# The Burr X Pareto Distribution: Properties, Applications and VaR Estimation

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## Abstract

**:**

## 1. Introduction

## 2. The New Model

**Lemma**

**1.**

**Proof.**

## 3. Expansions of pdf and cdf

## 4. Properties

#### 4.1. Moments

#### 4.2. Residual and Reversed Residual Life

#### 4.3. Order Statistics

## 5. Estimation Methods

#### 5.1. Maximum Likelihood Estimation

`R`(

`optim function`),

`SAS`(

`PROC NLMIXED`) or by solving the nonlinear likelihood equations obtained by differentiating ℓ. We note that since $x\in \left(\right)open="("\; close=")">\beta ,\infty $ the MLE of the $\beta $ parameter cannot be obtained in the usual way. Hence, the MLE of $\beta $ is the first order statistic ${X}_{\left(1\right)}$ (Johnson et al. 1994).

#### 5.2. Ordinary and Weighted Least Squares

## 6. Simulation Study

## 7. Real Data Modelling

## 8. Value-at-Risk Estimation with the BXP Distribution

#### 8.1. S&P-500

## 9. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Plots of the Burr XPareto (BXP) pdf (

**top**) and plots of the BXP hazard rate function (hrf) (

**bottom**).

**Figure 5.**Fitted pdfs (

**left**) and cdfs (

**right**) of the BXP and GP distribution for the S&P-500 dataset.

$(\mathit{\delta},\mathit{\alpha},\mathit{\beta})$ | $\mathit{\mu}$ | $\mathit{V}\mathit{a}\mathit{r}\left(\mathit{X}\right)$ | $\sqrt{{\mathit{\beta}}_{1}}$ | ${\mathit{\beta}}_{2}$ |
---|---|---|---|---|

(0.5, 0.5, 0.5) | 1.2801 | 1.1395 | 0.7311 | 4.4238 |

(1, 1, 1) | 1.6330 | 0.9671 | −1.2539 | 3.2132 |

(2, 2, 2) | 2.5365 | 1.7311 | −1.9644 | 4.3986 |

(1, 2, 3) | 2.9606 | 5.9323 | −0.8355 | 1.3785 |

(4, 2, 0.5) | 0.7411 | 0.0415 | −4.1218 | 17.7934 |

(10, 2, 0.25) | 0.4074 | 0.0011 | −6.3710 | 97.4674 |

(0.25, 5, 2) | 0.4962 | 1.2671 | 1.4287 | 2.6058 |

(0.9, 5, 1.8) | 1.0633 | 1.5440 | 0.0255 | 0.7191 |

**Table 2.**The empirical means, sds (given in $\left(\xb7\right)$), biases (given in $[\xb7]$) and MSEs (given in $\{\xb7\}$) for the special BXP distributions.

Parameters | $\mathit{n}=50$ | $\mathit{n}=100$ | $\mathit{n}=200$ | ||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{\delta},\mathit{\alpha},\mathit{\beta}$ | $\widehat{\mathit{\delta}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\beta}}$ | $\widehat{\mathit{\delta}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\beta}}$ | $\widehat{\mathit{\delta}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\beta}}$ |

3, 1.5, 2 | 3.0144 | 1.5495 | 2.0286 | 2.9995 | 1.5247 | 2.0159 | 3.0001 | 1.5125 | 2.0060 |

(0.1831) | (0.1585) | (0.1155) | (0.0399) | (0.1059) | (0.0757) | (0.0400) | (0.0648) | (0.0485) | |

[0.0144] | [0.0494] | [0.0286] | [−0.0005] | [0.0247] | [0.0160] | [0.0001] | [0.0125] | [0.0059] | |

{0.0330} | {0.0270} | {0.0140} | {0.0016} | {0.0117} | {0.0060} | {0.0016} | {0.0043} | {0.0023} | |

3, 2, 1 | 3.0772 | 2.0550 | 1.0040 | 3.0019 | 2.0093 | 1.0021 | 3.0016 | 2.0073 | 1.0013 |

