# Bayesian Inference and Data Analysis of the Unit–Power Burr X Distribution

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- ▪
- The UPBXD is a flexible model and can be used to describe a variety of datasets with a range between zero and one.
- ▪
- The new density function of the UBBXD takes several shapes, including unimodal, reversed J-shaped, U-shaped, left-skewed, and symmetric (see Section 2).
- ▪
- The HF shapes of the UPBXD can be increasing, J-shaped, or bathtub (U-HF) (see Section 2).
- ▪
- We derive some of the most important statistical characteristics of the UPBXD, such as the analytical expression for moments, the quantile function, incomplete moments, stochastic ordering, some uncertainty measures, and stress–strength reliability.
- ▪
- The parameter estimators of the UPBXD are explored using a Bayesian technique. The Bayesian credible intervals are also created.
- ▪
- To examine the effectiveness of estimators based on accuracy criteria, an exclusive simulation study was conducted.
- ▪
- Application to COVID-19 datasets from Saudi Arabia and the United Kingdom are used to show the superiority of the proposed model over other well-known models.

## 2. Unit Power Burr X Distribution

## 3. The UPBXD’s Properties

#### 3.1. Some Moments Measures

#### 3.2. Information Measures

#### 3.3. Stochastic Ordering

_{1}and W

_{2}, have CDFs that are ${F}_{{W}_{1}}(w)$ and ${F}_{{W}_{2}}(w)$, respectively, W

_{1}is said to be smaller than W

_{2}in the

- ▪
- Stochastic order (W
_{1}≤_{st}(W_{2})) if ${F}_{{W}_{1}}(w)$ ≥ ${F}_{{W}_{2}}(w)$ ∀w - ▪
- Hazard rate order (W
_{1}≤_{hr}(W_{2})) if ${h}_{{W}_{1}}(w)$ ≥ ${h}_{{W}_{2}}(w)$∀w - ▪
- Mean residual life order (W
_{1}≤_{mrl}(W_{2})) if ${m}_{{W}_{1}}(w)$ ≥ ${m}_{{W}_{2}}(w)$∀w - ▪
- Likelihood ratio order (W
_{1}≤_{lr}(W_{2})) if ${f}_{{W}_{1}}(w)/{f}_{{W}_{2}}(w)$ decreases in w.

_{i}, i = 1, 2 have the UPBXD with parameters $({a}_{i},{b}_{i},{\delta}_{i}).$ Further, assume that ${F}_{i}(w)$ and ${f}_{i}(w)$ indicate, respectively, W

_{i}’s CDF and PDF.

_{1}is said to be stochastically less than W

_{2}(W

_{1}≤

_{lr}W

_{2})

_{1}~UPBXD $({a}_{1},{b}_{1},{\delta}_{1})$ and W

_{2}~UPBXD $({a}_{2},{b}_{2},{\delta}_{2}),$ then the likelihood ratio ordering is as follows:

_{1}≤

_{lr}W

_{2}. Moreover, W

_{1}is said to be smaller than W

_{2}in other orderings such as SO (W

_{1}≤

_{st}W

_{2}), HF(W

_{1}≤

_{hr}W

_{2}), and mean residual order (W

_{1}≤

_{mrl}W

_{2}).

#### 3.4. Stress–Stress Reliability

_{2}and having random strength W

_{1}, with the system failing if W

_{2}is greater than W

_{1}, that is; R = P(W

_{2}< W

_{1}). Let us assume that W

_{1}∼UPBXD $({a}_{1},b,{\delta}_{1})$ and W

_{2}∼UPBXD $({a}_{2},b,{\delta}_{2})$ are two independent random variables. The SS reliability of the UPBXD is then calculated as follows:

## 4. Parameter Estimation

#### 4.1. Maximum Likelihood Method

#### 4.2. Bayesian Estimation

#### 4.2.1. Prior Information

#### 4.2.2. Posterior Distribution

#### 4.2.3. Markov Chain Monte Carlo

**Step 1:**Set the initial values ${a}^{(0)}=\widehat{a},\hspace{0.17em}\hspace{0.17em}{b}^{(0)}=\widehat{b},$ and ${\delta}^{(0)}=\widehat{\delta}.$

**Step 2:**Set I = 1.

**Step 3:**Generate ${a}^{\ast},\hspace{0.17em}\hspace{0.17em}{b}^{\ast}$ and ${\delta}^{\ast}$ from $N(\widehat{a},\hspace{0.17em}\hspace{0.17em}{V}_{\widehat{a}}),N(\widehat{b},{V}_{\widehat{b}})$ and $N(\widehat{\delta},\hspace{0.17em}\hspace{0.17em}{V}_{\widehat{\delta}}),$ respectively.

**Step 4:**Obtain ${\hslash}_{a}=\mathrm{min}\left[1,\frac{\pi \left({a}^{*}|{b}^{\left(I-1\right)},{\delta}^{\left(I-1\right)},\underset{\_}{w}\right)}{\pi \left({a}^{\left(I-1\right)}|{b}^{\left(I-1\right)},{\delta}^{\left(I-1\right)},\underset{\_}{w}\right)}\right],$${\hslash}_{b}=\mathrm{min}\left[1,\frac{\pi \left({b}^{*}|{a}^{\left(I-1\right)},{\delta}^{\left(I-1\right)},\underset{\_}{w}\right)}{\pi \left({b}^{\left(I-1\right)}|{a}^{\left(I-1\right)},{\delta}^{\left(I-1\right)},\underset{\_}{w}\right)}\right],$

**Step 5:**Generate samples U

_{j}j =1,2,3 from the uniform U(0, 1) distribution.

