# A New Sine Family of Generalized Distributions: Statistical Inference with Applications

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## Abstract

**:**

## 1. Introduction

## 2. The New Sine Family of Distributions

#### 2.1. Methodology

#### 2.2. Some Important Functional Forms of the New Sine Family of Distributions

- Reliability function: In probability theory, the reliability function is a function that offers the probability that a system or device will function correctly for a given amount of time, assuming that it has not failed up to that point. Intuitively, the reliability function offers the probability that the device or system will continue to function beyond time x given that it has not failed up to that point. The reliability function for AS-G FD can be expressed as$$R\left(x;\alpha ,\phi \right)=1-\frac{sin\left\{\frac{\pi \alpha}{2}G\left(x;\phi \right)\right\}}{sin\left(\frac{\pi \alpha}{2}\right)};\phantom{\rule{1.em}{0ex}}x\in \Re ,0<\alpha <1.$$
- Hazard function: In probability theory, the hazard function is a function that describes the rate at which an event occurs given that the event has not yet occurred up to a certain time. The hazard function is often used in survival analysis to model the failure rate of a system over time. The AS-G FD can be defined as$$h\left(x\right)=\frac{\pi \alpha}{2}\frac{g\left(x;\phi \right)cos\left\{\frac{\pi \alpha}{2}G\left(x;\phi \right)\right\}}{sin\left(\frac{\pi \alpha}{2}\right)-sin\left\{\frac{\pi \alpha}{2}G\left(x;\phi \right)\right\}};\phantom{\rule{1.em}{0ex}}x\in \Re .$$
- Odd function: Odd functions are a useful tool in probability theory for describing certain types of distributions and for simplifying calculations involving them. Here, the odd function for AS-G FD can be expressed as$$O\left(x\right)=\frac{sin\left\{\frac{\pi \alpha}{2}G\left(x;\phi \right)\right\}}{sin\left(\frac{\pi \alpha}{2}\right)-sin\left\{\frac{\pi \alpha}{2}G\left(x;\phi \right)\right\}};\phantom{\rule{1.em}{0ex}}x\in \Re .$$
- Failure rate average (FRA): The failure rate average function has important applications in reliability engineering and survival analysis, where it is used to model the behavior of systems and estimate their probability of failure over time. It can also be used to compare different systems’ reliability and identify the factors that affect their failure rates.$$K\left(x\right)=-\frac{1}{x}\left[log\left\{sin\left(\frac{\pi \alpha}{2}\right)-sin\left\{\frac{\pi \alpha}{2}G\left(x;\phi \right)\right\}\right\}-log\left\{sin\left(\frac{\pi \alpha}{2}\right)\right\}\right];\phantom{\rule{1.em}{0ex}}x\in \Re .$$

## 3. Properties of the New Sine Family of Distributions

#### 3.1. Linear Representation

#### 3.2. Critical Points of the New Sine Family of Distributions

#### 3.3. Quantile Function

#### 3.4. Moments

#### 3.5. Moment Generating Function

#### 3.6. Mean Residual Life Function

## 4. Alpha-Sine Weibull Distribution

#### Model Presentation

## 5. Properties of the Alpha-Sine Weibull Distribution

#### 5.1. Quantile Function

#### 5.2. Linear Expansion of Alpha-Sine Weibull Distribution

#### 5.3. Moments

#### 5.4. Moment Generating Function of Alpha-Sine Weibull Distribution

#### 5.5. Mean Waiting Time Function

## 6. Estimation Methods

#### 6.1. Maximum Likelihood Method

#### 6.2. Maximum Product of Spacings Method

#### 6.3. Least Squares Methods

#### 6.4. Minimum Distance Methods

## 7. Numerical Simulation

- Based on all estimation methods, the average estimate converges to the true values, which shows that these estimators are consistent.
- The AE tends to its initial values as the sample size increase, so we can say that our estimates are unbiased.
- For all methods, whenever the MSEs decrease, the sample size m increases.
- The MLE estimators perform better than all the other methods considered in this work.

