# The Truncated Cauchy Power Family of Distributions with Inference and Applications

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## Abstract

**:**

## 1. Introduction

## 2. The TCP-G Family

#### 2.1. Probability Density Function

#### 2.2. Hazard Rate Function

#### 2.3. Quantile Function

#### 2.4. Example: The Truncated Cauchy Power Weibull Distribution

## 3. Notable Properties

#### 3.1. Linear Representations

#### 3.2. On Moments and Related Functions

#### 3.3. Skewness and Kurtosis Based on Quantiles

#### 3.4. Rényi Entropy and q-Entropy

#### 3.5. Order Statistics

## 4. Estimation of the TCP-G Model Parameters

#### 4.1. Maximum Likelihood Method under SRS

#### 4.2. Maximum Likelihood Method under RSS

#### 4.3. Simulation Study

- Step 1: We consider $n=100$, 200 and 300.
- Step 2: The parameters values are selected as
- Set1: $(\alpha =0.5,\lambda =1.5,\theta =0.5)$,
- Set2: $(\alpha =1.2,\lambda =1.5,\theta =0.5)$,
- Set3: $(\alpha =1.2,\lambda =1.5,\theta =0.75)$,
- Set4: $(\alpha =0.5,\lambda =1.5,\theta =0.75)$.

- Step 3: For the chosen set of parameters and each sample of size n, the MLEs are computed under SRS and RSS as described in the above subsection.
- Step 4: Repeat the previous steps from 1 to 3, N times representing with different samples, where $N=1000$. Then, MSEs and REs are computed.
- Step 5: The LB, UB and AL for selected values of parameters are calculated based on SRS and RSS.

- For both of the sampling schemes, the MSEs decrease as n increases.
- For both of the sampling schemes, the AL of the CI become decreases as n increases.
- The estimates based on RSS have smaller MSE than the corresponding based on SRS. For this reason, in case of a high level of precision is required, RSS is preferable.

## 5. Application to Two Practical Data Sets

## 6. Concluding Remarks and Perspectives

- the cdf given by$$\begin{array}{c}\hfill F(x;\alpha ,\beta ,\xi )={\left\{\frac{4}{\pi}arctan\left[G{(x;\xi )}^{\alpha}\right]\right\}}^{\beta},\phantom{\rule{1.em}{0ex}}x\in \mathbb{R},\end{array}$$
- the cdf given by$$\begin{array}{c}\hfill F(x;\alpha ,\lambda ,\xi )=\frac{1}{arctan\left(\lambda \right)}arctan\left[\lambda G{(x;\xi )}^{\alpha}\right],\phantom{\rule{1.em}{0ex}}x\in \mathbb{R},\end{array}$$

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Plots of the mean and variance for the TCPW distribution: (

**a**) for fixed $\lambda $ and $\theta $ and varying $\alpha $ and (

**b**) for fixed $\theta $ and $\alpha $ and varying $\lambda $.

**Figure 4.**Plots of the MacGillivray skewness for selected values of the parameters when (

**a**) $\alpha $ increases and (

**b**) $\theta $ increases.

**Figure 5.**Plots of Galton skewness for selected values of the parameters when (

**a**) $\alpha $ varies and (

**b**) $\theta $ varies.

**Figure 6.**Plots of Moors kurtosis for selected values of the parameters when (

**a**) $\alpha $ varies and (

**b**) $\theta $ varies.

**Table 1.**Estimates, mean squared errors (MSEs) and relative efficients (REs) for Set1: $(\alpha =0.5,\lambda =1.5,\theta =0.5)$.

n | SRS | RSS | RE | ||
---|---|---|---|---|---|

MLE | MSE | MLE | MSE | ||

100 | 0.508 | 0.033 | 0.512 | 0.023 | 0.697 |

1.443 | 0.140 | 1.509 | 0.035 | 0.247 | |

0.564 | 0.044 | 0.524 | 0.012 | 0.266 | |

200 | 0.499 | 0.024 | 0.447 | 0.004 | 0.173 |

1.501 | 0.110 | 1.441 | 0.008 | 0.072 | |

0.539 | 0.022 | 0.552 | 0.004 | 0.181 | |

300 | 0.492 | 0.020 | 0.521 | 0.002 | 0.091 |

1.519 | 0.102 | 1.527 | 0.003 | 0.032 | |

0.533 | 0.016 | 0.486 | 0.001 | 0.052 |

**Table 2.**Estimates, mean squared errors (MSEs) and relative efficients (REs) for Set2: $(\alpha =1.2,\lambda =1.5,\theta =0.5)$.