**Step 6:**If ${U}_{1}\le {\hslash}_{a},{U}_{2}\le {\hslash}_{b},$ and ${U}_{3}\le {\hslash}_{\delta},$ then set ${a}^{(I)}={a}^{*},{b}^{(I)}={b}^{*},\hspace{0.17em}\hspace{0.17em}{\delta}^{(I)}={\delta}^{\ast}$; otherwise ${a}^{(I)}={a}^{(I-1)},{b}^{(I)}={b}^{(I-1)},$ and ${\delta}^{\left(I\right)}={\delta}^{\left(I-1\right)}.$

**Step 7:**Set I = I+ 1.

**Step 8:**Repeat steps 3–7 B times and obtain ${a}^{(I)},{b}^{(I)},$ and ${\delta}^{(I)},$ for I = 1, 2,..., B.

#### 4.2.4. Highest Posterior Density Interval

## 5. Simulation

- Algorithm for simulation: By establishing all simulation controls, we can build our model. The following actions must be finished in this stage in the correct order:
- Assume different values for the UPBXD parameter vector and sample size.
- Make the sample random values for the UPBXD using uniform and the QF in Equation (7).
- We calculated the accuracy measures for each Bayes estimates of the UPBXD parameters using MH algorithm.
- This experiment should be run (L-1) times.

#### 5.1. Simulation Results

- The estimates are asymptotically unbiased since they are more accurate as the sample size increases.
- The parameter estimates come from the best unbiased estimator when the MSE value is near zero.
- As the sample size grows, the MSE declines for each estimate, demonstrating consistency between the various estimates.
- When the true value of $\delta $ increases, the bias, MSE, and length of the credible confidence interval (LCCI) of all estimates decrease.
- The MSE and LCCI for the Bayesian estimates with positive weight for the asymmetric loss function are smaller than the Bayesian estimates with negative weight for asymmetric loss function.
- The LCCI for estimates obtains its largest value, based on the suggested method, as the true values of the parameters increase.
- An entropy loss function with positive weight is better than the other loss functions.

#### 5.2. Represention Results

## 6. Application of Real Data

#### 6.1. Analysis for First Data

#### 6.2. Analysis for Second Data

#### 6.3. Data Analysis via Bayesian Method

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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${{\mathit{\mu}}^{\prime}}_{\mathit{m}}$ | (i) | (ii) | (iii) | (iv) | (v) | (vi) | (vii) |
---|---|---|---|---|---|---|---|

${{\mu}^{\prime}}_{1}$ | 0.499 | 0.396 | 0.67 | 0.802 | 0.803 | 0.477 | 0.416 |

${{\mu}^{\prime}}_{2}$ | 0.264 | 0.231 | 0.458 | 0.653 | 0.688 | 0.239 | 0.18 |

${{\mu}^{\prime}}_{3}$ | 0.148 | 0.160 | 0.32 | 0.54 | 0.612 | 0.125 | 0.08 |

${{\mu}^{\prime}}_{4}$ | 0.087 | 0.122 | 0.228 | 0.454 | 0.556 | 0.069 | 0.037 |

${\sigma}^{2}$ | 0.015 | 0.074 | 0.009 | 0.011 | 0.044 | 0.011 | 0.0062 |

${\alpha}_{3}$ | 0.377 | 0.510 | 0.38 | −0.135 | −1.237 | 0.465 | 0.375 |

${\alpha}_{4}$ | 2.856 | 2.132 | 2.75 | 2.248 | 3.799 | 3.070 | 3.135 |

d | Measures | (i) | (ii) | (iii) | (iv) | (v) | (vi) | (vii) |
---|---|---|---|---|---|---|---|---|

0.5 | $\eta (d)$ | −0.619 | −0.308 | −0.819 | −0.829 | −0.499 | −0.669 | −0.944 |

$\mathrm{H}(d)$ | −1.115 | −0.64 | −1.35 | −1.36 | −0.948 | −1.177 | −1.475 | |

$\mathrm{K}(d)$ | −1.101 | 0.792 | −1.354 | −1.201 | −2.953 | −1.179 | −1.688 | |

1.5 | $\eta (d)$ | −0.877 | 1.01 | −1.034 | −0.94 | −1.814 | −0.927 | −1.224 |

$\mathrm{H}(d)$ | −1.16 | 0.976 | −1.405 | −1.257 | −2.835 | −1.236 | −1.72 | |

$\mathrm{K}(d)$ | −0.533 | −0.286 | −0.672 | −0.678 | −0.442 | −0.568 | −0.753 |

$\mathit{a}=0.5,$ $\mathit{b}=0.6$ | SELF | $\mathbf{LINEX}\left(\mathit{c}=-1.5\right)$ | $\mathbf{LINEX}\left(\mathit{c}=1.5\right)$ | $\mathbf{ELF}\left(\mathit{c}=-1.5\right)$ | $\mathbf{ELF}\left(\mathit{c}=1.5\right)$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\delta}$ | $\mathit{n}$ | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | |

0.5 | 40 | $a$ | 0.1392 | 0.0224 | 0.2075 | 0.1448 | 0.0242 | 0.2140 | 0.1335 | 0.0207 | 0.2021 | 0.1422 | 0.0233 | 0.2098 | 0.1237 | 0.0181 | 0.1985 |

$b$ | −0.0261 | 0.0008 | 0.0423 | −0.0253 | 0.0008 | 0.0420 | −0.0269 | 0.0007 | 0.0427 | −0.0256 | 0.0008 | 0.0421 | −0.0284 | 0.0007 | 0.0426 | ||

$\delta $ | 0.6123 | 0.4095 | 0.7079 | 0.6587 | 0.4793 | 0.7990 | 0.5592 | 0.3364 | 0.5954 | 0.6267 | 0.4298 | 0.7275 | 0.5222 | 0.2932 | 0.5536 | ||

80 | $a$ | 0.0900 | 0.0094 | 0.1390 | 0.0927 | 0.0099 | 0.1413 | 0.0872 | 0.0088 | 0.1365 | 0.0915 | 0.0097 | 0.1400 | 0.0818 | 0.0079 | 0.1331 | |

$b$ | −0.0136 | 0.0002 | 0.0262 | −0.0133 | 0.0002 | 0.0262 | −0.0139 | 0.0002 | 0.0265 | −0.0134 | 0.0002 | 0.0263 | −0.0144 | 0.0003 | 0.0267 | ||

