# Half Logistic Inverse Lomax Distribution with Applications

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Important Functions

#### 2.1. The Probability Density Function

#### 2.2. Reliability Functions

#### 2.3. Quantile Function with Applications

## 3. Statistical Properties

#### 3.1. Stochastic Orders

#### 3.2. Two Important Representations

#### 3.3. Moments

- When $\lambda $ and $\beta $ are constant and $\alpha $ increases, the considered measures increase.
- When $\lambda $ and $\alpha $ are constant and $\beta $ increases, the considered measures decrease.
- The HLIL distribution is mainly right-skewed with consequent variations on the skewness coefficient.
- The HLIL distribution is mainly leptokurtic; the value of the kurtosis can be close to 3 (the mesokurtic case) and have very large value; It is observed the value of $114.842$ for $\lambda =0.5$, $\alpha =0.5$ and $\beta =5$.

#### 3.4. Incomplete Moments

#### 3.5. Rényi and q-Entropies

#### 3.6. Order Statistics

## 4. Different Methods of Estimation and Simulation

#### 4.1. Maximum Likelihood Estimates (MLEs)

#### 4.2. Ordinary and Weighted Least Squares Estimates (LSEs) and (WLSEs)

#### 4.3. Percentile Estimates (PCEs)

#### 4.4. Cramér-Von Mises Minimum Distance Estimates (CVEs)

#### 4.5. Maximum Product of Spacings Estimates (MPSEs)

#### 4.6. Numerical Results

- For all estimates, as expected, the MSEs decrease as sample sizes increase.
- The MSEs of the MPSEs of $\mathbf{\varphi}$ take the lowest value among the corresponding MSEs for the other methods in almost all cases. Other details are given below.
- –
- For the MLEs, when $\beta $ increases, the MSEs for $\alpha $ and $\lambda $ decrease but the MSE for $\beta $ is increasing.
- –
- For the LSEs, when $\beta $ increases, the MSEs for $\alpha $, $\lambda $ and $\beta $ are increasing.
- –
- For the WLSEs, when $\beta $ increases, the MSEs for $\alpha $ and $\lambda $ are decreasing but the MSE for $\beta $ is increasing.
- –
- For the CVEs, when $\beta $ increases, the MSEs for $\alpha $ and $\lambda $ are decreasing but the MSE for $\beta $ is increasing
- –
- For the the PCEs, when $\beta $ increases, the MSEs for $\alpha $, $\lambda $ and $\beta $ are increasing except for $\alpha $ at $n=100$ where they decrease.
- –
- For the MPSEs, when $\beta $ increases, the MSEs for $\alpha $ and $\lambda $ are decreasing but the MSE for $\beta $ is increasing.

- when $\delta $ increases, the RBs of the Rényi entropy decrease.
- when $\beta $ increases, the RBs of the Rényi entropy decrease.

## 5. Applications

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof of**

**(7).**

**Proof of**

**(8).**

**Proof of**

**(9).**

**Proof of**

**(10).**

**Proof of**

**(13).**

**Proof of**

**(14).**

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**Figure 1.**Plots of the pdf of the half logistic inverse Lomax (HLIL) distribution for various choices of $\alpha $, $\beta $ and $\lambda $.

**Figure 3.**Plots of the hrf of the HLIL distribution for various choices of $\alpha $, $\beta $ and $\lambda $.

**Table 1.**Moments, variance, skewness and kurtosis of the HLIL distribution for various values of the parameters.

${\mathit{\mu}}_{\mathit{r}}^{\prime}$ | ($\mathit{\alpha}=0.5,\mathit{\beta}=5$) | ($\mathit{\alpha}=1,\mathit{\beta}=5$) | ($\mathit{\alpha}=1.5,\mathit{\beta}=5$) | ($\mathit{\alpha}=2,\mathit{\beta}=5$) | ($\mathit{\alpha}=2.5,\mathit{\beta}=5$) | ($\mathit{\alpha}=3,\mathit{\beta}=5$) |
---|---|---|---|---|---|---|