n | SRS | RSS | RE | ||
---|---|---|---|---|---|

MLE | MSE | MLE | MSE | ||

100 | 1.846 | 2.534 | 1.115 | 0.047 | 0.019 |

1.747 | 0.645 | 1.428 | 0.028 | 0.044 | |

0.481 | 0.027 | 0.541 | 0.005 | 0.198 | |

200 | 1.201 | 0.158 | 1.225 | 0.030 | 0.190 |

1.371 | 0.123 | 1.509 | 0.016 | 0.128 | |

0.519 | 0.008 | 0.500 | 0.002 | 0.242 | |

300 | 1.179 | 0.054 | 1.224 | 0.012 | 0.215 |

1.449 | 0.034 | 1.517 | 0.007 | 0.203 | |

0.521 | 0.004 | 0.498 | 0.001 | 0.151 |

**Table 3.**Estimates, mean squared errors (MSEs) and relative efficients (REs) for Set3: $(\alpha =1.2,\lambda =1.5,\theta =0.75)$.

n | SRS | RSS | RE | ||
---|---|---|---|---|---|

MLE | MSE | MLE | MSE | ||

100 | 1.253 | 0.761 | 1.326 | 0.154 | 0.202 |

1.412 | 0.373 | 1.562 | 0.060 | 0.161 | |

0.945 | 0.188 | 0.740 | 0.014 | 0.072 | |

200 | 1.271 | 0.268 | 1.209 | 0.022 | 0.084 |

1.547 | 0.186 | 1.501 | 0.012 | 0.064 | |

0.800 | 0.043 | 0.752 | 0.003 | 0.069 | |

300 | 1.148 | 0.091 | 1.118 | 0.010 | 0.115 |

1.426 | 0.047 | 1.431 | 0.007 | 0.137 | |

0.787 | 0.015 | 0.780 | 0.002 | 0.117 |

**Table 4.**Estimates, mean squared errors (MSEs) and relative efficients (REs) for Set4: $(\alpha =0.5,\lambda =1.5,\theta =0.75)$.

n | SRS | RSS | RE | ||
---|---|---|---|---|---|

MLE | MSE | MLE | MSE | ||

100 | 0.576 | 0.078 | 0.475 | 0.018 | 0.236 |

1.599 | 0.239 | 1.464 | 0.040 | 0.167 | |

0.848 | 0.108 | 0.843 | 0.051 | 0.470 | |

200 | 0.481 | 0.024 | 0.504 | 0.004 | 0.182 |

1.533 | 0.100 | 1.490 | 0.009 | 0.090 | |

0.859 | 0.057 | 0.751 | 0.004 | 0.069 | |

300 | 0.472 | 0.013 | 0.496 | 0.001 | 0.059 |

1.473 | 0.043 | 1.486 | 0.001 | 0.033 | |

0.795 | 0.026 | 0.754 | 0.001 | 0.037 |

**Table 5.**Lower bounds (LBs), upper bounds (UBs), and average lengths (ALs) based on simple random sampling (SRS) and ranked set sampling (RSS) for Set1: $(\alpha =0.5,\lambda =1.5,\theta =0.5)$.

n | SRS | RSS | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

90% | 95% | 90% | 95% | |||||||||

LB | UB | AL | LB | UB | AL | LB | UB | AL | LB | UB | AL | |

100 | 0.124 | 0.892 | 0.768 | 0.050 | 0.965 | 0.915 | 0.141 | 0.882 | 0.742 | 0.070 | 0.953 | 0.884 |

0.649 | 2.237 | 1.588 | 0.497 | 2.389 | 1.892 | 0.733 | 2.284 | 1.551 | 0.585 | 2.433 | 1.848 | |

0.205 | 0.923 | 0.718 | 0.136 | 0.992 | 0.856 | 0.233 | 0.816 | 0.583 | 0.177 | 0.871 | 0.695 | |

200 | 0.228 | 0.770 | 0.542 | 0.177 | 0.822 | 0.646 | 0.219 | 0.676 | 0.457 | 0.175 | 0.720 | 0.545 |

0.939 | 2.064 | 1.125 | 0.831 | 2.172 | 1.341 | 0.897 | 1.984 | 1.087 | 0.793 | 2.088 | 1.295 | |