## 8. Applications

#### 8.1. First Data Set

#### 8.2. Second DataSet

- Sine-inverse Weibull [4]:$$\begin{array}{cc}& F(x,\alpha ,\theta )=sin\left\{\frac{\pi}{2}{e}^{\left(-\alpha {x}^{-\theta}\right)}\right\}.\hfill \\ & f(x;\alpha ,\theta )=\frac{\pi}{2}\alpha \theta {x}^{-\theta -1}{e}^{\left(-\alpha {x}^{-\theta}\right)}cos\left\{\frac{\pi}{2}{e}^{\left(-\alpha {x}^{-\theta}\right)}\right\}\phantom{\rule{1.em}{0ex}}x>0,\alpha ,\theta >0.\hfill \end{array}$$
- The inverse Weibull distribution [26]:$$\begin{array}{cc}& F(x,\tau ,\theta )={\mathrm{e}}^{-{\left(\frac{\theta}{x}\right)}^{\tau}}.\hfill \\ & f(x;\tau ,\theta )=f\left(x\right)=\frac{\tau {(\theta /x)}^{\tau}{e}^{-{(\theta /x)}^{\tau}}}{x}\phantom{\rule{1.em}{0ex}}x>0,\tau ,\theta >0.\hfill \end{array}$$
- Weighted generalized quasi Lindley distribution (WGQLD) [27]:$$\begin{array}{cc}& F(x,\alpha ,\theta )=1-{\displaystyle \frac{\begin{array}{c}24+6{\alpha}^{2}[2+x\theta (2+x\theta )]\\ +6\alpha [6+x\theta (6+x\theta (3+x\theta )\left)\right]\\ +x\theta [24+x\theta (12+x\theta (4+x\theta )\left)\right]\end{array}}{12(1+\alpha )(2+\alpha )}}{\mathrm{e}}^{-\theta x}.\hfill \\ & f(x;\alpha ,\theta )={\displaystyle \frac{{\theta}^{3}{x}^{2}\xb7\left({\theta}^{2}{x}^{2}+6\alpha \theta x+6{\alpha}^{2}\right){\mathrm{e}}^{-\theta x}}{12\left(\alpha +1\right)\left(\alpha +2\right)}}\phantom{\rule{1.em}{0ex}}x>0,\alpha ,\theta >0.\hfill \end{array}$$
- Sine Burr XII distribution [11]:$$\begin{array}{cc}& F\left(x\right)=sin\left\{\frac{\pi}{2}\left[1-\frac{1}{{\left(1+{x}^{a}\right)}^{b}}\right]\right\}:a,b,x>0.\hfill \\ & f\left(x\right)=\frac{\pi}{2}\frac{ab{x}^{a+1}}{{\left(1+{x}^{a}\right)}^{b+1}}cos\left\{\frac{\pi}{2}\left[1-\frac{1}{{\left(1+{x}^{a}\right)}^{b}}\right]\right\},a,\phantom{\rule{0.166667em}{0ex}}b,x>0.\hfill \end{array}$$

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**The plots of skewness with constant $\alpha =0.5$ (

**left**) and constant $\delta =0.75$ (

**right**).

**Figure 5.**The plots of kurtosis with constant $\alpha =0.5$ (

**left**) and constant $\delta =0.75$ (

**right**).

**Figure 6.**Plots of estimated probability density functions and cumulative distribution functions for Dataset 1.

**Figure 7.**Plots of estimated probability density functions and cumulative distribution functions for Dataset 2.

Statistical Measure | Expression |
---|---|

Median | ${G}^{-1}\left({\displaystyle \frac{2}{\pi \alpha}}arcsin\left\{0.5sin\left({\displaystyle \frac{\pi \alpha}{2}}\right)\right\}\right)$ |

Lower Quartile | ${G}^{-1}\left({\displaystyle \frac{2}{\pi \alpha}}arcsin\left\{0.25sin\left({\displaystyle \frac{\pi \alpha}{2}}\right)\right\}\right)$ |

Upper Quartile | ${G}^{-1}\left({\displaystyle \frac{2}{\pi \alpha}}arcsin\left\{0.75sin\left({\displaystyle \frac{\pi \alpha}{2}}\right)\right\}\right)$ |

QD | $\frac{1}{2}}\left[{G}^{-1}\left({\displaystyle \frac{2}{\pi \alpha}}arcsin\left\{0.75sin\left({\displaystyle \frac{\pi \alpha}{2}}\right)\right\}\right)-{G}^{-1}\left({\displaystyle \frac{2}{\pi \alpha}}arcsin\left\{0.25sin\left({\displaystyle \frac{\pi \alpha}{2}}\right)\right\}\right)\right]$ |