$\delta $ | 0.4229 | 0.1878 | 0.3562 | 0.4479 | 0.2117 | 0.3996 | 0.3954 | 0.1632 | 0.3134 | 0.4323 | 0.1965 | 0.3721 | 0.3672 | 0.1405 | 0.2908 | ||

160 | $a$ | 0.0843 | 0.0078 | 0.1017 | 0.0865 | 0.0082 | 0.1032 | 0.0821 | 0.0074 | 0.0994 | 0.0856 | 0.0080 | 0.1026 | 0.0779 | 0.0067 | 0.0967 | |

$b$ | −0.0135 | 0.0002 | 0.0237 | −0.0126 | 0.0002 | 0.0237 | −0.0127 | 0.0002 | 0.0238 | −0.0136 | 0.0002 | 0.0237 | −0.0137 | 0.0002 | 0.0241 | ||

$\delta $ | 0.3906 | 0.1567 | 0.2496 | 0.4121 | 0.1751 | 0.2732 | 0.3670 | 0.1380 | 0.2186 | 0.3991 | 0.1638 | 0.2564 | 0.3411 | 0.1190 | 0.2017 | ||

1.2 | 40 | $a$ | 0.0804 | 0.0082 | 0.1604 | 0.0841 | 0.0088 | 0.1614 | 0.0767 | 0.0075 | 0.1585 | 0.0825 | 0.0085 | 0.1609 | 0.0695 | 0.0065 | 0.1600 |

$b$ | −0.0204 | 0.0008 | 0.0769 | −0.0187 | 0.0008 | 0.0775 | −0.0221 | 0.0009 | 0.0757 | −0.0195 | 0.0008 | 0.0772 | −0.0252 | 0.0010 | 0.0756 | ||

$\delta $ | 0.6959 | 0.5321 | 0.8034 | 0.7936 | 0.6999 | 0.9837 | 0.5905 | 0.3779 | 0.6559 | 0.7138 | 0.5604 | 0.8249 | 0.5986 | 0.3913 | 0.7033 | ||

80 | $a$ | 0.0417 | 0.0025 | 0.1130 | 0.0431 | 0.0027 | 0.1135 | 0.0403 | 0.0024 | 0.1112 | 0.0426 | 0.0026 | 0.1131 | 0.0372 | 0.0022 | 0.1105 | |

$b$ | −0.0076 | 0.0002 | 0.0494 | −0.0069 | 0.0002 | 0.0498 | −0.0082 | 0.0002 | 0.0491 | −0.0072 | 0.0002 | 0.0496 | −0.0094 | 0.0003 | 0.0486 | ||

$\delta $ | 0.3724 | 0.1515 | 0.4317 | 0.4044 | 0.1794 | 0.4884 | 0.3394 | 0.1252 | 0.3813 | 0.3792 | 0.1571 | 0.4387 | 0.3370 | 0.1238 | 0.3846 | ||

160 | $a$ | 0.0371 | 0.0018 | 0.0780 | 0.0380 | 0.0018 | 0.0789 | 0.0363 | 0.0017 | 0.0774 | 0.0377 | 0.0018 | 0.0785 | 0.0344 | 0.0016 | 0.0769 | |

$b$ | −0.0082 | 0.0002 | 0.0370 | −0.0079 | 0.0002 | 0.0371 | −0.0086 | 0.0002 | 0.0370 | −0.0080 | 0.0002 | 0.0371 | −0.0092 | 0.0002 | 0.0371 | ||

$\delta $ | 0.3429 | 0.1242 | 0.3101 | 0.3689 | 0.1443 | 0.3453 | 0.3161 | 0.1050 | 0.2732 | 0.3486 | 0.1284 | 0.3168 | 0.3135 | 0.1034 | 0.2735 | ||

3 | 40 | $a$ | 0.0224 | 0.0019 | 0.1450 | 0.0249 | 0.0020 | 0.1457 | 0.0198 | 0.0018 | 0.1438 | 0.0240 | 0.0020 | 0.1448 | 0.0143 | 0.0016 | 0.1431 |

$b$ | −0.0036 | 0.0011 | 0.1197 | −0.0013 | 0.0011 | 0.1215 | −0.0059 | 0.0011 | 0.1188 | −0.0023 | 0.0011 | 0.1199 | −0.0101 | 0.0011 | 0.1184 | ||

$\delta $ | 0.4244 | 0.2485 | 1.0159 | 0.4897 | 0.3280 | 1.1489 | 0.3576 | 0.1793 | 0.8598 | 0.4307 | 0.2556 | 1.0306 | 0.3919 | 0.2142 | 0.9512 | ||

80 | $a$ | 0.0091 | 0.0008 | 0.1013 | 0.0101 | 0.0008 | 0.1021 | 0.0080 | 0.0007 | 0.1011 | 0.0098 | 0.0008 | 0.1019 | 0.0057 | 0.0007 | 0.1005 | |

$b$ | −0.0010 | 0.0005 | 0.0904 | 0.0000 | 0.0005 | 0.0907 | −0.0019 | 0.0005 | 0.0898 | −0.0004 | 0.0005 | 0.0906 | −0.0037 | 0.0005 | 0.0899 | ||

$\delta $ | 0.1950 | 0.0567 | 0.5279 | 0.2113 | 0.0661 | 0.5690 | 0.1786 | 0.0481 | 0.4905 | 0.1967 | 0.0577 | 0.5322 | 0.1866 | 0.0523 | 0.5095 | ||

160 | $a$ | 0.0086 | 0.0005 | 0.0767 | 0.0091 | 0.0005 | 0.0766 | 0.0080 | 0.0004 | 0.0757 | 0.0089 | 0.0005 | 0.0766 | 0.0067 | 0.0004 | 0.0753 | |

$b$ | −0.0016 | 0.0004 | 0.0789 | −0.0010 | 0.0004 | 0.0791 | −0.0022 | 0.0004 | 0.0783 | −0.0013 | 0.0004 | 0.0791 | −0.0032 | 0.0004 | 0.0791 | ||

