# The Truncated Burr X-G Family of Distributions: Properties and Applications to Actuarial and Financial Data

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## Abstract

**:**

## 1. Introduction

## 2. Presentation of the TBX-G Family

#### 2.1. Distributional Functions

#### 2.1.1. Definition of the pdf

#### 2.1.2. Definition of the hrf

#### 2.1.3. Definition of the qf

#### 2.2. Special Survival Distributions

#### 2.2.1. TBX Exponential Distribution

#### 2.2.2. TBX Rayleigh Distribution

#### 2.2.3. TBX Lindley Distribution

## 3. Mathematical Properties of the TBX-G Family

#### 3.1. Asymptotic Study

#### 3.2. First-Order Stochastic Dominance Study

**Proposition**

**1.**

**Proof.**

#### 3.3. Series Expansion Study

**Proposition**

**2.**

**Proof.**

- TBXE distribution, for $x>0$, we have:$${v}_{G}(x;m,p,\mathbf{\omega})={g}_{E}{(x;\mathbf{\omega})}^{p}{\overline{G}}_{E}{(x;\mathbf{\omega})}^{m}={\beta}^{p}{e}^{-(p+m)\beta x};$$
- TBXR distribution, for $x>0$, we have:$${v}_{G}(x;m,p,\mathbf{\omega})={g}_{R}{(x;\mathbf{\omega})}^{p}{\overline{G}}_{R}{(x;\mathbf{\omega})}^{m}={\rho}^{p}{x}^{p}{e}^{-(p+m)\rho {x}^{2}/2};$$
- TBXL distribution, for $x>0$, we have:$$\begin{array}{cc}\hfill {v}_{G}(x;m,p,\mathbf{\omega})& ={g}_{L}{(x;\mathbf{\omega})}^{p}{\overline{G}}_{L}{(x;\mathbf{\omega})}^{m}=\frac{{a}^{2p}}{{(1+a)}^{p}}{(1+x)}^{p}{\left[1+\frac{ax}{1+a}\right]}^{m}{e}^{-(p+m)ax}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{a}^{2p}}{{(1+a)}^{m+p}}\sum _{k=0}^{m}\left(\genfrac{}{}{0pt}{}{m}{k}\right){a}^{k}{(1+x)}^{k+p}{e}^{-(p+m)ax}.\hfill \end{array}$$

#### 3.4. Tsallis Entropy Study

#### 3.5. Moment Study

#### 3.6. Risk Measures

## 4. Statistical and Inferential Approaches

#### 4.1. Methodology

#### 4.2. Simulation

## 5. Applications to Actuarial and Financial Data

#### 5.1. Data Fitting

#### 5.2. Estimation of $Va{R}_{q}$ and $E{S}_{q}$

## 6. Concluding Notes and Perspectives

#### 6.1. Concluding Notes

#### 6.2. Perspectives

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Plots of the pdf (

**left**) and hrf (

**right**) of the TBXE distribution for specific parameter values.

**Figure 2.**Three-dimensional plots of $SK$ (

**left**) and $KU$ (

**right**) for the TBXE distribution for $\beta =1$.

**Figure 3.**Plots of the pdf (

**left**) and hrf (

**right**) for the TBXR distribution for some parameter values.

**Figure 4.**Three-dimensional plots of $SK$ (

**left**) and $KU$ (

**right**) of the TBXR distribution for $\rho =1$.

**Figure 5.**Plots of the pdf (

**left**) and hrf (

**right**) of the TBXL distribution for chosen parameter values.

**Figure 6.**Three-dimensional plots of $SK$ (

**left**) and $KU$ (

**right**) of the TBXL distribution for $a=1$.

**Figure 7.**Plots of $Va{R}_{q}$ (

**left**) and $E{S}_{q}$ (

**right**) of the TBXE distribution for some parameter values.

**Figure 8.**(

**a**) TTT plot; (

**b**) box plot; (

**c**) plot of the estimated pdf over the histogram; (

**d**) plot of the estimated cdf over the empirical cdf for the TBXE model for D1.

**Figure 9.**(

**a**) TTT plot; (

**b**) box plot; (

**c**) plot of the estimated pdf over the histogram; (

**d**) plot of the estimated cdf over the empirical cdf for the TBXE model for D2.

**Figure 10.**Plots of the estimated $Va{R}_{q}$ (

**left**) and estimated $E{S}_{q}$ (

**right**) of the considered models for D1.

**Figure 11.**Plots of the estimated $Va{R}_{q}$ (

**left**) and estimated $E{S}_{q}$ (

**right**) of the considered models for D2.

**Table 1.**Actual values, AEs, and RMSEs of the simulated data from the TBXE distribution for some parameter values.

Sample Size | Actual Values | AE | RMSE | ||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{n}$ | $\mathit{\alpha}$ | $\mathit{\theta}$ | $\mathit{\beta}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\beta}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\beta}}$ |

