# A New Extension of the Kumaraswamy Exponential Model with Modeling of Food Chain Data

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## Abstract

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## 1. Introduction

- The new KMKE distribution gives more flexibility than the SEWE model and other well-known statistical models for food chain data as we prove in Section 7.
- The new recommended distribution is quite versatile and comprises three sub-models.
- The shapes of the pdf for the KMKE model can be decreasing, right skewness and uni-modal. However, the hazard rate function (hrf) for the KMKE model can be decreasing, increasing and j-shaped.
- Numerous statistical and computational characteristics of the recently proposed model are investigated.
- The parameters of the KMKE model are estimated utilizing maximum likelihood and Bayesian techniques.

## 2. Relevant Literature

## 3. The Construction of the Kavya–Manoharan Kumaraswamy Exponential Model

## 4. Statistical and Computational Features

#### 4.1. Quantile Function

#### 4.2. Moments

## 5. Estimation Methods

#### 5.1. Maximum Likelihood Estimation

#### 5.2. Bayesian Estimation

## 6. Simulation

#### 6.1. Simulation Study

#### 6.2. Final Thoughts on the Simulation Results

- The Bayesian estimation is superior to the MLE in every situation, we observe.
- The Bayesian estimation with positive weight asymmetric loss function is superior to the Bayesian estimation with negative weight asymmetric loss function, as we note.
- We note that the Bayesian estimation method with positive weight asymmetric loss function is better than the other estimation method.
- The Bayesian estimation with symmetric loss function is superior to the Bayesian estimation with negative weight asymmetric loss function, in some simulations.
- Bayesian credible and HPD intervals are the shortest LCI.

## 7. Modeling Food Data

**Firstly:**The food chain in the UK from 2000 to 2019 is shown in the first dataset, which can be found at https://www.gov.uk/government/statistics/food-chain-productivity and was accessed on 18 July 2022. Furthermore, this data has been cited in [9]. The data are as follows: “102.9, 104.1, 104.8, 105.5, 107.2, 108.6, 104.7, 105.8, 103.4, 104.1, 100, 99.9, 98.5, 100.1, 101.9, 101.4, 103.1, 103.2, 104.2, 109”. The results of this data are attached in Table 7 and Table 8, and Figure 9, Figure 10, Figure 11 and Figure 12.

**Secondly**, as one component of factor total productivity (FTP), food and drink wholesaling in the UK from 2000 to 2019, see https://www.gov.uk/government/statistics/food-chain-productivity, accessed on 18 July 2022. Furthermore, this data has been cited in [9]. The data are as follows: “101.1, 104.2, 104.6, 106.3, 100,101.7,99.6, 101, 102.7, 104.8, 109.1, 112, 114.4, 105.6, 107.1, 107.5, 108.6, 107.5, 106.6, 112.5”. The results of this data are attached in Table 9 and Table 10, and Figure 13, Figure 14, Figure 15 and Figure 16.

## 8. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## List of Symbols

Z | Random variable |

$G\left(z\right)$ | Cumulative distribution function of Kumaraswamy generated family |

$H\left(z\right)$ | Cumulative distribution function of exponential distribution |

$g\left(z;\alpha ,\phantom{\rule{0.277778em}{0ex}}\beta ,\gamma \right)$ | Probability density function of Kumaraswamy exponential distribution |

$G\left(z;\alpha ,\phantom{\rule{0.277778em}{0ex}}\beta ,\gamma \right)$ | Probability density function of Kumaraswamy exponential distribution |

$\alpha $ | Scale parameter |

$\beta $ | Shape parameter |

$\gamma $ | Shape parameter |

$F\left(z;\alpha ,\phantom{\rule{0.277778em}{0ex}}\beta ,\gamma \right)$ | Cumulative distribution function of the Kavya–Manoharan Kumaraswamy exponential distribution |

$f\left(z;\alpha ,\phantom{\rule{0.277778em}{0ex}}\beta ,\gamma \right)$ | Probability density function of the Kavya–Manoharan Kumaraswamy exponential distribution |

$S\left(z;\alpha ,\phantom{\rule{0.277778em}{0ex}}\beta ,\gamma \right)$ | Reliability function of the Kavya–Manoharan Kumaraswamy exponential distribution |

$h\left(z;\alpha ,\phantom{\rule{0.277778em}{0ex}}\beta ,\gamma \right)$ | Hazard rate function of the Kavya–Manoharan Kumaraswamy exponential distribution |

$\tau \left(z;\alpha ,\phantom{\rule{0.277778em}{0ex}}\beta ,\gamma \right)$ | Reversed hazard rate function of the Kavya–Manoharan Kumaraswamy exponential distribution |

