of Kevlar 49/Epoxy Strands data.

Distribution | Statistics | |
---|---|---|

${\mathit{W}}^{\ast}$ | ${\mathit{A}}^{\ast}$ | |

$KLLoGR(a,b,\alpha ,\beta )$ | 0.1635 | 0.9753 |

$ELLoGW(a,b,\alpha ,\beta )$ | 0.1319 | 0.8073 |

$ELLoGE(a,b,\alpha ,\beta )$ | 0.1447 | 0.8635 |

$ELLoGR(a,b,\alpha ,\beta )$ | 0.2610 | 1.4415 |

$LLoGW(a,b,\alpha ,\beta )$ | 0.1070 | 0.7446 |

$LLoGR(a,b,\alpha ,\beta )$ | 0.1776 | 1.1049 |

$BetaMW(a,b,\alpha ,\gamma ,\lambda )$ | 0.1955 | 1.1190 |

$GLLoGW(c,\alpha ,\beta ,\delta ,\theta )$ | 0.1322 | 0.7996 |

$BBE2(a,b,\theta ,\lambda )$ | 0.12446 | 0.77445 |

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**MDPI and ACS Style**

Hassan, O.H.M.; Elbatal, I.; Al-Nefaie, A.H.; El-Saeed, A.R.
Statistical Inference of the Beta Binomial Exponential 2 Distribution with Application to Environmental Data. *Axioms* **2022**, *11*, 740.
https://doi.org/10.3390/axioms11120740

**AMA Style**

Hassan OHM, Elbatal I, Al-Nefaie AH, El-Saeed AR.
Statistical Inference of the Beta Binomial Exponential 2 Distribution with Application to Environmental Data. *Axioms*. 2022; 11(12):740.
https://doi.org/10.3390/axioms11120740

**Chicago/Turabian Style**

Hassan, Osama H. Mahmoud, Ibrahim Elbatal, Abdullah H. Al-Nefaie, and Ahmed R. El-Saeed.
2022. "Statistical Inference of the Beta Binomial Exponential 2 Distribution with Application to Environmental Data" *Axioms* 11, no. 12: 740.
https://doi.org/10.3390/axioms11120740