Coefficient of QD | $\frac{\left[{G}^{-1}\left({\displaystyle \frac{2}{\pi \alpha}}arcsin\left\{0.75sin\left({\displaystyle \frac{\pi \alpha}{2}}\right)\right\}\right)-{G}^{-1}\left({\displaystyle \frac{2}{\pi \alpha}}arcsin\left\{0.25sin\left({\displaystyle \frac{\pi \alpha}{2}}\right)\right\}\right)\right]}{\left[{G}^{-1}\left({\displaystyle \frac{2}{\pi \alpha}}arcsin\left\{0.75sin\left({\displaystyle \frac{\pi \alpha}{2}}\right)\right\}\right)+{G}^{-1}\left({\displaystyle \frac{2}{\pi \alpha}}arcsin\left\{0.25sin\left({\displaystyle \frac{\pi \alpha}{2}}\right)\right\}\right)\right]}$ |

Skewness [20] | $\frac{Q\left({\displaystyle \frac{3}{4}};\alpha ,\phi \right)-2Q\left({\displaystyle \frac{1}{2}};\alpha ,\phi \right)+Q\left({\displaystyle \frac{1}{4}};\alpha ,\phi \right)}{Q\left({\displaystyle \frac{3}{4}};\alpha ,\phi \right)-Q\left({\displaystyle \frac{1}{4}};\alpha ,\phi \right)}$ |

Kurtosis [21] | $\frac{Q\left({\displaystyle \frac{7}{8}};\alpha ,\phi \right)-Q\left({\displaystyle \frac{5}{8}};\alpha ,\phi \right)-Q\left({\displaystyle \frac{1}{8}};\alpha ,\phi \right)+Q\left({\displaystyle \frac{3}{8}};\alpha ,\phi \right)}{Q\left({\displaystyle \frac{3}{4}};\alpha ,\phi \right)-Q\left({\displaystyle \frac{1}{4}};\alpha ,\phi \right)}$ |

Sample Size | MLE | MPS | LSE | WLS | CVE | ADE | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AE | MSE | AE | MSE | AE | MSE | AE | MSE | AE | MSE | AE | MSE | ||

30 | $\widehat{\alpha}$ | 0.4282 | 0.2229 | 0.5727 | 0.2056 | 0.4468 | 0.2089 | 0.4707 | 0.2165 | 0.3965 | 0.2279 | 0.4661 | 0.2144 |

$\widehat{\delta}$ | 2.0909 | 0.3833 | 2.2162 | 0.4673 | 2.1140 | 0.2179 | 2.1152 | 0.1477 | 2.0552 | 0.0475 | 2.1046 | 0.1377 | |

$\widehat{\lambda}$ | 3.1459 | 0.3697 | 2.7929 | 0.2730 | 2.9844 | 0.3670 | 3.0017 | 0.3144 | 3.2108 | 0.4036 | 3.0294 | 0.2770 | |

60 | $\widehat{\alpha}$ | 0.5516 | 0.0155 | 0.5333 | 0.1673 | 0.4501 | 0.1589 | 0.4570 | 0.1585 | 0.3957 | 0.1626 | 0.4503 | 0.1587 |

$\widehat{\delta}$ | 2.0105 | 0.0031 | 2.1136 | 0.0490 | 2.0668 | 0.0376 | 2.0637 | 0.0282 | 2.0285 | 0.0191 | 2.0594 | 0.0284 | |

$\widehat{\lambda}$ | 3.0861 | 0.0258 | 2.8456 | 0.1225 | 2.9515 | 0.1485 | 2.9718 | 0.1242 | 3.0802 | 0.1362 | 2.9820 | 0.1158 | |

100 | $\widehat{\alpha}$ | 0.5429 | 0.0129 | 0.5160 | 0.1593 | 0.4173 | 0.1547 | 0.4282 | 0.1496 | 0.3918 | 0.1458 | 0.4378 | 0.1519 |

$\widehat{\delta}$ | 2.0054 | 0.0016 | 2.0937 | 0.0366 | 2.0448 | 0.0194 | 2.0446 | 0.0189 | 2.0211 | 0.0134 | 2.0490 | 0.0206 | |