$\delta $ | 0.1804 | 0.0409 | 0.3497 | 0.1911 | 0.0459 | 0.3734 | 0.1696 | 0.0361 | 0.3294 | 0.1815 | 0.0414 | 0.3519 | 0.1748 | 0.0384 | 0.3404 |

$\mathit{a}=0.5,$ $\mathit{b}=1.7$ | SELF | $\mathbf{LINEX}\left(\mathit{c}=-1.5\right)$ | $\mathbf{LINEX}\left(\mathit{c}=1.5\right)$ | $\mathbf{ELF}\left(\mathit{c}=-1.5\right)$ | $\mathbf{ELF}\left(\mathit{c}=1.5\right)$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\delta}$ | n | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | |

0.5 | 40 | $a$ | 0.1212 | 0.0179 | 0.2155 | 0.1257 | 0.0191 | 0.2199 | 0.1167 | 0.0167 | 0.2110 | 0.1236 | 0.0185 | 0.2172 | 0.1086 | 0.0147 | 0.2073 |

$b$ | −0.0308 | 0.0013 | 0.0722 | −0.0281 | 0.0012 | 0.0716 | −0.0336 | 0.0015 | 0.0742 | −0.0303 | 0.0013 | 0.0722 | −0.0336 | 0.0015 | 0.0744 | ||

$\delta $ | 0.5944 | 0.3901 | 0.7286 | 0.6374 | 0.4534 | 0.8182 | 0.5457 | 0.3238 | 0.6240 | 0.6078 | 0.4086 | 0.7470 | 0.5113 | 0.2844 | 0.5813 | ||

80 | $a$ | 0.0845 | 0.0084 | 0.1382 | 0.0870 | 0.0089 | 0.1402 | 0.0820 | 0.0079 | 0.1365 | 0.0859 | 0.0087 | 0.1386 | 0.0771 | 0.0071 | 0.1335 | |

$b$ | −0.0229 | 0.0009 | 0.0662 | −0.0215 | 0.0008 | 0.0663 | −0.0242 | 0.0010 | 0.0661 | −0.0226 | 0.0009 | 0.0662 | −0.0242 | 0.0010 | 0.0661 | ||

$\delta $ | 0.4267 | 0.1911 | 0.3714 | 0.4511 | 0.2147 | 0.4076 | 0.3995 | 0.1666 | 0.3243 | 0.4359 | 0.1996 | 0.3810 | 0.3717 | 0.1439 | 0.2986 | ||

160 | $a$ | 0.0754 | 0.0062 | 0.0897 | 0.0771 | 0.0065 | 0.0910 | 0.0736 | 0.0059 | 0.0889 | 0.0764 | 0.0064 | 0.0904 | 0.0701 | 0.0054 | 0.0877 | |

$b$ | −0.0196 | 0.0005 | 0.0476 | −0.0189 | 0.0005 | 0.0477 | −0.0203 | 0.0006 | 0.0476 | −0.0195 | 0.0005 | 0.0477 | −0.0203 | 0.0006 | 0.0476 | ||

$\delta $ | 0.3890 | 0.1550 | 0.2339 | 0.4099 | 0.1725 | 0.2573 | 0.3660 | 0.1368 | 0.2117 | 0.3973 | 0.1618 | 0.2414 | 0.3404 | 0.1182 | 0.1901 | ||

1.2 | 40 | $a$ | 0.0722 | 0.0069 | 0.1623 | 0.0751 | 0.0074 | 0.1645 | 0.0693 | 0.0065 | 0.1600 | 0.0739 | 0.0072 | 0.1631 | 0.0635 | 0.0057 | 0.1576 |

$b$ | −0.0311 | 0.0033 | 0.1871 | −0.0241 | 0.0030 | 0.1885 | −0.0380 | 0.0037 | 0.1841 | −0.0297 | 0.0032 | 0.1865 | −0.0380 | 0.0037 | 0.1853 | ||

$\delta $ | 0.7333 | 0.5923 | 0.8827 | 0.8377 | 0.7792 | 1.0665 | 0.6187 | 0.4166 | 0.7014 | 0.7523 | 0.6236 | 0.9119 | 0.6288 | 0.4342 | 0.7495 | ||

80 | $a$ | 0.0423 | 0.0026 | 0.1102 | 0.0436 | 0.0027 | 0.1111 | 0.0410 | 0.0025 | 0.1094 | 0.0431 | 0.0026 | 0.1108 | 0.0383 | 0.0022 | 0.1089 | |

$b$ | −0.0179 | 0.0020 | 0.1693 | −0.0152 | 0.0020 | 0.1689 | −0.0207 | 0.0021 | 0.1678 | −0.0174 | 0.0020 | 0.1694 | −0.0207 | 0.0022 | 0.1681 | ||

$\delta $ | 0.3962 | 0.1709 | 0.4626 | 0.4310 | 0.2032 | 0.5201 | 0.3599 | 0.1403 | 0.4013 | 0.4036 | 0.1774 | 0.4722 | 0.3577 | 0.1390 | 0.4080 | ||

160 | $a$ | 0.0365 | 0.0018 | 0.0839 | 0.0373 | 0.0019 | 0.0847 | 0.0357 | 0.0017 | 0.0831 | 0.0370 | 0.0018 | 0.0845 | 0.0340 | 0.0016 | 0.0822 | |

$b$ | −0.0163 | 0.0017 | 0.1472 | −0.0146 | 0.0016 | 0.1465 | −0.0180 | 0.0018 | 0.1486 | −0.0160 | 0.0017 | 0.1468 | −0.0180 | 0.0018 | 0.1488 | ||

$\delta $ | 0.3476 | 0.1271 | 0.3037 | 0.3738 | 0.1475 | 0.3367 | 0.3205 | 0.1076 | 0.2707 | 0.3533 | 0.1314 | 0.3102 | 0.3179 | 0.1060 | 0.2717 | ||