0.317 | 0.761 | 0.444 | 0.274 | 0.804 | 0.530 | 0.338 | 0.766 | 0.429 | 0.297 | 0.807 | 0.511 | |

300 | 0.284 | 0.700 | 0.416 | 0.244 | 0.740 | 0.496 | 0.313 | 0.729 | 0.416 | 0.273 | 0.769 | 0.496 |

1.073 | 1.965 | 0.892 | 0.987 | 2.051 | 1.063 | 1.098 | 1.955 | 0.858 | 1.016 | 2.037 | 1.022 | |

0.362 | 0.704 | 0.341 | 0.330 | 0.736 | 0.407 | 0.344 | 0.627 | 0.284 | 0.317 | 0.654 | 0.338 |

**Table 6.**Lower bounds (LBs), upper bounds (UBs), and average lengths (ALs) based on simple random sampling (SRS) and ranked set sampling (RSS) for Set2: $(\alpha =1.2,\lambda =1.5,\theta =0.5)$.

n | SRS | RSS | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

90% | 95% | 90% | 95% | |||||||||

LB | UB | AL | LB | UB | AL | LB | UB | AL | LB | UB | AL | |

100 | −0.097 | 3.789 | 3.886 | −0.469 | 4.161 | 4.630 | 0.172 | 2.058 | 1.886 | −0.008 | 2.239 | 2.247 |

0.848 | 2.646 | 1.798 | 0.676 | 2.818 | 2.142 | 0.600 | 2.257 | 1.657 | 0.441 | 2.416 | 1.974 | |

0.235 | 0.726 | 0.491 | 0.188 | 0.774 | 0.585 | 0.261 | 0.820 | 0.559 | 0.208 | 0.874 | 0.666 | |

200 | 0.524 | 1.879 | 1.355 | 0.394 | 2.008 | 1.614 | 0.544 | 1.906 | 1.362 | 0.414 | 2.037 | 1.623 |

0.835 | 1.908 | 1.073 | 0.732 | 2.011 | 1.279 | 0.953 | 2.065 | 1.112 | 0.847 | 2.172 | 1.325 | |

0.346 | 0.692 | 0.346 | 0.312 | 0.725 | 0.413 | 0.333 | 0.666 | 0.333 | 0.301 | 0.698 | 0.397 | |

300 | 0.626 | 1.733 | 1.108 | 0.520 | 1.839 | 1.320 | 0.663 | 1.784 | 1.122 | 0.555 | 1.892 | 1.337 |

0.986 | 1.912 | 0.926 | 0.897 | 2.001 | 1.104 | 1.059 | 1.975 | 0.916 | 0.971 | 2.063 | 1.092 | |

0.373 | 0.669 | 0.296 | 0.345 | 0.697 | 0.352 | 0.362 | 0.634 | 0.272 | 0.336 | 0.660 | 0.324 |

**Table 7.**Lower bounds (LBs), upper bounds (UBs), and average lengths (ALs) based on simple random sampling (SRS) and ranked set sampling (RSS) for Set3: $(\alpha =1.2,\lambda =1.5,\theta =0.75)$.

n | SRS | RSS | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

90% | 95% | 90% | 95% | |||||||||

LB | UB | AL | LB | UB | AL | LB | UB | AL | LB | UB | AL | |

100 | 0.138 | 2.368 | 2.231 | −0.076 | 2.582 | 2.658 | 0.216 | 2.436 | 2.219 | 0.004 | 2.648 | 2.644 |

0.607 | 2.218 | 1.611 | 0.453 | 2.372 | 1.919 | 0.740 | 2.385 | 1.645 | 0.583 | 2.542 | 1.960 | |

0.431 | 1.459 | 1.028 | 0.332 | 1.557 | 1.225 | 0.380 | 1.101 | 0.721 | 0.311 | 1.170 | 0.859 | |

200 | 0.432 | 2.110 | 1.677 | 0.272 | 2.270 | 1.999 | 0.536 | 1.882 | 1.346 | 0.407 | 2.011 | 1.603 |

0.936 | 2.159 | 1.223 | 0.819 | 2.276 | 1.457 | 0.946 | 2.056 | 1.110 | 0.840 | 2.162 | 1.322 | |

0.505 | 1.095 | 0.590 | 0.449 | 1.151 | 0.703 | 0.502 | 1.002 | 0.500 | 0.454 | 1.050 | 0.596 | |

300 | 0.601 | 1.694 | 1.093 | 0.497 | 1.799 | 1.302 | 0.625 | 1.611 | 0.986 | 0.531 | 1.706 | 1.174 |