(0.2928) | (0.2053) | (0.0443) | (0.0212) | (0.0976) | (0.0211) | (0.0203) | (0.0851) | (0.0182) | |

[0.0772] | [0.0550] | [0.0040] | [0.0019] | [0.0093] | [0.0021] | [0.0016] | [0.0073] | [0.0013] | |

{0.0900} | {0.0443} | {0.0020} | {0.0004} | {0.0095} | {0.0004} | {0.0004} | {0.0072} | {0.0003} | |

5, 0.5, 5 | 5.0863 | 0.5111 | 5.1216 | 5.0044 | 0.5012 | 5.0065 | 4.9954 | 0.4996 | 4.9970 |

(0.2792) | (0.0290) | (0.3641) | (0.0404) | (0.0095) | (0.0490) | (0.0400) | (0.0084) | (0.0439) | |

[0.0863] | [0.0111] | [0.1216] | [0.0044] | [0.0012] | [0.0065] | [−0.0046] | [−0.0004] | [−0.0030] | |

{0.0838} | {0.0010} | {0.1447} | {0.0072} | {0.00008} | {0.0071} | {0.0054} | {0.00007} | {0.0070} | |

10, 30, 20 | 10.0407 | 30.0438 | 20.0024 | 10.0009 | 30.0013 | 19.9998 | 9.9984 | 29.9980 | 20.0001 |

(0.2318) | (0.2809) | (0.0101) | (0.0110) | (0.0130) | (0.0086) | (0.0101) | (0.0120) | (0.0059) | |

[0.0406] | [0.0438] | [0.0024] | [0.0009] | [0.0013] | [−0.0002] | [−0.0016] | [−0.0020] | [0.0001] | |

{0.0543} | {0.0793} | {0.0001} | {0.0001} | {0.0001} | {0.00007} | {0.0001} | {0.0001} | {0.00004} | |

4, 0.5, 0.5 | 3.9077 | 0.5147 | 0.5265 | 4.0179 | 0.5121 | 0.5203 | 4.0012 | 0.5052 | 0.5079 |

(0.1261) | (0.0532) | (0.0926) | (0.1010) | (0.0411) | (0.0711) | (0.0878) | (0.0246) | (0.0440) | |

[−0.0923] | [0.0147] | [0.0265] | [0.0179] | [0.0121] | [0.0203] | [0.0012] | [0.0052] | [0.0079] | |

{0.0356} | {0.0030} | {0.0100} | {0.0164} | {0.0018} | {0.0054} | {0.0076} | {0.0006} | {0.0019} |

**Table 3.**MLEs and their standard errors (in parentheses) for both datasets. P, Pareto; PAT, P ArcTan; KwP, Kumaraswamy P; WP, Weibull P; BP, Beta P; EP, Exponentiated P.

Leukaemia Data | ||||

Model | $\widehat{\mathit{\delta}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{a}}$ | $\widehat{\mathit{\beta}}$ |

BXP | 0.8505 | 0.1900 | 1 | |

(0.1785) | (0.0146) | |||

PAT | 0.8603 | 12.6124 | 1 | |

(0.1428) | (6.6619) | |||

KwP | 2.3992 | 0.0007 | 1,828,015 | 1 |

(0.0291) | (0.0001) | (5.9317) | ||

WP | 1.8274 | 0.1994 | 1 | |

(0.2846) | (0.0145) | |||

BP | 51.9800 | 0.0239 | 3.8540 | 1 |

(0.1240) | (0.0048) | (0.6551) | ||

EP | 4.3606 | 0.7089 | 1 | |

(1.3221) | (0.1192) | |||

P | 0.3319 | 1 | ||

(0.0596) | ||||

Earthquake Data | ||||

BXP | 1.9916 | 0.1678 | 9 | |

(0.5622) | (0.0117) | |||

PAT | 1.1704 | 168.1574 | 9 | |

(0.0667) | (5.9619) | |||

WP | 2.9843 | 0.1408 | 9 | |

(0.4949) | (0.0074) | |||

BP | 60.8341 | 0.0428 | 12.5592 | 9 |

(1.0981) | (0.0053) | (0.9570) | ||

EP | 26.9837 | 0.8707 | 9 | |

(5.7196) | (0.0770) | |||

P | 0.2264 | 9 | ||

(0.0472) |

**Table 4.**Goodness-of-fit statistics for both datasets. CAIC, Corrected Akaike Information Criterion; HQIC, Hannan–Quinn Information Criterion.