$H\left(z;\alpha ,\phantom{\rule{0.277778em}{0ex}}\beta ,\gamma \right)$ | Cumulative hazard rate function of the Kavya–Manoharan Kumaraswamy exponential distribution |

$Q\left(u\right)$ | Quantile function |

${\mu}_{w}^{{}^{\prime}}$ | The ${w}_{th}$ moment |

${M}_{Z}\left(t\right)$ | Moment generating function |

${\eta}_{m}\left(t\right)$ | The ${m}_{th}$ incomplete moment |

${\tau}_{m}\left(t\right)$ | The ${m}_{th}$ conditional moment |

$lnL$ | Log-likelihood function |

n | Sample size |

${w}_{j}$ | Shape parameter of hyper-parameter |

${\mathcal{\nabla}}_{j}$ | Scale parameter of hyper-parameter |

N | The number of samples |

$\mathbb{C}$ | Constant of posterior distribution |

${L}_{S}$ | Squared-error loss function |

${\tilde{\mathsf{\Omega}}}^{S}$ | Bayesian estimator under SELF |

${E}_{\mathsf{\Omega}}$ | Average expectation |

${L}_{L}$ | LINEX loss function |

${\tilde{\mathsf{\Omega}}}^{L}$ | Bayesian estimator under LINEX |

c | Shape parameter of LINEX loss function |

${L}_{E}$ | Entropy loss function |

${\tilde{\mathsf{\Omega}}}_{E}$ | Bayesian estimator under entropy |

## Appendix A

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**Figure 6.**Heatmaps of MSE values for parameters of the KMWE distribution with different sample cases: $\alpha =0.5,\beta =0.4$.

**Figure 7.**Heatmaps of MSE values for parameters of the KMWE distribution with different sample cases: $\alpha =0.5,\beta =1.5$.

**Figure 8.**Heatmaps of MSE values for parameters of KMWE distribution with different sample cases: $\alpha =0.5,\beta =1.5$.

**Figure 10.**MLE of cdf, and pdf with empirical and histogram, QQ and PP of the KMWE model for food chain data.

**Figure 14.**MLE of cdf, and pdf with empirical and histogram, QQ and PP of the KMWE model for food and drink wholesaling data.

**Figure 15.**MCMC plot and convergence line for parameters of the KMWE model for food and drink wholesaling data.

**Figure 16.**Histogram plot with normal curve for parameters of the KMWE model for food and drink wholesaling data.

Model | Modeling | Authors |
---|---|---|

The new suggested model (KMKE model) | Food chain data | New |

K-Weibull model | Failure times data | [50] |

K-generalized Rayleigh model | Engineering data | [51] |

K-modified Weibull model | Failure times data | [52] |

K-transmuted exponentiated modified Weibull model | Medical data | [53] |

K-transmuted modified Weibull model | Failure times data | [54] |

K-Gompertz Makeham model | Physics data | [55] |

K-Gumbel model | Engineering data | [56] |

K-generalized gamma model | Industrial and medical data | [57] |

K-generalized power Lomax model | Physics data | [58] |

K-Burr XII model | Engineering, physics and medical data | [59] |

K-generalized inverse Lomax model | Reliability and survival data | [60] |

K-Dagum model | Income and lifetime data | [61] |

Modified K model | Engineering data | [62] |

Transmuted K-Lindley model | Medical data | [63] |

K-Marshall–Olkin exponential model | Medical data | [64] |

K-half logistic model | Physics and medical data | [65] |

K-log logistic model | Medical data | [66] |

K-Marshall–Olkin log-logistic model | Physics data | [67] |

Modified K Weibull model | Reliability and engineering data | [68] |

K-inverted Topp–Leone model | COVID-19 data | [69] |

Kavya–Manoharan-K model | COVID-19 and physics data | [70] |

Transmuted K model | Medical and environmental data | [71] |

Generalized inverted K-G | Physics data | [72] |

Topp–Leone generalized inverted K model | Physics data | [73] |

K log-logistic Weibull model | Failure times data | [74] |

Exponentiated inverse K model | Economic data | [75] |

Beta K Burr Type X model | Physics and medical data | [76] |

Marshall–Olkin extended inverted K model | Physics, failure and medical data | [77] |

K generalized Kappa model | Geological data | [78] |

Cubic rank transmuted K model | Food and industrial data | [79] |

K Marshall–Olkin log-logistic model | Physics data | [67] |

Odd generalized exponential K model | Geological and environmental data | [80] |

K exponentiated U-quadratic model | Medical data | [81] |

K odd Burr-G | Physics and engineering data | [82] |

Exponentiated generalized K model | Environmental, agriculture and engineering data | [83] |