0.961 | 1.890 | 0.929 | 0.872 | 1.979 | 1.107 | 0.991 | 1.870 | 0.879 | 0.907 | 1.954 | 1.047 | |

0.560 | 1.014 | 0.454 | 0.516 | 1.057 | 0.541 | 0.570 | 0.989 | 0.419 | 0.530 | 1.030 | 0.500 |

**Table 8.**Lower bounds (LBs), upper bounds (UBs), and average lengths (ALs) based on simple random sampling (SRS) and ranked set sampling (RSS) for Set4: $(\alpha =0.5,\lambda =1.5,\theta =0.75)$.

n | SRS | RSS | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

90% | 95% | 90% | 95% | |||||||||

LB | UB | AL | LB | UB | AL | LB | UB | AL | LB | UB | AL | |

100 | 0.137 | 1.014 | 0.877 | 0.053 | 1.098 | 1.045 | 0.115 | 0.834 | 0.719 | 0.046 | 0.903 | 0.856 |

0.812 | 2.386 | 1.575 | 0.661 | 2.537 | 1.876 | 0.657 | 2.271 | 1.615 | 0.502 | 2.426 | 1.924 | |

0.359 | 1.337 | 0.978 | 0.266 | 1.431 | 1.165 | 0.329 | 1.357 | 1.028 | 0.231 | 1.455 | 1.225 | |

200 | 0.247 | 0.715 | 0.469 | 0.202 | 0.760 | 0.558 | 0.251 | 0.757 | 0.507 | 0.202 | 0.806 | 0.604 |

1.004 | 2.063 | 1.059 | 0.902 | 2.164 | 1.262 | 0.956 | 2.025 | 1.070 | 0.853 | 2.128 | 1.275 | |

0.538 | 1.181 | 0.643 | 0.477 | 1.242 | 0.766 | 0.472 | 1.031 | 0.559 | 0.418 | 1.084 | 0.666 | |

300 | 0.270 | 0.674 | 0.404 | 0.232 | 0.713 | 0.481 | 0.293 | 0.700 | 0.407 | 0.254 | 0.739 | 0.485 |

1.019 | 1.927 | 0.908 | 0.932 | 2.014 | 1.081 | 1.050 | 1.923 | 0.873 | 0.966 | 2.006 | 1.040 | |

0.535 | 1.056 | 0.522 | 0.485 | 1.106 | 0.622 | 0.526 | 0.983 | 0.458 | 0.482 | 1.027 | 0.545 |

n | Mean | Median | Standard Deviation | Skewness | Kurtosis |
---|---|---|---|---|---|

24 | 71.47 | 61.68 | 36.85 | 0.94 | 0.35 |

n | Mean | Median | Standard Deviation | Skewness | Kurtosis |
---|---|---|---|---|---|

20 | 1.9 | 1.7 | 0.7 | 1.59 | 2.35 |

**Table 11.**Goodness-of-fit measures, maximum likelihood estimates (MLEs) and standard errors (SEs) (in parentheses) for the first data set.

Model | CVM | AD | KS | p-value | MLEs | ||||
---|---|---|---|---|---|---|---|---|---|

TCPW | 0.0384 | 0.2194 | 0.1078 | 0.9429 | 5.1975 | 0.0279 | 1.0104 | - | - |

($\alpha ,\lambda ,\theta $) | (1.2660) | (0.0483) | (0.3225) | - | - | ||||

KwWE | 0.0717 | 0.3868 | 0.1530 | 0.6272 | 7.8198 | 21.5152 | 1.4692 | 0.4015 | 0.0051 |

($a,b,\alpha ,\beta ,\lambda $) | (3.9916) | (0.0998) | (1.0216) | (0.3623) | (0.0019) | ||||

KwW | 0.0411 | 0.2305 | 0.1131 | 0.9145 | 12.8249 | 2.7789 | 0.2028 | 0.5722 | - |

($\lambda ,c,a,b$) | (2.5960) | (9.9083) | (4.1252) | (9.706634) | - | ||||

BW | 0.0402 | 0.2282 | 0.1106 | 0.9280 | 11.9919 | 3.4218 | 0.1125 | 0.6320 | - |

($\alpha ,\beta ,c,\gamma $) | (19.5339) | (20.2379) | (0.4676) | (1.1721) | - | ||||

W | 0.0615 | 0.3282 | 0.2437 | 0.1156 | 0.0021 | 1.4348 | - | - | - |

($\lambda ,\theta $) | (0.0004) | (0.06016) | - | - | - |

**Table 12.**Goodness-of-fit measures, maximum likelihood estimates (MLEs) and standard errors (SEs) for the second data set.