${\mu}_{1}^{\prime}$ | 0.191 | 0.723 | 1.359 | 2.033 | 2.725 | 3.426 |

${\mu}_{2}^{\prime}$ | 0.14 | 1.17 | 3.487 | 7.191 | 12.312 | 18.864 |

${\mu}_{3}^{\prime}$ | 0.26 | 3.842 | 16.612 | 45.03 | 95.671 | 175.138 |

${\mu}_{4}^{\prime}$ | 1.399 | 31.772 | 186.592 | 640.474 | 1648 | 3545 |

${\sigma}^{2}$ | 0.103 | 0.648 | 1.64 | 3.056 | 4.886 | 7.126 |

skewness | 5.835 | 3.955 | 3.53 | 3.366 | 3.287 | 3.242 |

kurtosis | 114.842 | 56.072 | 46.356 | 42.979 | 41.411 | 40.557 |

**Table 2.**Moments, variance, skewness and kurtosis of the HLIL distribution for various values of the parameters.

${\mathit{\mu}}_{\mathit{r}}^{\prime}$ | ($\mathit{\alpha}=3,\mathit{\beta}=6$) | ($\mathit{\alpha}=3,\mathit{\beta}=8$) | ($\mathit{\alpha}=3,\mathit{\beta}=10$) | ($\mathit{\alpha}=3,\mathit{\beta}=12$) | ($\mathit{\alpha}=3,\mathit{\beta}=15$) | ($\mathit{\alpha}=3,\mathit{\beta}=20$) |
---|---|---|---|---|---|---|

${\mu}_{1}^{\prime}$ | 2.923 | 2.331 | 1.988 | 1.76 | 1.53 | 1.292 |

${\mu}_{2}^{\prime}$ | 12.846 | 7.597 | 5.31 | 4.058 | 2.989 | 2.077 |

${\mu}_{3}^{\prime}$ | 84.961 | 33.621 | 18.34 | 11.753 | 7.142 | 3.982 |

${\mu}_{4}^{\prime}$ | 923.833 | 204.7 | 81.428 | 42.193 | 20.498 | 8.918 |

${\sigma}^{2}$ | 4.304 | 2.163 | 1.357 | 0.959 | 0.648 | 0.408 |

skewness | 2.493 | 1.83 | 1.51 | 1.313 | 1.124 | 0.936 |

kurtosis | 19.977 | 10.753 | 7.947 | 6.611 | 5.554 | 4.697 |

$\mathit{\alpha}$ | $\mathit{\lambda}$ | $\mathit{\beta}$ | Rényi Entropy | ||
---|---|---|---|---|---|

$\mathit{\delta}=\mathbf{1.2}$ | $\mathit{\delta}=\mathbf{1.5}$ | $\mathit{\delta}=\mathbf{2}$ | |||

1.2 | 0.5 | 1.5 | 141.958 | 78.722 | 57.638 |

1.5 | 0.5 | 1.5 | 141.07 | 78.279 | 57.343 |

2 | 0.5 | 1.5 | 139.945 | 77.716 | 56.968 |

3 | 0.5 | 1.5 | 138.36 | 76.924 | 56.44 |

4 | 0.5 | 1.5 | 137.236 | 76.362 | 56.065 |

5 | 0.5 | 1.5 | 136.364 | 75.926 | 55.774 |

1.5 | 0.5 | 1.8 | 166.843 | 91.16 | 65.929 |

1.5 | 0.5 | 2 | 184.062 | 99.767 | 71.667 |

1.5 | 0.5 | 3 | 270.459 | 142.963 | 100.464 |

1.5 | 0.5 | 4 | 357.156 | 186.311 | 129.363 |

1.5 | 0.5 | 5 | 444.021 | 229.744 | 158.318 |

$\mathit{\alpha}$ | $\mathit{\lambda}$ | $\mathit{\beta}$ | q-Entropy | ||
---|---|---|---|---|---|

$\mathit{q}=\mathbf{1.2}$ | $\mathit{q}=\mathbf{1.5}$ | $\mathit{q}=\mathbf{2}$ | |||

1.2 | 2 | 1.5 | 1.027 | 0.757 | 0.529 |

1.5 | 2 | 1.5 | 1.230 | 0.914 | 0.643 |

2 | 2 | 1.5 | 1.462 | 1.075 | 0.743 |

3 | 2 | 1.5 | 1.752 | 1.254 | 0.833 |

4 | 2 | 1.5 | 1.938 | 1.357 | 0.876 |

5 | 2 | 1.5 | 2.074 | 1.426 | 0.901 |

1.5 | 2 | 1.8 | 1.002 | 0.759 | 0.546 |

1.5 | 2 | 2 | 0.873 | 0.665 | 0.481 |

1.5 | 2 | 3 | 0.385 | 0.267 | 0.155 |

1.5 | 2 | 4 | 0.048 | −0.048 | −0.160 |

1.5 | 2 | 5 | −0.209 | −0.312 | −0.464 |

**Table 5.**Estimates and mean square errors (MSEs) of HLIL distribution for maximum likelihood (ML), least squares (LS), weighted least squares (WLS), percentile (PC), and maximum product of spacing (MPS) estimates for Set1: $(\alpha =2,\lambda =0.5,\beta =2)$.