30 | 0.3 | 0.7 | 1.2 | 0.5537 | 0.8124 | 1.1760 | 0.6695 | 0.2679 | 0.3346 |

80 | 0.5279 | 0.7408 | 1.1247 | 0.6429 | 0.1302 | 0.2472 | |||

130 | 0.4911 | 0.7296 | 1.1310 | 0.6031 | 0.0989 | 0.2173 | |||

180 | 0.4759 | 0.7232 | 1.1310 | 0.5765 | 0.0818 | 0.1976 | |||

230 | 0.4587 | 0.7161 | 1.1448 | 0.5442 | 0.0721 | 0.1795 | |||

280 | 0.4276 | 0.7161 | 1.1448 | 0.5144 | 0.0642 | 0.1662 | |||

30 | 1.3 | 2.7 | 0.6 | 1.0669 | 3.4192 | 0.6530 | 0.7543 | 4.7441 | 0.1966 |

80 | 1.1540 | 2.7689 | 0.6120 | 0.6604 | 0.8594 | 0.1576 | |||

130 | 1.1975 | 2.6818 | 0.6018 | 0.5846 | 0.6084 | 0.1398 | |||

180 | 1.2315 | 2.6669 | 0.5973 | 0.5241 | 0.5153 | 0.1284 | |||

230 | 1.2481 | 2.6672 | 0.5971 | 0.4783 | 0.4576 | 0.1203 | |||

280 | 1.2640 | 2.6628 | 0.5944 | 0.4472 | 0.4174 | 0.1134 | |||

30 | 0.8 | 0.8 | 0.9 | 0.6695 | 0.9178 | 0.9476 | 0.6962 | 0.3263 | 0.2809 |

80 | 0.7074 | 0.8290 | 0.8894 | 0.6706 | 0.1497 | 0.2144 | |||

130 | 0.7187 | 0.8125 | 0.8125 | 0.6477 | 0.1135 | 0.1934 | |||

180 | 0.7339 | 0.8066 | 0.8760 | 0.6190 | 0.0952 | 0.1819 | |||

230 | 0.7268 | 0.8042 | 0.8802 | 0.5901 | 0.0846 | 0.1694 | |||

280 | 0.7380 | 0.8020 | 0.8779 | 0.5683 | 0.0761 | 0.1610 | |||

30 | 1.5 | 0.9 | 0.9 | 0.9432 | 1.0338 | 1.2239 | 0.7431 | 0.3945 | 0.4349 |

80 | 1.1262 | 0.9287 | 1.0851 | 0.6901 | 0.1767 | 0.3340 | |||

130 | 1.2224 | 0.9121 | 1.0370 | 0.6314 | 0.1347 | 0.3021 | |||

180 | 1.2828 | 0.9060 | 1.0103 | 0.5801 | 0.1152 | 0.2839 | |||

230 | 1.3237 | 0.8999 | 0.9885 | 0.5365 | 0.1001 | 0.2634 | |||

280 | 1.3556 | 0.8984 | 0.9749 | 0.5114 | 0.0924 | 0.2560 | |||

30 | 1.2 | 0.8 | 0.6 | 0.7988 | 0.8992 | 0.7166 | 0.7192 | 0.3198 | 0.2379 |

80 | 0.9468 | 0.8176 | 0.6501 | 0.6862 | 0.1493 | 0.1816 | |||

130 | 1.0275 | 0.8052 | 0.6271 | 0.6413 | 0.1133 | 0.1653 | |||

180 | 1.0539 | 0.7964 | 0.6208 | 0.6045 | 0.0944 | 0.1535 | |||

230 | 1.0804 | 0.7953 | 0.6154 | 0.5765 | 0.0847 | 0.1478 | |||

280 | 1.0974 | 0.7944 | 0.6129 | 0.5420 | 0.0766 | 0.1401 | |||

30 | 1.0 | 1.0 | 0.8 | 0.7735 | 1.1469 | 0.8711 | 0.7156 | 0.4412 | 0.2654 |

80 | 0.8561 | 1.0253 | 0.8068 | 0.6855 | 0.1976 | 0.2076 | |||

130 | 0.8889 | 1.0075 | 0.7974 | 0.6465 | 0.1514 | 0.1881 | |||