$\widehat{\lambda}$ | 3.0741 | 0.0222 | 2.8854 | 0.0665 | 2.9601 | 0.0819 | 2.9766 | 0.0678 | 3.0436 | 0.0701 | 2.9799 | 0.0655 | |

150 | $\widehat{\alpha}$ | 0.5333 | 0.0100 | 0.5197 | 0.1487 | 0.4428 | 0.1360 | 0.4476 | 0.1369 | 0.4175 | 0.1318 | 0.4463 | 0.1386 |

$\widehat{\delta}$ | 2.0021 | 0.0006 | 2.0819 | 0.0287 | 2.0388 | 0.0154 | 2.0411 | 0.0155 | 2.0230 | 0.0125 | 2.0409 | 0.0157 | |

$\widehat{\lambda}$ | 3.0633 | 0.0190 | 2.9047 | 0.0439 | 2.9641 | 0.0539 | 2.9732 | 0.0442 | 3.0188 | 0.0451 | 2.9764 | 0.0427 | |

200 | $\widehat{\alpha}$ | 0.5237 | 0.0071 | 0.5464 | 0.1465 | 0.4247 | 0.1372 | 0.4404 | 0.1344 | 0.4234 | 0.1303 | 0.4439 | 0.1322 |

$\widehat{\delta}$ | 2.0003 | 0.0001 | 2.0910 | 0.0290 | 2.0323 | 0.0129 | 2.0366 | 0.0137 | 2.0256 | 0.0121 | 2.0366 | 0.0135 | |

$\widehat{\lambda}$ | 3.0531 | 0.0159 | 2.9242 | 0.0342 | 2.9802 | 0.0411 | 2.9868 | 0.0325 | 3.0209 | 0.0342 | 2.9892 | 0.0327 | |

500 | $\widehat{\alpha}$ | 0.5027 | 0.0008 | 0.4988 | 0.1292 | 0.4153 | 0.1220 | 0.4110 | 0.1198 | 0.3952 | 0.1173 | 0.4115 | 0.1188 |

$\widehat{\delta}$ | 2.0000 | 0.0000 | 2.0624 | 0.0207 | 2.0232 | 0.0096 | 2.0205 | 0.0095 | 2.0122 | 0.0081 | 2.0200 | 0.0094 | |

$\widehat{\lambda}$ | 3.0297 | 0.0089 | 2.9533 | 0.0145 | 2.9794 | 0.0166 | 2.9861 | 0.0138 | 3.0008 | 0.0137 | 2.9871 | 0.0136 |

Sample Size | MLE | MPS | LSE | WLS | CVE | ADE | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AE | MSE | AE | MSE | AE | MSE | AE | MSE | AE | MSE | AE | MSE | ||

30 | $\widehat{\alpha}$ | 0.2849 | 0.1535 | 0.4189 | 0.1723 | 0.3544 | 0.1660 | 0.3813 | 0.1785 | 0.3276 | 0.1644 | 0.3751 | 0.1642 |

$\widehat{\delta}$ | 1.0271 | 0.0322 | 1.1228 | 0.2186 | 1.0844 | 0.3253 | 1.0866 | 0.0735 | 1.0361 | 0.0235 | 1.0658 | 0.0312 | |

$\widehat{\lambda}$ | 1.5828 | 0.0631 | 1.4216 | 0.0535 | 1.4915 | 0.0732 | 1.5027 | 0.0642 | 1.6071 | 0.0870 | 1.5224 | 0.0585 | |

60 | $\widehat{\alpha}$ | 0.4456 | 0.0091 | 0.4559 | 0.1658 | 0.3692 | 0.1467 | 0.3912 | 0.1513 | 0.3620 | 0.1387 | 0.3953 | 0.1506 |

$\widehat{\delta}$ | 1.0211 | 0.0049 | 1.1065 | 0.0401 | 1.0575 | 0.0192 | 1.0652 | 0.0222 | 1.0406 | 0.0167 | 1.0647 | 0.0230 | |

$\widehat{\lambda}$ | 1.5492 | 0.0098 | 1.4396 | 0.0255 | 1.4911 | 0.0357 | 1.4999 | 0.0295 | 1.5527 | 0.0348 | 1.5045 | 0.0271 | |