Size-biased K model | Engineering data | [84] |

K generalized power Weibull model | Engineering data | [85] |

Exponentiated K-Dagum model | Income and lifetime data | [61] |

Model | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ |
---|---|---|---|

KMKE | - | - | - |

KM- Topp–Leone exponential | - | 2 | - |

KM- exponentiated exponential | - | - | 1 |

KM- exponential | - | 1 | 1 |

$\mathit{\alpha}$ = 0.5, $\mathit{\beta}$ = 0.4 | MLE | SELF | LINEX c = −1.2 | LINEX c = 1.2 | ELF c = −1.2 | ELF c = 1.2 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\gamma $ | n | Bias | MSE | LACI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | |

0.5 | 40 | $\alpha $ | 0.1445 | 0.1540 | 1.4311 | 0.0940 | 0.0510 | 0.7638 | 0.1177 | 0.0620 | 0.8008 | 0.0705 | 0.0415 | 0.7209 | 0.1066 | 0.0546 | 0.7633 | 0.0237 | 0.0381 | 0.7249 |

$\beta $ | 0.1306 | 0.0269 | 0.3887 | 0.0919 | 0.0184 | 0.3302 | 0.1023 | 0.0224 | 0.3430 | 0.0816 | 0.0150 | 0.3127 | 0.0984 | 0.0202 | 0.3340 | 0.0571 | 0.0108 | 0.3083 | ||

$\gamma $ | 0.1640 | 0.1442 | 1.4311 | 0.0791 | 0.0395 | 0.6464 | 0.0979 | 0.0480 | 0.7020 | 0.0607 | 0.0322 | 0.5960 | 0.0889 | 0.0424 | 0.6670 | 0.0279 | 0.0278 | 0.5870 | ||

70 | $\alpha $ | 0.0608 | 0.0681 | 0.9954 | 0.0492 | 0.0172 | 0.4605 | 0.0560 | 0.0188 | 0.4672 | 0.0424 | 0.0158 | 0.4544 | 0.0532 | 0.0178 | 0.4605 | 0.0228 | 0.0147 | 0.4560 | |

$\beta $ | 0.1179 | 0.0183 | 0.2602 | 0.0582 | 0.0066 | 0.2139 | 0.0618 | 0.0071 | 0.2180 | 0.0547 | 0.0060 | 0.2106 | 0.0607 | 0.0069 | 0.2157 | 0.0454 | 0.0050 | 0.2083 | ||

$\gamma $ | 0.1596 | 0.0922 | 0.9954 | 0.0413 | 0.0117 | 0.3662 | 0.0468 | 0.0127 | 0.3674 | 0.0360 | 0.0108 | 0.3598 | 0.0445 | 0.0121 | 0.3647 | 0.0251 | 0.0098 | 0.3621 | ||

150 | $\alpha $ | 0.0449 | 0.0451 | 0.8142 | 0.0437 | 0.0087 | 0.3183 | 0.0470 | 0.0092 | 0.3259 | 0.0404 | 0.0081 | 0.3147 | 0.0456 | 0.0089 | 0.3212 | 0.0223 | 0.0074 | 0.3126 | |

$\beta $ | 0.1165 | 0.0159 | 0.1896 | 0.0578 | 0.0052 | 0.1589 | 0.0597 | 0.0054 | 0.1625 | 0.0456 | 0.0049 | 0.1565 | 0.0591 | 0.0053 | 0.1606 | 0.0451 | 0.0043 | 0.1543 | ||

$\gamma $ | 0.1436 | 0.0662 | 0.8142 | 0.0344 | 0.0062 | 0.2708 | 0.0370 | 0.0066 | 0.2759 | 0.0318 | 0.0059 | 0.2647 | 0.0360 | 0.0064 | 0.2731 | 0.0246 | 0.0054 | 0.2616 | ||

1.7 | 40 | $\alpha $ | 0.3260 | 0.1931 | 1.1554 | 0.1486 | 0.0729 | 0.8499 | 0.1805 | 0.0935 | 0.9256 | 0.1169 | 0.0558 | 0.7739 | 0.1639 | 0.0796 | 0.8640 | 0.0607 | 0.0481 | 0.8048 |

$\beta $ | 0.0864 | 0.0113 | 0.2430 | 0.0565 | 0.0104 | 0.2146 | 0.0682 | 0.0110 | 0.2254 | 0.0458 | 0.0091 | 0.2041 | 0.0609 | 0.0105 | 0.2250 | 0.0339 | 0.0091 | 0.2036 | ||