n | MLE | LSE | WLSE | CVE | PCE | MPSE | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | |

1.964 | 1.985 | 2.34 | 3.311 | 2.33 | 0.96 | 2.625 | 3.398 | 2.083 | 1.443 | 2.257 | 2.645 | |

100 | 0.993 | 1.205 | 0.664 | 0.875 | 0.681 | 0.232 | 0.846 | 1.088 | 0.615 | 0.657 | 0.779 | 0.559 |

2.223 | 0.155 | 2.098 | 0.216 | 1.969 | 0.185 | 2.01 | 0.223 | 2.334 | 0.72 | 2.037 | 0.104 | |

2.051 | 0.922 | 2.091 | 0.343 | 2.171 | 0.275 | 2.465 | 1.143 | 1.876 | 0.379 | 1.874 | 0.913 | |

200 | 0.646 | 0.138 | 0.506 | 0.042 | 0.575 | 0.087 | 0.712 | 0.376 | 0.474 | 0.142 | 0.717 | 0.184 |

2.082 | 0.045 | 2.131 | 0.142 | 2.064 | 0.094 | 2.045 | 0.108 | 2.333 | 0.611 | 2.066 | 0.046 | |

2.175 | 0.68 | 2.1 | 0.269 | 2.035 | 0.221 | 2.063 | 0.192 | 1.873 | 0.303 | 1.938 | 0.573 | |

300 | 0.552 | 0.076 | 0.538 | 0.041 | 0.538 | 0.07 | 0.512 | 0.025 | 0.458 | 0.091 | 0.64 | 0.112 |

2.028 | 0.029 | 2.074 | 0.088 | 2.021 | 0.073 | 2.076 | 0.09 | 2.275 | 0.414 | 2.034 | 0.025 | |

2.04 | 0.061 | 2.063 | 0.076 | 1.968 | 0.018 | 2.136 | 0.09 | 1.947 | 0.108 | 1.976 | 0.035 | |

1000 | 0.518 | 0.013 | 0.527 | 0.011 | 0.472 | 0.005 | 0.553 | 0.015 | 0.48 | 0.037 | 0.503 | 0.008 |

2.011 | 0.023 | 1.994 | 0.022 | 2.06 | 0.019 | 1.961 | 0.016 | 2.137 | 0.175 | 1.974 | 0.023 | |

2.046 | 0.026 | 2.086 | 0.061 | 1.979 | 0.015 | 2.036 | 0.016 | 1.979 | 0.052 | 1.942 | 0.023 | |

2000 | 0.519 | 0.006 | 0.539 | 0.01 | 0.489 | 0.004 | 0.511 | 0.004 | 0.488 | 0.018 | 0.474 | 0.004 |

1.994 | 0.008 | 1.979 | 0.011 | 2.03 | 0.013 | 2.012 | 0.012 | 2.099 | 0.14 | 2.037 | 0.009 |

**Table 6.**Estimates and MSEs of HLIL distribution for ML, LS, WLS, CV, PC and MPS estimates for Set2: $(\alpha =2,\lambda =0.5,\beta =3)$.

n | MLE | LSE | WLSE | CVE | PCE | MPSE | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | |

1.957 | 1.01 | 3.141 | 3.554 | 2.302 | 0.621 | 2.447 | 2.234 | 2.023 | 1.266 | 2.182 | 1.843 | |

100 | 0.783 | 0.449 | 1.05 | 0.931 | 0.639 | 0.1 | 0.705 | 0.505 | 0.64 | 0.761 | 0.682 | 0.227 |

3.198 | 0.223 | 2.966 | 0.505 | 2.951 | 0.295 | 2.939 | 0.363 | 3.32 | 1.647 | 3.037 | 0.156 | |

1.951 | 0.646 | 2.571 | 1.413 | 2.156 | 0.27 | 2.224 | 0.316 | 2.076 | 0.898 | 1.854 | 0.539 | |

200 | 0.659 | 0.134 | 0.761 | 0.268 | 0.583 | 0.062 | 0.635 | 0.1 | 0.579 | 0.394 | 0.696 | 0.19 |

3.103 | 0.079 | 2.857 | 0.262 | 2.975 | 0.265 | 2.958 | 0.351 | 3.39 | 1.216 | 3.075 | 0.086 | |