100 | $\widehat{\alpha}$ | 0.4443 | 0.0066 | 0.4371 | 0.1590 | 0.3933 | 0.1348 | 0.3974 | 0.1376 | 0.3694 | 0.1267 | 0.4026 | 0.1380 |

$\widehat{\delta}$ | 1.0214 | 0.0032 | 1.0922 | 0.0333 | 1.0584 | 0.0173 | 1.0603 | 0.0187 | 1.0403 | 0.0139 | 1.0611 | 0.0192 | |

$\widehat{\lambda}$ | 1.5381 | 0.0057 | 1.4635 | 0.0143 | 1.4995 | 0.0197 | 1.5059 | 0.0160 | 1.5389 | 0.0182 | 1.5087 | 0.0157 | |

150 | $\widehat{\alpha}$ | 0.4456 | 0.0068 | 0.4645 | 0.1582 | 0.3779 | 0.1307 | 0.3865 | 0.1320 | 0.3649 | 0.1234 | 0.3895 | 0.1330 |

$\widehat{\delta}$ | 1.0136 | 0.0020 | 1.0971 | 0.0329 | 1.0486 | 0.0138 | 1.0509 | 0.0149 | 1.0363 | 0.0116 | 1.0520 | 0.0157 | |

$\widehat{\lambda}$ | 1.5297 | 0.0045 | 1.4626 | 0.0106 | 1.4918 | 0.0140 | 1.4976 | 0.0114 | 1.5198 | 0.0122 | 1.4988 | 0.0110 | |

200 | $\widehat{\alpha}$ | 0.4410 | 0.0061 | 0.4412 | 0.1528 | 0.3754 | 0.1280 | 0.3748 | 0.1271 | 0.3583 | 0.1200 | 0.3764 | 0.1296 |

$\widehat{\delta}$ | 1.0118 | 0.0018 | 1.0898 | 0.0296 | 1.0505 | 0.0129 | 1.0492 | 0.0137 | 1.0379 | 0.0111 | 1.0508 | 0.0146 | |

$\widehat{\lambda}$ | 1.5245 | 0.0037 | 1.4695 | 0.0080 | 1.4901 | 0.0101 | 1.4969 | 0.0082 | 1.5137 | 0.0086 | 1.4972 | 0.0080 | |

500 | $\widehat{\alpha}$ | 0.4380 | 0.0057 | 0.4484 | 0.1420 | 0.3764 | 0.1161 | 0.3778 | 0.1151 | 0.3629 | 0.1094 | 0.3789 | 0.1153 |

$\widehat{\delta}$ | 1.0012 | 0.0002 | 1.0815 | 0.0264 | 1.0413 | 0.0110 | 1.0416 | 0.0121 | 1.0330 | 0.0098 | 1.0418 | 0.0122 | |

$\widehat{\lambda}$ | 1.5115 | 0.0017 | 1.4802 | 0.0034 | 1.4932 | 0.0039 | 1.4960 | 0.0032 | 1.5034 | 0.0032 | 1.4964 | 0.0032 |

Sample Size | MLE | MPS | LSE | WLS | CVE | ADE | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AE | MSE | AE | MSE | AE | MSE | AE | MSE | AE | MSE | AE | MSE | ||

30 | $\widehat{\alpha}$ | 0.6597 | 0.0195 | 0.5390 | 0.2054 | 0.5081 | 0.1784 | 0.5212 | 0.1858 | 0.4882 | 0.1821 | 0.5191 | 0.1895 |

$\widehat{\delta}$ | 2.0642 | 0.1340 | 2.6170 | 8.4422 | 2.3168 | 2.3140 | 2.3242 | 1.8946 | 2.2091 | 2.7088 | 2.2942 | 1.1570 | |

$\widehat{\lambda}$ | 1.0537 | 0.0140 | 0.9369 | 0.0276 | 0.9900 | 0.0380 | 0.9979 | 0.0330 | 1.0665 | 0.0433 | 1.0081 | 0.0285 | |

60 | $\widehat{\alpha}$ | 0.6494 | 0.0099 | 0.5570 | 0.1611 | 0.4988 | 0.1622 | 0.5099 | 0.1605 | 0.4692 | 0.1636 | 0.5016 | 0.1645 |

$\widehat{\delta}$ | 2.0634 | 0.0127 | 2.2948 | 0.4264 | 2.1652 | 0.3995 | 2.1645 | 0.2667 | 2.0678 | 0.1926 | 2.1487 | 0.2479 | |