Leukaemia Data | |||||
---|---|---|---|---|---|

Model | AIC | CAIC | BIC | HQIC | KS |

BXP | 295.0115 | 295.4401 | 297.8795 | 295.9464 | 0.1328 |

PAT | 301.1477 | 301.5763 | 304.0157 | 302.0826 | 0.1398 |

KwP | 298.9148 | 299.8037 | 303.2167 | 300.3171 | 0.1486 |

WP | 295.2830 | 295.7116 | 298.1510 | 296.2179 | 0.1418 |

BP | 301.5970 | 302.4859 | 305.8990 | 302.9994 | 0.1494 |

EP | 300.9643 | 301.3929 | 303.8323 | 301.8992 | 0.1630 |

P | 319.1294 | 319.2673 | 320.5634 | 319.5968 | 0.2733 |

Earthquake Data | |||||

BXP | 381.9004 | 382.5004 | 384.1714 | 382.4715 | 0.0817 |

PAT | 383.7187 | 384.3187 | 385.9897 | 384.2899 | 0.0971 |

WP | 382.3901 | 382.9901 | 384.6610 | 382.9612 | 0.0962 |

BP | 384.5029 | 385.7661 | 387.9094 | 385.3597 | 0.0819 |

EP | 384.3233 | 384.9233 | 386.5943 | 384.8944 | 0.1038 |

P | 420.6338 | 420.8243 | 421.7693 | 420.9194 | 0.4218 |

Descriptive Statistics | S&P-500 |
---|---|

Number of observations | 1465 |

Minimum | −0.0402 |

Maximum | 0.0383 |

Mean | 0.0004 |

Median | 0.0004 |

Std.Deviation | 0.007 |

Skewness | −0.322 |

Kurtosis | 5.403 |

Jarque–Bera | 377.839 (<0.001) |

Ljung–Box | 28.516 (0.098) |

**Table 6.**MLEs, corresponding standard errors (in second line) and goodness-of-fit statistics for the S&P-500.

Models | Parameters | Goodness-of-Fit | |||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{\xi}$ | $\mathit{\delta}$ | $\mathit{\sigma}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | $-\mathit{\ell}$ | KS | ${\mathit{A}}^{*}$ | ${\mathit{W}}^{*}$ | |

BXP | 3.2480 | 0.1893 | 4.89818 × 10${}^{-5}$ | −93.4016 | 0.1427 | 0.3809 | 0.0556 | ||

1.0266 | 0.0120 | - | |||||||

GP | 0.0847 | 0.0057 | −88.7171 | 0.1498 | 0.4039 | 0.0661 | |||

0.1996 | 0.0015 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Korkmaz, M.Ç.; Altun, E.; Yousof, H.M.; Afify, A.Z.; Nadarajah, S.
The Burr X Pareto Distribution: Properties, Applications and VaR Estimation. *J. Risk Financial Manag.* **2018**, *11*, 1.
https://doi.org/10.3390/jrfm11010001

**AMA Style**

Korkmaz MÇ, Altun E, Yousof HM, Afify AZ, Nadarajah S.
The Burr X Pareto Distribution: Properties, Applications and VaR Estimation. *Journal of Risk and Financial Management*. 2018; 11(1):1.
https://doi.org/10.3390/jrfm11010001

**Chicago/Turabian Style**

Korkmaz, Mustafa Ç., Emrah Altun, Haitham M. Yousof, Ahmed Z. Afify, and Saralees Nadarajah.
2018. "The Burr X Pareto Distribution: Properties, Applications and VaR Estimation" *Journal of Risk and Financial Management* 11, no. 1: 1.
https://doi.org/10.3390/jrfm11010001