$\gamma $ | 0.0968 | 0.1019 | 1.1554 | 0.0503 | 0.0901 | 1.1350 | 0.0874 | 0.1006 | 1.1495 | 0.0131 | 0.0831 | 1.1221 | 0.0575 | 0.0908 | 1.1262 | 0.0132 | 0.0890 | 1.1664 | ||

70 | $\alpha $ | 0.2740 | 0.1419 | 1.0138 | 0.0604 | 0.0186 | 0.4642 | 0.0685 | 0.0205 | 0.4716 | 0.0524 | 0.0169 | 0.4578 | 0.0651 | 0.0194 | 0.4639 | 0.0361 | 0.0158 | 0.4654 | |

$\beta $ | 0.0811 | 0.0090 | 0.1934 | 0.0354 | 0.0044 | 0.1487 | 0.0377 | 0.0054 | 0.1495 | 0.0331 | 0.0036 | 0.1472 | 0.0369 | 0.0047 | 0.1491 | 0.0277 | 0.0029 | 0.1474 | ||

$\gamma $ | 0.1487 | 0.1005 | 1.0138 | 0.0212 | 0.0210 | 0.5670 | 0.0298 | 0.0219 | 0.5699 | 0.0126 | 0.0203 | 0.5507 | 0.0229 | 0.0211 | 0.5651 | 0.0129 | 0.0206 | 0.5597 | ||

150 | $\alpha $ | 0.2688 | 0.1075 | 0.7365 | 0.0514 | 0.0091 | 0.3213 | 0.0551 | 0.0098 | 0.3290 | 0.0478 | 0.0085 | 0.3141 | 0.0536 | 0.0094 | 0.3249 | 0.0404 | 0.0077 | 0.3116 | |

$\beta $ | 0.0740 | 0.0066 | 0.1318 | 0.0294 | 0.0016 | 0.1023 | 0.0302 | 0.0016 | 0.1029 | 0.0286 | 0.0015 | 0.1012 | 0.0300 | 0.0016 | 0.1026 | 0.0263 | 0.0014 | 0.0989 | ||

$\gamma $ | 0.0710 | 0.0825 | 0.7365 | 0.0179 | 0.0102 | 0.3891 | 0.0218 | 0.0105 | 0.3893 | 0.0140 | 0.0099 | 0.3890 | 0.0187 | 0.0102 | 0.3891 | 0.0142 | 0.0100 | 0.3902 | ||

3 | 40 | $\alpha $ | 0.4328 | 0.2818 | 1.2054 | 0.1428 | 0.0775 | 0.8571 | 0.1717 | 0.0980 | 0.9234 | 0.1143 | 0.0604 | 0.7936 | 0.1498 | 0.0805 | 0.8620 | 0.0558 | 0.0523 | 0.8153 |

$\beta $ | 0.0941 | 0.0116 | 0.2039 | 0.0829 | 0.0105 | 0.1237 | 0.1090 | 0.1034 | 0.1925 | 0.0582 | 0.0094 | 0.1923 | 0.0855 | 0.0106 | 0.2004 | 0.0498 | 0.0094 | 0.1822 | ||

$\gamma $ | 0.2276 | 0.1485 | 1.2054 | 0.0333 | 0.1037 | 1.1906 | 0.0652 | 0.1101 | 1.2040 | 0.0011 | 0.0996 | 1.1774 | 0.0351 | 0.1038 | 1.1910 | 0.0134 | 0.1033 | 1.2095 | ||

70 | $\alpha $ | 0.4255 | 0.2423 | 0.9707 | 0.0574 | 0.0208 | 0.5085 | 0.0641 | 0.0224 | 0.5166 | 0.0507 | 0.0192 | 0.4998 | 0.0594 | 0.0211 | 0.5095 | 0.0346 | 0.0183 | 0.5118 | |

$\beta $ | 0.0908 | 0.0100 | 0.1639 | 0.0333 | 0.0074 | 0.1338 | 0.0358 | 0.0103 | 0.1349 | 0.0310 | 0.0053 | 0.1330 | 0.0340 | 0.0077 | 0.1342 | 0.0264 | 0.0042 | 0.1305 | ||

$\gamma $ | 0.1816 | 0.1177 | 0.9707 | 0.0048 | 0.0227 | 0.5639 | 0.0118 | 0.0230 | 0.5656 | -0.0023 | 0.0226 | 0.5664 | 0.0051 | 0.0227 | 0.5640 | 0.0004 | 0.0227 | 0.5683 | ||