$\widehat{\lambda}$ | 1.0276 | 0.0055 | 0.9435 | 0.0135 | 0.9803 | 0.0164 | 0.9865 | 0.0139 | 1.0219 | 0.0152 | 0.9883 | 0.0125 | |

100 | $\widehat{\alpha}$ | 0.6442 | 0.0088 | 0.5808 | 0.1450 | 0.5099 | 0.1456 | 0.5104 | 0.1474 | 0.4700 | 0.1500 | 0.5034 | 0.1487 |

$\widehat{\delta}$ | 2.0510 | 0.0102 | 2.2622 | 0.3124 | 2.1219 | 0.2023 | 2.1178 | 0.1835 | 2.0400 | 0.1417 | 2.1039 | 0.1735 | |

$\widehat{\lambda}$ | 1.0250 | 0.0050 | 0.9625 | 0.0075 | 0.9858 | 0.0086 | 0.9913 | 0.0073 | 1.0135 | 0.0078 | 0.9944 | 0.0070 | |

150 | $\widehat{\alpha}$ | 0.6378 | 0.0076 | 0.5631 | 0.1385 | 0.4894 | 0.1464 | 0.4867 | 0.1480 | 0.4572 | 0.1500 | 0.4871 | 0.1483 |

$\widehat{\delta}$ | 2.0552 | 0.0110 | 2.2113 | 0.2558 | 2.0807 | 0.1505 | 2.0728 | 0.1324 | 2.0186 | 0.1090 | 2.0729 | 0.1341 | |

$\widehat{\lambda}$ | 1.0140 | 0.0028 | 0.9649 | 0.0054 | 0.9829 | 0.0063 | 0.9873 | 0.0051 | 1.0024 | 0.0051 | 0.9883 | 0.0050 | |

200 | $\widehat{\alpha}$ | 0.6272 | 0.0054 | 0.5662 | 0.1418 | 0.4872 | 0.1449 | 0.4817 | 0.1453 | 0.4546 | 0.1477 | 0.4877 | 0.1450 |

$\widehat{\delta}$ | 2.0500 | 0.0100 | 2.2230 | 0.2529 | 2.0804 | 0.1423 | 2.0684 | 0.1301 | 2.0218 | 0.1096 | 2.0768 | 0.1367 | |

$\widehat{\lambda}$ | 1.0108 | 0.0022 | 0.9718 | 0.0041 | 0.9895 | 0.0048 | 0.9929 | 0.0039 | 1.0045 | 0.0040 | 0.9931 | 0.0038 | |

500 | $\widehat{\alpha}$ | 0.608 | 0.0016 | 0.5878 | 0.1189 | 0.4732 | 0.1353 | 0.4847 | 0.1241 | 0.4688 | 0.1234 | 0.4885 | 0.1239 |

$\widehat{\delta}$ | 2.000 | 0.0000 | 2.2199 | 0.2269 | 2.0446 | 0.0999 | 2.0459 | 0.0965 | 2.0175 | 0.0843 | 2.0511 | 0.1001 | |

$\widehat{\lambda}$ | 1.000 | 0.0000 | 0.9808 | 0.0018 | 0.9936 | 0.0018 | 0.9949 | 0.0015 | 1.0000 | 0.0015 | 0.9949 | 0.0015 |

Sample Size | MLE | MPS | LSE | WLS | CVE | ADE | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AE | MSE | AE | MSE | AE | MSE | AE | MSE | AE | MSE | AE | MSE | ||

30 | $\widehat{\alpha}$ | 0.7483 | 0.0274 | 0.7345 | 0.1700 | 0.6787 | 0.1822 | 0.6691 | 0.1730 | 0.5878 | 0.2083 | 0.6681 | 0.1844 |

$\widehat{\delta}$ | 3.0782 | 0.0368 | 3.5572 | 3.0963 | 3.4365 | 2.7615 | 3.3212 | 1.4609 | 3.1723 | 0.7521 | 3.3279 | 1.4390 | |

$\widehat{\lambda}$ | 2.5959 | 0.0441 | 2.2631 | 0.2335 | 2.4062 | 0.2695 | 2.4325 | 0.2375 | 2.6046 | 0.2761 | 2.4583 | 0.2202 | |