3 | 40 | $a$ | 0.0202 | 0.0018 | 0.1484 | 0.0222 | 0.0019 | 0.1503 | 0.0182 | 0.0017 | 0.1483 | 0.0215 | 0.0019 | 0.1493 | 0.0138 | 0.0016 | 0.1477 |

$b$ | −0.0037 | 0.0100 | 0.3923 | 0.0087 | 0.0104 | 0.3968 | −0.0161 | 0.0100 | 0.3886 | −0.0013 | 0.0100 | 0.3898 | −0.0160 | 0.0102 | 0.3926 | ||

$\delta $ | 0.4503 | 0.2804 | 1.0727 | 0.5244 | 0.3753 | 1.2260 | 0.3748 | 0.1981 | 0.9209 | 0.4575 | 0.2888 | 1.0900 | 0.4138 | 0.2398 | 1.0071 | ||

80 | $a$ | 0.0106 | 0.0008 | 0.1023 | 0.0113 | 0.0008 | 0.1019 | 0.0098 | 0.0008 | 0.1018 | 0.0110 | 0.0008 | 0.1019 | 0.0081 | 0.0008 | 0.1018 | |

$b$ | −0.0056 | 0.0052 | 0.2814 | −0.0015 | 0.0052 | 0.2815 | −0.0096 | 0.0053 | 0.2807 | −0.0048 | 0.0052 | 0.2808 | −0.0096 | 0.0053 | 0.2821 | ||

$\delta $ | 0.2074 | 0.0618 | 0.5202 | 0.2251 | 0.0722 | 0.5520 | 0.1897 | 0.0522 | 0.4859 | 0.2092 | 0.0628 | 0.5225 | 0.1983 | 0.0568 | 0.5058 | ||

160 | $a$ | 0.0082 | 0.0004 | 0.0708 | 0.0085 | 0.0004 | 0.0709 | 0.0078 | 0.0004 | 0.0706 | 0.0084 | 0.0004 | 0.0709 | 0.0069 | 0.0004 | 0.0702 | |

$b$ | −0.0037 | 0.0025 | 0.1941 | −0.0018 | 0.0025 | 0.1921 | −0.0056 | 0.0025 | 0.1949 | −0.0033 | 0.0025 | 0.1941 | −0.0056 | 0.0025 | 0.1953 | ||

$\delta $ | 0.1895 | 0.0457 | 0.3920 | 0.2014 | 0.0517 | 0.4220 | 0.1775 | 0.0400 | 0.3662 | 0.1908 | 0.0463 | 0.3952 | 0.1833 | 0.0428 | 0.3771 |

$\mathit{a}=2,$ $\mathit{b}=0.6$ | SELF | $\mathbf{LINEX}\left(\mathit{c}=-1.5\right)$ | $\mathbf{LINEX}\left(\mathit{c}=1.5\right)$ | $\mathbf{ELF}\left(\mathit{c}=-1.5\right)$ | $\mathbf{ELF}\left(\mathit{c}=1.5\right)$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\delta}$ | n | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | |

0.5 | 40 | $a$ | 0.1442 | 0.0330 | 0.4135 | 0.1607 | 0.0393 | 0.4250 | 0.1275 | 0.0273 | 0.3941 | 0.1468 | 0.0339 | 0.4181 | 0.1313 | 0.0286 | 0.4004 |

$b$ | −0.0388 | 0.0021 | 0.0635 | −0.0380 | 0.0020 | 0.0621 | −0.0396 | 0.0021 | 0.0652 | −0.0382 | 0.0020 | 0.0625 | −0.0411 | 0.0023 | 0.0681 | ||

$\delta $ | 0.5679 | 0.3597 | 0.7317 | 0.6074 | 0.4169 | 0.8179 | 0.5232 | 0.2998 | 0.6075 | 0.5804 | 0.3765 | 0.7545 | 0.4907 | 0.2634 | 0.5613 | ||

80 | $a$ | 0.0678 | 0.0096 | 0.2757 | 0.0735 | 0.0107 | 0.2852 | 0.0622 | 0.0087 | 0.2721 | 0.0687 | 0.0098 | 0.2764 | 0.0634 | 0.0089 | 0.2724 | |

$b$ | −0.0275 | 0.0009 | 0.0399 | −0.0271 | 0.0008 | 0.0395 | −0.0279 | 0.0009 | 0.0402 | −0.0272 | 0.0009 | 0.0397 | −0.0286 | 0.0009 | 0.0410 | ||

$\delta $ | 0.3992 | 0.1676 | 0.3463 | 0.4199 | 0.1861 | 0.3752 | 0.3764 | 0.1482 | 0.3121 | 0.4073 | 0.1745 | 0.3559 | 0.3518 | 0.1293 | 0.2869 | ||

160 | $a$ | 0.0582 | 0.0062 | 0.2080 | 0.0614 | 0.0067 | 0.2116 | 0.0550 | 0.0058 | 0.2065 | 0.0587 | 0.0063 | 0.2084 | 0.0556 | 0.0058 | 0.2069 | |

$b$ | −0.0282 | 0.0008 | 0.0342 | −0.0279 | 0.0009 | 0.0342 | −0.0285 | 0.0010 | 0.0341 | −0.0280 | 0.0009 | 0.0334 | −0.0291 | 0.0010 | 0.0332 | ||

$\delta $ | 0.3702 | 0.1410 | 0.2369 | 0.3886 | 0.1557 | 0.2640 | 0.3501 | 0.1257 | 0.2096 | 0.3776 | 0.1468 | 0.2473 | 0.3270 | 0.1094 | 0.1937 | ||

1.2 | 40 | $a$ | 0.0995 | 0.0178 | 0.3423 | 0.1126 | 0.0210 | 0.3483 | 0.0866 | 0.0150 | 0.3373 | 0.1016 | 0.0182 | 0.3429 | 0.0892 | 0.0156 | 0.3408 |

$b$ | −0.0373 | 0.0104 | 0.0748 | −0.0353 | 0.0132 | 0.0747 | −0.0400 | 0.0051 | 0.0744 | −0.0363 | 0.0109 | 0.0746 | −0.0429 | 0.0062 | 0.0747 | ||