1.86 | 0.371 | 2.306 | 0.651 | 2.182 | 0.113 | 2.297 | 0.18 | 2.041 | 0.288 | 1.763 | 0.3 | |

300 | 0.626 | 0.066 | 0.7 | 0.253 | 0.605 | 0.025 | 0.638 | 0.041 | 0.575 | 0.127 | 0.67 | 0.117 |

3.092 | 0.051 | 2.86 | 0.257 | 2.806 | 0.122 | 2.888 | 0.215 | 3.134 | 1.039 | 3.082 | 0.057 | |

2.039 | 0.05 | 2.126 | 0.048 | 2.128 | 0.044 | 2.202 | 0.08 | 2.013 | 0.097 | 1.982 | 0.028 | |

1000 | 0.52 | 0.013 | 0.583 | 0.014 | 0.579 | 0.014 | 0.611 | 0.02 | 0.528 | 0.037 | 0.506 | 0.008 |

3.015 | 0.042 | 2.792 | 0.104 | 2.874 | 0.108 | 2.836 | 0.095 | 2.992 | 0.327 | 2.977 | 0.041 | |

2.044 | 0.022 | 2.117 | 0.039 | 2.113 | 0.024 | 2.136 | 0.042 | 2.003 | 0.043 | 1.957 | 0.021 | |

2000 | 0.521 | 0.006 | 0.589 | 0.014 | 0.588 | 0.014 | 0.578 | 0.012 | 0.513 | 0.017 | 0.478 | 0.005 |

2.984 | 0.032 | 2.835 | 0.06 | 2.837 | 0.048 | 2.843 | 0.066 | 3.003 | 0.271 | 3.069 | 0.035 |

n | $\mathit{\delta}=0.5$ | $\mathit{\delta}=1.5$ | $\mathit{\delta}=2.5$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Exact Value | Estimates | RB | Exact Value | Estimates | RB | Exact Value | Estimates | RB | |

100 | −16.078 | 0.32 | 110.569 | 0.117 | 68.301 | 0.104 | |||

200 | −13.631 | 0.119 | 103.169 | 0.042 | 64.19 | 0.037 | |||

300 | −12.18 | −12.627 | 0.037 | 99.018 | 100.113 | 0.011 | 61.884 | 62.492 | 0.01 |

1000 | −12.406 | 0.019 | 99.467 | 0.005 | 62.134 | 0.004 | |||

2000 | −12.162 | 0.001 | 98.741 | 0.003 | 61.73 | 0.002 |

n | $\mathit{\delta}=0.5$ | $\mathit{\delta}=1.5$ | $\mathit{\delta}=2.5$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Exact Value | Estimates | RB | Exact Value | Estimates | RB | Exact Value | Estimates | RB | |

100 | −30.078 | 0.132 | 152.297 | 0.073 | 91.483 | 0.068 | |||

200 | −28.502 | 0.072 | 147.556 | 0.04 | 88.849 | 0.037 | |||

300 | −26.58 | −28.237 | 0.062 | 141.838 | 146.74 | 0.035 | 85.673 | 88.396 | 0.032 |