60 | $\widehat{\alpha}$ | 0.7210 | 0.0063 | 0.6983 | 0.1259 | 0.6762 | 0.1393 | 0.6420 | 0.1358 | 0.5789 | 0.1566 | 0.6587 | 0.1363 |

$\widehat{\delta}$ | 3.0507 | 0.0152 | 3.2034 | 0.1991 | 3.1919 | 0.3088 | 3.1130 | 0.1402 | 3.0399 | 0.1163 | 3.1341 | 0.1548 | |

$\widehat{\lambda}$ | 2.5813 | 0.0244 | 2.3468 | 0.1024 | 2.4330 | 0.1317 | 2.4604 | 0.1028 | 2.5513 | 0.1109 | 2.4620 | 0.0987 | |

100 | $\widehat{\alpha}$ | 0.7054 | 0.0016 | 0.6886 | 0.1179 | 0.6692 | 0.1197 | 0.6293 | 0.1264 | 0.5736 | 0.1445 | 0.6494 | 0.1217 |

$\widehat{\delta}$ | 3.0354 | 0.0106 | 3.1517 | 0.1021 | 3.1244 | 0.1366 | 3.0739 | 0.0787 | 3.0172 | 0.0689 | 3.0901 | 0.0833 | |

$\widehat{\lambda}$ | 2.5756 | 0.0227 | 2.3837 | 0.0631 | 2.4531 | 0.0757 | 2.4717 | 0.0618 | 2.5291 | 0.0638 | 2.4711 | 0.0590 | |

150 | $\widehat{\alpha}$ | 0.7027 | 0.0008 | 0.6884 | 0.1119 | 0.6650 | 0.1186 | 0.6020 | 0.1345 | 0.5654 | 0.1444 | 0.6330 | 0.1253 |

$\widehat{\delta}$ | 3.0246 | 0.0074 | 3.1333 | 0.0846 | 3.1112 | 0.1146 | 3.0444 | 0.0621 | 3.0061 | 0.0571 | 3.0696 | 0.0679 | |

$\widehat{\lambda}$ | 2.5552 | 0.0166 | 2.3985 | 0.0393 | 2.4481 | 0.0500 | 2.4685 | 0.0377 | 2.5075 | 0.0375 | 2.4649 | 0.0371 | |

200 | $\widehat{\alpha}$ | 0.7006 | 0.0002 | 0.6686 | 0.1123 | 0.6532 | 0.1147 | 0.5924 | 0.1309 | 0.5587 | 0.1392 | 0.6294 | 0.1191 |

$\widehat{\delta}$ | 3.0168 | 0.0050 | 3.1048 | 0.0724 | 3.0849 | 0.0863 | 3.0256 | 0.0514 | 2.9926 | 0.0492 | 3.0551 | 0.0578 | |

$\widehat{\lambda}$ | 2.5450 | 0.0135 | 2.4154 | 0.0282 | 2.4574 | 0.0366 | 2.4756 | 0.0275 | 2.5059 | 0.0279 | 2.4706 | 0.0272 | |

500 | $\widehat{\alpha}$ | 0.7000 | 0.0000 | 0.6829 | 0.0887 | 0.6505 | 0.0995 | 0.5927 | 0.1110 | 0.5671 | 0.1176 | 0.6345 | 0.0976 |

$\widehat{\delta}$ | 3.0000 | 0.0000 | 3.0887 | 0.0561 | 3.0578 | 0.0586 | 3.0039 | 0.0359 | 2.9809 | 0.0339 | 3.0369 | 0.0428 | |

$\widehat{\lambda}$ | 2.5126 | 0.0038 | 2.4351 | 0.0132 | 2.4620 | 0.0144 | 2.4740 | 0.0107 | 2.4876 | 0.0101 | 2.4687 | 0.0112 |

Datasets | Minimum | One Quntile | Median | Mean | Three Quntile | Maximum | Skew | Kurt |
---|---|---|---|---|---|---|---|---|

Dataset 1 | 0.070 | 1.170 | 2.490 | 3.494 | 5.840 | 13.300 | 1.152 | 3.890 |

Dataset 2 | 2.998 | 21.187 | 51.385 | 55.123 | 75.435 | 138.500 | 0.555 | 2.108 |