$\delta $ | 0.7256 | 0.5834 | 0.9188 | 0.8305 | 0.7724 | 1.1304 | 0.6121 | 0.4104 | 0.7213 | 0.7446 | 0.6146 | 0.9436 | 0.6218 | 0.4277 | 0.7719 | ||

80 | $a$ | 0.0395 | 0.0057 | 0.2532 | 0.0438 | 0.0062 | 0.2577 | 0.0352 | 0.0053 | 0.2509 | 0.0402 | 0.0058 | 0.2542 | 0.0360 | 0.0054 | 0.2511 | |

$b$ | −0.0198 | 0.0006 | 0.0506 | −0.0191 | 0.0005 | 0.0511 | −0.0205 | 0.0006 | 0.0497 | −0.0194 | 0.0005 | 0.0509 | −0.0218 | 0.0006 | 0.0492 | ||

$\delta $ | 0.3838 | 0.1612 | 0.4450 | 0.4172 | 0.1913 | 0.4972 | 0.3493 | 0.1327 | 0.3917 | 0.3909 | 0.1672 | 0.4565 | 0.3470 | 0.1313 | 0.3945 | ||

160 | $a$ | 0.0352 | 0.0035 | 0.1802 | 0.0375 | 0.0037 | 0.1816 | 0.0329 | 0.0033 | 0.1780 | 0.0356 | 0.0036 | 0.1805 | 0.0334 | 0.0034 | 0.1788 | |

$b$ | −0.0194 | 0.0006 | 0.0366 | −0.0191 | 0.0007 | 0.0365 | −0.0198 | 0.0008 | 0.0366 | −0.0192 | 0.0007 | 0.0365 | −0.0206 | 0.0009 | 0.0368 | ||

$\delta $ | 0.3437 | 0.1244 | 0.2940 | 0.3693 | 0.1441 | 0.3334 | 0.3172 | 0.1055 | 0.2635 | 0.3493 | 0.1285 | 0.3026 | 0.3146 | 0.1039 | 0.2647 | ||

3 | 40 | $a$ | 0.0382 | 0.0061 | 0.2646 | 0.0461 | 0.0069 | 0.2666 | 0.0304 | 0.0054 | 0.2582 | 0.0395 | 0.0062 | 0.2640 | 0.0318 | 0.0055 | 0.2594 |

$b$ | −0.0038 | 0.0167 | 0.0996 | −0.0002 | 0.0318 | 0.1001 | −0.0081 | 0.0036 | 0.0979 | −0.0025 | 0.0187 | 0.0993 | −0.0113 | 0.0051 | 0.0978 | ||

$\delta $ | 0.4644 | 0.2952 | 1.0815 | 0.5388 | 0.3941 | 1.2013 | 0.3883 | 0.2092 | 0.9174 | 0.4716 | 0.3039 | 1.0866 | 0.4279 | 0.2530 | 1.0157 | ||

80 | $a$ | 0.0103 | 0.0031 | 0.2140 | 0.0134 | 0.0032 | 0.2145 | 0.0071 | 0.0030 | 0.2110 | 0.0108 | 0.0031 | 0.2143 | 0.0077 | 0.0030 | 0.2116 | |

$b$ | −0.0058 | 0.0005 | 0.0801 | −0.0047 | 0.0005 | 0.0800 | −0.0068 | 0.0005 | 0.0796 | −0.0052 | 0.0005 | 0.0802 | −0.0086 | 0.0005 | 0.0795 | ||

$\delta $ | 0.2169 | 0.0675 | 0.5525 | 0.2362 | 0.0797 | 0.5974 | 0.1974 | 0.0564 | 0.5100 | 0.2188 | 0.0687 | 0.5577 | 0.2069 | 0.0618 | 0.5310 | ||

160 | $a$ | 0.0115 | 0.0014 | 0.1356 | 0.0129 | 0.0014 | 0.1365 | 0.0100 | 0.0013 | 0.1352 | 0.0117 | 0.0014 | 0.1357 | 0.0102 | 0.0013 | 0.1353 | |

$b$ | −0.0051 | 0.0002 | 0.0534 | −0.0046 | 0.0002 | 0.0532 | −0.0055 | 0.0002 | 0.0531 | −0.0048 | 0.0002 | 0.0533 | −0.0063 | 0.0002 | 0.0529 | ||

$\delta $ | 0.1894 | 0.0461 | 0.3862 | 0.2014 | 0.0521 | 0.4096 | 0.1773 | 0.0404 | 0.3600 | 0.1907 | 0.0467 | 0.3896 | 0.1832 | 0.0431 | 0.3755 |

$\mathit{a}=2,$ $\mathit{b}=1.7$ | SELF | $\mathbf{LINEX}\left(\mathit{c}=-1.5\right)$ | $\mathbf{LINEX}\left(\mathit{c}=1.5\right)$ | $\mathbf{ELF}\left(\mathit{c}=-1.5\right)$ | $\mathbf{ELF}\left(\mathit{c}=1.5\right)$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\delta}$ | $\mathit{n}$ | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | |

0.5 | 40 | $a$ | 0.1255 | 0.0232 | 0.3228 | 0.1388 | 0.0274 | 0.3337 | 0.1122 | 0.0194 | 0.3150 | 0.1276 | 0.0238 | 0.3237 | 0.1151 | 0.0203 | 0.3166 |

$b$ | −0.0538 | 0.0036 | 0.0995 | −0.0507 | 0.0032 | 0.0975 | −0.0568 | 0.0040 | 0.1018 | −0.0532 | 0.0035 | 0.0990 | −0.0569 | 0.0040 | 0.1021 | ||

$\delta $ | 0.5381 | 0.3172 | 0.6043 | 0.5710 | 0.3603 | 0.6771 | 0.5012 | 0.2719 | 0.5360 | 0.5489 | 0.3305 | 0.6253 | 0.4723 | 0.2413 | 0.5051 | ||