180 | 0.9068 | 1.0008 | 0.7930 | 0.6066 | 0.1279 | 0.1739 | |||

230 | 0.9156 | 0.9984 | 0.7931 | 0.5716 | 0.1137 | 0.1634 | |||

280 | 0.9347 | 0.9945 | 0.7885 | 0.5448 | 0.1036 | 0.1553 | |||

30 | 1.2 | 0.8 | 0.9 | 0.8092 | 0.9033 | 1.0767 | 0.7121 | 0.3203 | 0.3530 |

80 | 0.9476 | 0.8189 | 0.9739 | 0.6910 | 0.1504 | 0.2734 | |||

130 | 1.0226 | 0.8027 | 0.9405 | 0.6481 | 0.1143 | 0.2457 | |||

180 | 1.0763 | 0.7996 | 0.9251 | 0.6020 | 0.0951 | 0.2319 | |||

230 | 1.0842 | 0.7964 | 0.9221 | 0.5752 | 0.0844 | 0.2212 | |||

280 | 1.1167 | 0.7933 | 0.9107 | 0.5486 | 0.0764 | 0.2123 | |||

30 | 0.5 | 1.8 | 1.5 | 0.9069 | 2.3930 | 1.4504 | 0.7638 | 1.4810 | 0.3880 |

80 | 0.6344 | 2.0147 | 1.4201 | 0.7482 | 0.5264 | 0.2887 | |||

130 | 0.5924 | 1.9439 | 1.4294 | 0.6190 | 0.3896 | 0.2482 | |||

180 | 0.5499 | 1.9137 | 1.4450 | 0.5578 | 0.3264 | 0.2088 | |||

230 | 0.5349 | 1.8937 | 1.4489 | 0.5239 | 0.2821 | 0.1848 | |||

280 | 0.5096 | 1.8782 | 1.4604 | 0.4914 | 0.2556 | 0.1630 |

Model | Parameters | MLEs (D1) | SEs (D1) | MLEs (D2) | SEs (D2) |
---|---|---|---|---|---|

TBXE | $\alpha $ | 2.6510 | 0.282 | 1.8533 | 0.3379 |

$\theta $ | 10.4527 | 4.99 | 5.1464 | 2.0897 | |

$\beta $ | 0.0152 | 0.0040 | 0.1191 | 0.0356 | |

BX | $\alpha $ | 0.0200 | 0.0012 | 0.0644 | 0.0056 |

$\theta $ | 1.9912 | 0.4252 | 1.0310 | 0.1844 | |

EE | $\alpha $ | 0.05 | 0.006 | 0.1786 | 0.0232 |

$\beta $ | 16.08 | 5.251 | 5.5321 | 1.4350 | |

E | $\beta $ | 0.0142 | 0.0020 | 0.0741 | 0.0096 |

MOE | $\alpha $ | 0.0664 | 0.0082 | 0.2092 | 0.0308 |

a | 72.1333 | 41.0232 | 11.5647 | 5.2019 | |

EW | $\alpha $ | 0.4333 | 0.4872 | 1.5481 | 0.9126 |

$\beta $ | 0.6043 | 0.1992 | 0.4706 | 0.1308 | |

a | 130.3842 | 219.0512 | 88.6904 | 8.4074 | |

OWE | $\alpha $ | 0.0028 | 0.0005 | 0.0164 | 0.0185 |

a | 14.2800 | 7.2812 | 6.6161 | 5.4439 | |

b | 1.9155 | 0.1843 | 1.5472 | 1.5625 | |

W | $\alpha $ | 0.0029 | 0.0006 | 0.0069 | 0.0028 |

$\beta $ | 1.3611 | 0.0552 | 1.8215 | 0.1339 | |

TLE | $\alpha $ | 0.0242 | 0.0028 | 0.0893 | 0.0116 |

a | 15.9758 | 5.1955 | 5.5322 | 1.4347 |

Model | $-\widehat{\mathit{\ell}}$ | AIC | CAIC | BIC | HQIC | A${}^{\ast}$ | W${}^{\ast}$ | K.S | p-Value |
---|---|---|---|---|---|---|---|---|---|