Model | CVM | AD | KS | p-value | MLEs | ||||
---|---|---|---|---|---|---|---|---|---|

TCPW | 0.0337 | 0.1959 | 0.1086 | 0.9722 | 200.3272 | 3.7521 | 0.6613 | - | - |

($\alpha ,\lambda ,\theta $) | (9.9442) | (1.2310) | (0.2927) | - | - | ||||

KwWE | 0.0483 | 0.2819 | 0.1380 | 0.8408 | 57.5128 | 0.4407 | 34.5503 | 1.0974 | 0.0965 |

($a,b,\alpha ,\beta ,\lambda $) | (7.3437) | (0.4832) | (7.2847) | (0.5249) | (0.1460) | ||||

KwW | 0.0425 | 0.2458 | 0.1274 | 0.9012 | 68.9084 | 0.3396 | 2.9571 | 1.3003 | - |

($\lambda ,c,a,b$) | (2.4681) | (0.3679) | (1.1769) | (0.6407) | - | ||||

BW | 0.0407 | 0.2344 | 0.1265 | 0.9057 | 78.7504 | 0.3148 | 3.3232 | 1.2708 | - |

($\alpha ,\beta ,c,\gamma $) | (19.5339) | (20.2379) | (0.4676) | (1.1721) | - | ||||

W | 0.1857 | 1.0928 | 0.1849 | 0.5007 | 0.1215 | 2.7869 | - | - | - |

($\lambda ,\theta $) | (0.0562) | (0.4272) | - | - | - |

**Table 13.**The $-\widehat{\ell}$, Akaike information criterion (AIC), corrected Akaike information criterion (CAIC), Bayesian information criterion (BIC), and Hannan–Quinn information criterion (HQIC) for the first data set.

Model | $-\widehat{\mathit{\ell}}$ | AIC | CAIC | BIC | HQIC |
---|---|---|---|---|---|

TCPW | 117.2952 | 240.5904 | 241.7904 | 244.1246 | 241.5280 |

KwWE | 117.9379 | 245.8759 | 249.2092 | 251.7662 | 247.4386 |

KwW | 117.3225 | 242.6459 | 244.7502 | 247.3572 | 243.8951 |

BW | 117.3125 | 242.6249 | 244.7302 | 247.3371 | 243.8751 |

W | 120.9310 | 245.8621 | 246.4335 | 248.2182 | 246.4872 |

**Table 14.**The $-\widehat{\ell}$, Akaike information criterion (AIC), corrected Akaike information criterion (CAIC), Bayesian information criterion (BIC), and Hannan–Quinn information criterion (HQIC) for the second data set.

Model | $-\widehat{\mathit{\ell}}$ | AIC | CAIC | BIC | HQIC |
---|---|---|---|---|---|

TCPW | 15.6075 | 37.2151 | 38.7151 | 40.2023 | 37.7982 |

KwWE | 15.9309 | 41.8619 | 46.1476 | 46.8405 | 42.8337 |

KwW | 15.7235 | 39.4471 | 42.11383 | 43.4300 | 40.2246 |

BW | 15.6801 | 39.3603 | 42.0272 | 43.3432 | 40.1378 |

W | 20.5864 | 45.1728 | 45.8786 | 47.1642 | 45.5615 |

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**MDPI and ACS Style**

Aldahlan, M.A.; Jamal, F.; Chesneau, C.; Elgarhy, M.; Elbatal, I.
The Truncated Cauchy Power Family of Distributions with Inference and Applications. *Entropy* **2020**, *22*, 346.
https://doi.org/10.3390/e22030346

**AMA Style**

Aldahlan MA, Jamal F, Chesneau C, Elgarhy M, Elbatal I.
The Truncated Cauchy Power Family of Distributions with Inference and Applications. *Entropy*. 2020; 22(3):346.
https://doi.org/10.3390/e22030346

**Chicago/Turabian Style**

Aldahlan, Maha A., Farrukh Jamal, Christophe Chesneau, Mohammed Elgarhy, and Ibrahim Elbatal.
2020. "The Truncated Cauchy Power Family of Distributions with Inference and Applications" *Entropy* 22, no. 3: 346.
https://doi.org/10.3390/e22030346