80 | $a$ | 0.0715 | 0.0094 | 0.2549 | 0.0774 | 0.0105 | 0.2621 | 0.0657 | 0.0084 | 0.2483 | 0.0724 | 0.0096 | 0.2567 | 0.0668 | 0.0086 | 0.2506 | |

$b$ | −0.0505 | 0.0032 | 0.0934 | −0.0485 | 0.0030 | 0.0927 | −0.0525 | 0.0034 | 0.0944 | −0.0501 | 0.0032 | 0.0932 | −0.0526 | 0.0034 | 0.0946 | ||

$\delta $ | 0.4003 | 0.1681 | 0.3332 | 0.4211 | 0.1868 | 0.3669 | 0.3775 | 0.1487 | 0.3008 | 0.4083 | 0.1751 | 0.3439 | 0.3531 | 0.1297 | 0.2722 | ||

160 | $a$ | 0.0580 | 0.0054 | 0.1784 | 0.0608 | 0.0058 | 0.1805 | 0.0551 | 0.0050 | 0.1763 | 0.0584 | 0.0055 | 0.1784 | 0.0557 | 0.0051 | 0.1770 | |

$b$ | −0.0364 | 0.0016 | 0.0611 | −0.0355 | 0.0015 | 0.0604 | −0.0372 | 0.0017 | 0.0622 | −0.0362 | 0.0016 | 0.0610 | −0.0372 | 0.0017 | 0.0623 | ||

$\delta $ | 0.3624 | 0.1346 | 0.2258 | 0.3790 | 0.1475 | 0.2470 | 0.3442 | 0.1211 | 0.2064 | 0.3692 | 0.1398 | 0.2338 | 0.3225 | 0.1062 | 0.1864 | ||

1.2 | 40 | $a$ | 0.1031 | 0.0180 | 0.3187 | 0.1156 | 0.0213 | 0.3285 | 0.0906 | 0.0150 | 0.3132 | 0.1050 | 0.0184 | 0.3206 | 0.0932 | 0.0157 | 0.3156 |

$b$ | −0.0780 | 0.0089 | 0.2048 | −0.0696 | 0.0077 | 0.2064 | −0.0863 | 0.0103 | 0.2051 | −0.0763 | 0.0086 | 0.2049 | −0.0866 | 0.0103 | 0.2065 | ||

$\delta $ | 0.7317 | 0.5927 | 0.9007 | 0.8358 | 0.7821 | 1.0799 | 0.6180 | 0.4171 | 0.7011 | 0.7505 | 0.6241 | 0.9342 | 0.6282 | 0.4349 | 0.7433 | ||

80 | $a$ | 0.0423 | 0.0053 | 0.2295 | 0.0464 | 0.0058 | 0.2322 | 0.0381 | 0.0048 | 0.2286 | 0.0429 | 0.0053 | 0.2303 | 0.0389 | 0.0049 | 0.2297 | |

$b$ | −0.0347 | 0.0026 | 0.1457 | −0.0319 | 0.0024 | 0.1454 | −0.0376 | 0.0028 | 0.1452 | −0.0341 | 0.0026 | 0.1456 | −0.0376 | 0.0028 | 0.1455 | ||

$\delta $ | 0.3935 | 0.1676 | 0.4409 | 0.4283 | 0.1995 | 0.4871 | 0.3573 | 0.1376 | 0.3851 | 0.4009 | 0.1741 | 0.4527 | 0.3549 | 0.1362 | 0.3910 | ||

160 | $a$ | 0.0339 | 0.0032 | 0.1765 | 0.0361 | 0.0033 | 0.1796 | 0.0317 | 0.0030 | 0.1745 | 0.0342 | 0.0032 | 0.1768 | 0.0321 | 0.0030 | 0.1751 | |

$b$ | −0.0333 | 0.0021 | 0.1238 | −0.0316 | 0.0020 | 0.1235 | −0.0350 | 0.0023 | 0.1249 | −0.0330 | 0.0021 | 0.1237 | −0.0350 | 0.0023 | 0.1250 | ||

$\delta $ | 0.3408 | 0.1224 | 0.3073 | 0.3661 | 0.1417 | 0.3402 | 0.3147 | 0.1039 | 0.2688 | 0.3464 | 0.1265 | 0.3149 | 0.3121 | 0.1023 | 0.2702 | ||

3 | 40 | $a$ | 0.0365 | 0.0058 | 0.2445 | 0.0440 | 0.0065 | 0.2480 | 0.0291 | 0.0052 | 0.2413 | 0.0377 | 0.0059 | 0.2444 | 0.0305 | 0.0053 | 0.2415 |

$b$ | −0.0165 | 0.0067 | 0.2990 | −0.0058 | 0.0067 | 0.3036 | −0.0271 | 0.0069 | 0.2941 | −0.0144 | 0.0066 | 0.2979 | −0.0271 | 0.0070 | 0.2965 | ||

$\delta $ | 0.4806 | 0.3234 | 1.1672 | 0.5638 | 0.4389 | 1.3260 | 0.3958 | 0.2241 | 1.0117 | 0.4887 | 0.3334 | 1.1839 | 0.4399 | 0.2746 | 1.0958 | ||

80 | $a$ | 0.0158 | 0.0025 | 0.1799 | 0.0188 | 0.0027 | 0.1825 | 0.0128 | 0.0024 | 0.1786 | 0.0163 | 0.0026 | 0.1801 | 0.0133 | 0.0024 | 0.1791 | |

$b$ | −0.0097 | 0.0059 | 0.2519 | −0.0053 | 0.0053 | 0.2507 | −0.0140 | 0.0065 | 0.2519 | −0.0087 | 0.0056 | 0.2512 | −0.0142 | 0.0073 | 0.2523 | ||

$\delta $ | 0.2242 | 0.0715 | 0.5563 | 0.2439 | 0.0842 | 0.5915 | 0.2043 | 0.0598 | 0.5135 | 0.2262 | 0.0727 | 0.5598 | 0.2141 | 0.0655 | 0.5382 | ||