1.33 | 1.43 | 1.01 | 1.62 | 3.15 | 1.05 | 7.72 | 0.2 | 6.03 | 0.25 | 7.83 | 0.25 | 0.88 | 6.29 | 0.94 |

5.84 | 3.23 | 3.7 | 1.26 | 2.64 | 1.17 | 2.49 | 1.62 | 2.1 | 0.14 | 2.57 | 3.85 | 7.02 | 5.04 | 7.27 |

1.53 | 6.7 | 0.07 | 2.01 | 10.35 | 5.42 | 13.3 |

14.712 | 32.644 | 61.979 | 65.521 | 105.50 | 114.60 | 120.40 |

138.50 | 8.610 | 11.741 | 54.535 | 55.047 | 58.928 | 63.391 |

105.18 | 113.02 | 2.998 | 5.016 | 15.628 | 23.040 | 27.851 |

37.843 | 38.050 | 48.226 |

Datasets | Estimate | SE | $\mathit{l}(\mathit{x};\xb7)$ |
---|---|---|---|

Dataset 1 | $\widehat{\alpha}$ = 0.0003 | 1.1977 | −83.265 |

$\widehat{\lambda}$ = 1.0495 | 0.1381 | ||

$\widehat{\delta}$ = 3.55905 | 0.5862 | ||

Dataset 2 | $\widehat{\alpha}$ = 0.002 | 1.083 | −119.119 |

$\widehat{\lambda}$ = 59.518 | 9.820 | ||

$\widehat{\delta}$ = 1.300 | 0.216 |

Model | AIC | AICc | BIC | HQIC | K-S | p-Value |
---|---|---|---|---|---|---|

AS-W | 172.5304 | 173.2577 | 177.3632 | 174.2342 | 0.0907 | 0.9212 |

Sine-inverse Weibull | 184.3137 | 184.6666 | 187.5355 | 185.4495 | 0.15862 | 0.3096 |

Inverse Weibull | 190.8537 | 191.2066 | 194.0755 | 191.9896 | 0.1897 | 0.1394 |

WGQLD | 206.7907 | 207.1436 | 210.0125 | 207.9265 | 0.2682 | 0.0097 |

Sine Burr XII | 181.3963 | 181.7493 | 184.6181 | 182.5322 | 0.1423 | 0.4417 |

Model | AIC | AICc | BIC | HQIC | K-S | p-Value |
---|---|---|---|---|---|---|

AS-W | 244.239 | 245.439 | 247.7732 | 245.1767 | 0.1271 | 0.7875 |

Sine-inverse Weibull | 251.187 | 251.7585 | 253.5431 | 251.8121 | 0.1546 | 0.5622 |

Inverse Weibull | 255.0592 | 255.6306 | 257.4153 | 255.6843 | 0.1778 | 0.3884 |

WGQLD | 252.8124 | 253.3839 | 255.1686 | 253.4375 | 0.1950 | 0.2824 |

Sine Burr XII | 284.8518 | 285.4232 | 287.2079 | 285.4768 | 0.3609 | 0.0026 |

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## Share and Cite

**MDPI and ACS Style**

Benchiha, S.; Sapkota, L.P.; Al Mutairi, A.; Kumar, V.; Khashab, R.H.; Gemeay, A.M.; Elgarhy, M.; Nassr, S.G.
A New Sine Family of Generalized Distributions: Statistical Inference with Applications. *Math. Comput. Appl.* **2023**, *28*, 83.
https://doi.org/10.3390/mca28040083

**AMA Style**

Benchiha S, Sapkota LP, Al Mutairi A, Kumar V, Khashab RH, Gemeay AM, Elgarhy M, Nassr SG.
A New Sine Family of Generalized Distributions: Statistical Inference with Applications. *Mathematical and Computational Applications*. 2023; 28(4):83.
https://doi.org/10.3390/mca28040083

**Chicago/Turabian Style**

Benchiha, SidAhmed, Laxmi Prasad Sapkota, Aned Al Mutairi, Vijay Kumar, Rana H. Khashab, Ahmed M. Gemeay, Mohammed Elgarhy, and Said G. Nassr.
2023. "A New Sine Family of Generalized Distributions: Statistical Inference with Applications" *Mathematical and Computational Applications* 28, no. 4: 83.
https://doi.org/10.3390/mca28040083