TBXE | 265.329 | 536.658 | 537.102 | 542.839 | 539.066 | 0.716 | 0.125 | 0.111 | 0.471 |

BX | 275.364 | 554.728 | 554.947 | 558.849 | 556.334 | 2.416 | 0.444 | 0.182 | 0.044 |

EE | 267.487 | 538.973 | 539.191 | 543.094 | 540.578 | 1.090 | 0.201 | 0.113 | 0.447 |

E | 304.967 | 611.934 | 612.006 | 613.995 | 612.737 | 1.755 | 0.324 | 0.387 | 0.00001 |

MOE | 274.318 | 552.636 | 552.854 | 556.757 | 554.241 | 2.218 | 0.400 | 0.140 | 0.204 |

EW | 266.190 | 538.379 | 538.824 | 544.561 | 540.787 | 0.896 | 0.163 | 0.117 | 0.408 |

OWE | 281.963 | 569.927 | 570.371 | 576.108 | 572.334 | 3.312 | 0.608 | 0.187 | 0.034 |

W | 291.235 | 586.470 | 586.688 | 590.591 | 588.075 | 2.065 | 0.380 | 0.332 | 0.00001 |

TLE | 267.487 | 538.973 | 539.191 | 543.094 | 540.578 | 1.091 | 0.202 | 0.113 | 0.445 |

Model | $-\widehat{\mathit{\ell}}$ | AIC | CAIC | BIC | HQIC | A${}^{\ast}$ | W${}^{\ast}$ | K.S | p-Value |
---|---|---|---|---|---|---|---|---|---|