160 | $a$ | 0.0110 | 0.0012 | 0.1251 | 0.0124 | 0.0012 | 0.1255 | 0.0096 | 0.0012 | 0.1245 | 0.0112 | 0.0012 | 0.1251 | 0.0098 | 0.0012 | 0.1247 | |

$b$ | −0.0107 | 0.0019 | 0.1672 | −0.0089 | 0.0018 | 0.1683 | −0.0124 | 0.0019 | 0.1684 | −0.0103 | 0.0019 | 0.1673 | −0.0124 | 0.0019 | 0.1686 | ||

$\delta $ | 0.1970 | 0.0475 | 0.3568 | 0.2092 | 0.0537 | 0.3804 | 0.1847 | 0.0417 | 0.3325 | 0.1982 | 0.0481 | 0.3603 | 0.1906 | 0.0445 | 0.3454 |

a | b | $\mathit{\delta}$ | KSS | p-Value | WS | AS | ||
---|---|---|---|---|---|---|---|---|

UPBXD | Estimates | 2.2717 | 0.5609 | 2813.2886 | 0.0778 | 0.9693 | 0.0327 | 0.2364 |

SE | 0.0708 | 0.0692 | 469.2354 | |||||

TLPTLE | Estimates | 693.1774 | 0.6471 | 0.6476 | 0.0938 | 0.8800 | 0.0479 | 0.3110 |

SE | 1626.8445 | 0.0939 | 0.8346 | |||||

TLGE | Estimates | 0.3682 | 20.4075 | 179.7044 | 0.1403 | 0.4378 | 0.0965 | 0.5911 |

SE | 0.2843 | 2.7038 | 178.4010 | |||||

K | Estimates | 3.3085 | 125.2161 | 0.1821 | 0.1621 | 0.0421 | 0.2793 | |

SE | 0.2821 | 49.4480 | ||||||

Beta | Estimates | 20.8174 | 76.5218 | 0.1127 | 0.7089 | 0.0636 | 0.3992 | |

SE | 4.8690 | 18.0555 | ||||||

UW | Estimates | 0.0203 | 7.7557 | 0.1633 | 0.2624 | 0.1824 | 1.0950 | |

SE | 0.0110 | 0.9132 | ||||||

UEHL | Estimates | 6.0655 | 3670.3422 | 0.0792 | 0.9641 | 0.0330 | 0.2393 | |

SE | 0.7918 | 405.8862 | ||||||

MOK | Estimates | 703.3130 | 1.3097 | 45.4476 | 0.0811 | 0.9567 | 0.0336 | 0.2554 |

SE | 4615.4897 | 1.5712 | 72.4483 |

a | b | $\mathit{\delta}$ | KSS | p-Value | WS | AS | ||
---|---|---|---|---|---|---|---|---|

UPBXD | Estimates | 2.3471 | 0.3354 | 2154.8742 | 0.1100 | 0.8512 | 0.1115 | 0.7238 |

SE | 0.0716 | 0.0478 | 402.9847 | |||||

TLPTLE | Estimates | 3760.8372 | 0.1127 | 0.0216 | 0.2571 | 0.0404 | 0.4656 | 2.6059 |

SE | 5117.2571 | 0.0287 | 0.0151 | |||||

TLGE | Estimates | 0.2968 | 10.3896 | 13.8963 | 0.2097 | 0.1471 | 0.2804 | 1.6718 |

SE | 0.3076 | 2.3410 | 15.4726 | |||||

K | Estimates | 2.9163 | 125.0007 | 0.1329 | 0.6570 | 0.1492 | 0.9456 | |

SE | 0.4689 | 94.0394 | ||||||

Beta | Estimates | 3.9277 | 19.1899 | 0.1925 | 0.2202 | 0.2609 | 1.5673 | |

SE | 1.0090 | 5.1885 | ||||||

UW | Estimates | 0.0904 | 3.2548 | 0.2468 | 0.0548 | 0.4622 | 2.5905 | |

SE | 0.0386 | 0.4231 | ||||||

UEHL | Estimates | 2.9789 | 69.5723 | 0.1294 | 0.6888 | 0.1476 | 0.9369 | |

SE | 0.4808 | 53.4192 | ||||||

MOK | Estimates | 0.0124 | 3.5483 | 6.7014 | 0.1506 | 0.5020 | 0.3117 | 1.8656 |

SE | 0.0299 | 0.6026 | 14.9920 |

Data | Estimates | SE | Lower | Upper | |
---|---|---|---|---|---|

Saudi Arabia | $a$ | 2.3235 | 0.0619 | 1.9541 | 2.7490 |

$b$ | 0.5556 | 0.0558 | 0.3990 | 0.7320 | |

$\delta $ | 2979.2033 | 2.4931 | 2974.2166 | 2984.1337 | |

The United Kingdom | $a$ | 2.3852 | 0.0560 | 2.0108 | 2.8278 |

$b$ | 0.3353 | 0.0314 | 0.2064 | 0.4830 | |

$\delta $ | 2154.8743 | 0.0787 | 2154.7169 | 2155.0299 |

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## Share and Cite

**MDPI and ACS Style**

Fayomi, A.; Hassan, A.S.; Baaqeel, H.; Almetwally, E.M.
Bayesian Inference and Data Analysis of the Unit–Power Burr X Distribution. *Axioms* **2023**, *12*, 297.
https://doi.org/10.3390/axioms12030297

**AMA Style**

Fayomi A, Hassan AS, Baaqeel H, Almetwally EM.
Bayesian Inference and Data Analysis of the Unit–Power Burr X Distribution. *Axioms*. 2023; 12(3):297.
https://doi.org/10.3390/axioms12030297

**Chicago/Turabian Style**

Fayomi, Aisha, Amal S. Hassan, Hanan Baaqeel, and Ehab M. Almetwally.
2023. "Bayesian Inference and Data Analysis of the Unit–Power Burr X Distribution" *Axioms* 12, no. 3: 297.
https://doi.org/10.3390/axioms12030297