1000 | −26.808 | 0.009 | 142.427 | 0.004 | 86 | 0.004 | |||

2000 | −26.363 | 0.008 | 141.198 | 0.004 | 85.361 | 0.004 |

**Table 9.**Maximum Likelihood Estimates (MLEs) with their standard errors (SEs) for the first data set.

Model | MLEs and SEs | |||
---|---|---|---|---|

HLIL | 3.3774 | 7.9710 | 0.8085 | - |

($\alpha ,\beta ,\lambda $) | (0.1673) | (0.0245) | (0.4986) | - |

EPL | 25.2967 | 0.7590 | 12.7190 | 7.9758 |

($\alpha ,\beta ,\lambda ,a$) | (4.0719) | (0.0341) | (12.6748) | (0.1906) |

WIL | 0.0024 | 2.0529 | 0.0604 | 2.5469 |

($a,b,\lambda ,\beta $) | (0.0005) | (0.2048) | (0.0216) | (0.0501) |

TIL | 11.6699 | 0.3946 | 1.0885 | - |

($\alpha ,\beta ,\lambda $) | (4.6202) | (0.6050) | (3.0847) | - |

IL | 0.0210 | 54.8749 | - | - |

($\alpha ,\lambda $) | (0.0348) | (0.0946) | - | - |

Model | MLEs and SEs | |||
---|---|---|---|---|

HLIL | 2.2493 | 139.8835 | 0.0532 | - |

($\alpha ,\beta ,\lambda $) | (0.2886) | (7.0477) | (0.0337) | - |

KwL | 3.2372 | 11.9966 | 3.2456 | 9.1600 |

($\alpha ,\beta ,a,b$) | (2.4802) | (16.4620) | (1.0686) | (5.3663) |

BL | 165.2269 | 5.1558 | 3.5609 | 43.6788 |

($c,k,a,b$) | (5.9677) | (0.6954) | (0.5250) | (8.0152) |

WIL | 0.0027 | 1.5655 | 0.0606 | 1.1832 |

($a,b,\lambda ,\beta $) | (0.0004) | (0.2026) | (0.0179) | (0.8351) |

TIL | 3.9207 | 1.4443 | 1.0833 | - |

($\alpha ,\beta ,\lambda $) | (3.5674) | (0.6032) | (0.8878) | - |

IL | 0.5031 | 4.0306 | - | - |

($\alpha ,\lambda $) | (0.1892) | (1.2907) | - | - |

Model | MLEs and SEs | |||
---|---|---|---|---|

HLIL | 37.9868 | 12.1643 | 0.2301 | - |

($\alpha ,\beta ,\lambda $) | (3.9893) | (1.2670) | (0.2570) | - |

EPL | 25.5454 | 0.6161 | 96.1370 | 15.6028 |

($\alpha ,\beta ,\lambda ,a$) | (2.7281) | (0.7262) | (0.3338) | (3.0301) |

WIL | 0.0026 | 0.6577 | 0.0116 | 0.5668 |

($a,b,\lambda ,\beta $) | (0.0005) | (0.0476) | (0.0044) | (0.2457) |

TIL | 56.5719 | 1.1760 | 7.0916 | - |

($\alpha ,\beta ,\lambda $) | (5.3292) | (0.8181) | (0.0656) | - |

IL | 1.5381 | 40.4936 | - | - |

($\beta ,\lambda $) | (1.2669) | (3.3980) | - | - |

**Table 12.**The Akaike information criterion (AIC), Bayesian information criterion (BIC), Anderson-Darling (AD), and Cramér-von Mises (CVM) values for the first data set.

Model | AIC | BIC | AD | CVM |
---|---|---|---|---|

HLIL | 82.3694 | 86.5730 | 0.1070 | 0.0147 |

EPL | 84.8304 | 90.4352 | 0.1760 | 0.0272 |

WIL | 96.1966 | 101.8014 | 0.2459 | 0.0329 |

TIL | 86.9198 | 91.1234 | 0.4658 | 0.0751 |

IL | 96.8127 | 99.6151 | 0.5066 | 0.0817 |

Model | AIC | BIC | AD | CVM |
---|---|---|---|---|

HLIL | 268.8231 | 276.1510 | 0.5765 | 0.0567 |

KwL | 280.4553 | 290.2259 | 1.0633 | 0.1175 |

BL | 285.2487 | 295.0193 | 1.3961 | 0.1662 |

WIL | 331.6880 | 341.4586 | 0.5802 | 0.0552 |

TIL | 328.0861 | 335.4140 | 3.9985 | 0.5999 |

IL | 365.4879 | 370.3732 | 5.4250 | 0.8625 |

Model | AIC | BIC | AD | CVM |
---|---|---|---|---|

HLIL | 537.6881 | 543.8695 | 0.7968 | 0.1396 |

EPL | 565.9949 | 574.2367 | 1.0505 | 0.1939 |

WIL | 670.4852 | 678.7270 | 1.5208 | 0.2809 |

TIL | 559.5491 | 565.7304 | 0.8901 | 0.1580 |

IL | 613.7669 | 617.8878 | 1.2920 | 0.2223 |

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## Share and Cite

**MDPI and ACS Style**

Al-Marzouki, S.; Jamal, F.; Chesneau, C.; Elgarhy, M.
Half Logistic Inverse Lomax Distribution with Applications. *Symmetry* **2021**, *13*, 309.
https://doi.org/10.3390/sym13020309

**AMA Style**

Al-Marzouki S, Jamal F, Chesneau C, Elgarhy M.
Half Logistic Inverse Lomax Distribution with Applications. *Symmetry*. 2021; 13(2):309.
https://doi.org/10.3390/sym13020309

**Chicago/Turabian Style**

Al-Marzouki, Sanaa, Farrukh Jamal, Christophe Chesneau, and Mohammed Elgarhy.
2021. "Half Logistic Inverse Lomax Distribution with Applications" *Symmetry* 13, no. 2: 309.
https://doi.org/10.3390/sym13020309