TBXE | 265.3401 | 383.0907 | 383.5271 | 389.3233 | 385.5237 | 0.3621 | 0.0623 | 0.0703 | 0.9321 |

BX | 275.3641 | 399.3927 | 399.6070 | 403.5478 | 401.0147 | 1.9904 | 0.3112 | 0.1763 | 0.0509 |

EE | 267.4865 | 386.4471 | 386.6614 | 390.6021 | 388.0690 | 0.8708 | 0.1442 | 0.1148 | 0.4180 |

E | 304.9673 | 611.9345 | 612.0060 | 613.9950 | 612.7371 | 1.7555 | 0.3236 | 0.3869 | 0.0000 |

MOE | 274.3689 | 552.7378 | 552.9560 | 556.8587 | 554.3430 | 2.2232 | 0.4015 | 0.1498 | 0.1481 |

EW | 266.2673 | 538.5346 | 538.9790 | 544.7159 | 540.9423 | 0.9065 | 0.1651 | 0.1148 | 0.4282 |

OWE | 199.4381 | 404.8762 | 405.3125 | 411.1088 | 407.3091 | 2.1691 | 0.3368 | 0.1446 | 0.1694 |

W | 197.2967 | 398.5934 | 398.8077 | 402.7485 | 400.2154 | 1.8483 | 0.2897 | 0.1392 | 0.2025 |

TLE | 191.2235 | 386.4471 | 386.6614 | 390.6021 | 388.0690 | 0.8708 | 0.1442 | 0.1148 | 0.4183 |

q | TBXE | BX | EE | EW | W | E |
---|---|---|---|---|---|---|

0.55 | 78.32 | 42.40 | 68.33 | 67.06 | 61.48 | 56.43 |

0.60 | 84.53 | 46.69 | 71.52 | 70.33 | 68.02 | 64.76 |

0.65 | 91.98 | 51.44 | 74.99 | 73.95 | 75.17 | 74.20 |

0.70 | 100.92 | 56.81 | 78.84 | 78.04 | 83.13 | 85.09 |

0.75 | 112.17 | 63.03 | 83.23 | 82.80 | 92.21 | 97.98 |

0.80 | 127.27 | 70.52 | 88.43 | 88.57 | 102.89 | 113.75 |

0.85 | 149.78 | 80.02 | 94.94 | 95.99 | 116.11 | 134.08 |

0.90 | 191.17 | 93.20 | 103.85 | 106.51 | 133.86 | 162.74 |

0.95 | 357.62 | 115.42 | 118.67 | 124.94 | 162.42 | 211.72 |

q | TBXE | BX | EE | EW | W | E |
---|---|---|---|---|---|---|

0.55 | 51.68 | 18.44 | 51.84 | 51.27 | 31.64 | 24.50 |

0.60 | 54.16 | 19.86 | 53.34 | 52.72 | 34.40 | 27.50 |

0.65 | 56.79 | 21.33 | 54.87 | 54.21 | 37.26 | 30.72 |

0.70 | 59.63 | 22.88 | 56.45 | 55.77 | 40.24 | 34.21 |

0.75 | 62.76 | 24.53 | 58.08 | 57.41 | 43.40 | 38.02 |

0.80 | 66.31 | 26.31 | 59.81 | 59.17 | 46.77 | 42.24 |

0.85 | 70.53 | 28.27 | 61.68 | 61.10 | 50.45 | 47.01 |

0.90 | 75.96 | 30.51 | 63.76 | 63.31 | 54.56 | 52.59 |

0.95 | 84.91 | 33.21 | 66.21 | 66.01 | 59.41 | 59.53 |

q | TBXE | BX | EE | EW | W | E |
---|---|---|---|---|---|---|

0.55 | 69.34 | 6.37 | 12.76 | 0.10 | 13.58 | 10.78 |

0.60 | 75.44 | 7.30 | 13.60 | 0.12 | 14.64 | 12.37 |

0.65 | 82.31 | 8.34 | 14.51 | 0.13 | 15.78 | 14.17 |

0.70 | 90.22 | 9.55 | 15.53 | 0.15 | 17.01 | 16.25 |

0.75 | 99.57 | 10.97 | 16.70 | 0.18 | 18.38 | 18.71 |

0.80 | 110.98 | 12.71 | 18.09 | 0.20 | 19.95 | 21.72 |

0.85 | 125.59 | 14.95 | 19.83 | 0.24 | 21.83 | 25.60 |

0.90 | 145.60 | 18.10 | 22.23 | 0.28 | 24.28 | 31.07 |

0.95 | 176.39 | 23.49 | 26.23 | 0.35 | 28.06 | 40.43 |

q | TBXE | BX | EE | EW | W | E |
---|---|---|---|---|---|---|

0.55 | 42.77 | 2.80 | 8.59 | 0.04 | 8.01 | 4.68 |

0.60 | 45.23 | 3.14 | 8.98 | 0.05 | 8.51 | 5.25 |

0.65 | 47.82 | 3.50 | 9.37 | 0.05 | 9.03 | 5.87 |

0.70 | 50.56 | 3.88 | 9.77 | 0.06 | 9.55 | 6.53 |

0.75 | 53.50 | 4.31 | 10.19 | 0.07 | 10.10 | 7.26 |

0.80 | 56.73 | 4.78 | 10.64 | 0.07 | 10.66 | 8.07 |

0.85 | 60.33 | 5.31 | 11.13 | 0.08 | 11.26 | 8.98 |

0.90 | 64.48 | 5.92 | 11.67 | 0.09 | 11.91 | 10.04 |

0.95 | 69.49 | 6.69 | 12.32 | 0.10 | 12.65 | 11.37 |

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## Share and Cite

**MDPI and ACS Style**

Bantan, R.A.R.; Chesneau, C.; Jamal, F.; Elbatal, I.; Elgarhy, M.
The Truncated Burr X-G Family of Distributions: Properties and Applications to Actuarial and Financial Data. *Entropy* **2021**, *23*, 1088.
https://doi.org/10.3390/e23081088

**AMA Style**

Bantan RAR, Chesneau C, Jamal F, Elbatal I, Elgarhy M.
The Truncated Burr X-G Family of Distributions: Properties and Applications to Actuarial and Financial Data. *Entropy*. 2021; 23(8):1088.
https://doi.org/10.3390/e23081088

**Chicago/Turabian Style**

Bantan, Rashad A. R., Christophe Chesneau, Farrukh Jamal, Ibrahim Elbatal, and Mohammed Elgarhy.
2021. "The Truncated Burr X-G Family of Distributions: Properties and Applications to Actuarial and Financial Data" *Entropy* 23, no. 8: 1088.
https://doi.org/10.3390/e23081088