150 | $\alpha $ | 0.3773 | 0.1607 | 0.5305 | 0.0514 | 0.0102 | 0.3328 | 0.0545 | 0.0109 | 0.3401 | 0.0482 | 0.0096 | 0.3286 | 0.0523 | 0.0104 | 0.3340 | 0.0314 | 0.0088 | 0.3255 | |

$\beta $ | 0.0825 | 0.0075 | 0.1040 | 0.0253 | 0.0011 | 0.0783 | 0.0258 | 0.0011 | 0.0791 | 0.0249 | 0.0010 | 0.0778 | 0.0255 | 0.0011 | 0.0785 | 0.0235 | 0.0010 | 0.0774 | ||

$\gamma $ | 0.1440 | 0.0258 | 0.5305 | 0.0031 | 0.0111 | 0.3980 | 0.0105 | 0.0113 | 0.3982 | 0.0080 | 0.0110 | 0.3977 | 0.0041 | 0.0111 | 0.3981 | 0.0004 | 0.0111 | 0.3957 |

$\mathit{\alpha}=0.5,\mathit{\beta}=1.5$ | MLE | SELF | LINEX c = −1.2 | LINEX c = 1.2 | ELF c = −1.2 | ELF c = 1.2 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\gamma $ | n | Bias | MSE | LACI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | |

0.5 | 40 | $\alpha $ | 0.1630 | 0.1298 | 1.2599 | 0.0606 | 0.0401 | 0.7178 | 0.0751 | 0.0464 | 0.7495 | 0.0463 | 0.0346 | 0.6763 | 0.0644 | 0.0411 | 0.7205 | 0.0167 | 0.0309 | 0.6627 |

$\beta $ | 0.6417 | 0.5919 | 1.6645 | 0.0864 | 0.0951 | 1.0774 | 0.1182 | 0.1123 | 1.1213 | 0.0550 | 0.0843 | 1.0349 | 0.0899 | 0.0954 | 1.0806 | 0.0477 | 0.0937 | 1.0550 | ||

$\gamma $ | 0.1161 | 0.1152 | 1.2599 | 0.0751 | 0.0523 | 0.7778 | 0.0936 | 0.0614 | 0.8249 | 0.0570 | 0.0444 | 0.7511 | 0.0799 | 0.0537 | 0.7748 | 0.0200 | 0.0398 | 0.7325 | ||

70 | $\alpha $ | 0.0417 | 0.0482 | 0.8452 | 0.0205 | 0.0104 | 0.3666 | 0.0246 | 0.0110 | 0.3716 | 0.0164 | 0.0099 | 0.3610 | 0.0218 | 0.0105 | 0.3683 | 0.0064 | 0.0095 | 0.3620 | |

$\beta $ | 0.5048 | 0.3385 | 1.1347 | 0.0398 | 0.0237 | 0.5758 | 0.0472 | 0.0254 | 0.5835 | 0.0325 | 0.0224 | 0.5670 | 0.0406 | 0.0239 | 0.5784 | 0.0312 | 0.0227 | 0.5702 | ||

$\gamma $ | 0.1842 | 0.1144 | 0.8452 | 0.0299 | 0.0140 | 0.4264 | 0.0348 | 0.0148 | 0.4326 | 0.0249 | 0.0132 | 0.4221 | 0.0314 | 0.0141 | 0.4265 | 0.0128 | 0.0127 | 0.4249 | ||

150 | $\alpha $ | 0.1410 | 0.0391 | 0.8045 | 0.0168 | 0.0052 | 0.2569 | 0.0187 | 0.0054 | 0.2621 | 0.0149 | 0.0051 | 0.2558 | 0.0174 | 0.0053 | 0.2578 | 0.0061 | 0.0049 | 0.2524 | |

$\beta $ | 0.5564 | 0.3282 | 1.0533 | 0.0355 | 0.0110 | 0.3667 | 0.0389 | 0.0119 | 0.3691 | 0.0320 | 0.0103 | 0.3612 | 0.0358 | 0.0111 | 0.3669 | 0.0305 | 0.0103 | 0.3626 | ||

$\gamma $ | 0.0604 | 0.0546 | 1.0449 | 0.0236 | 0.0082 | 0.2991 | 0.0262 | 0.0092 | 0.3013 | 0.0211 | 0.0073 | 0.2977 | 0.0244 | 0.0083 | 0.2986 | 0.0125 | 0.0069 | 0.2997 | ||

1.7 | 40 | $\alpha $ | 0.2562 | 0.2002 | 1.4386 | 0.0786 | 0.0271 | 0.5305 | 0.0895 | 0.0312 | 0.5599 | 0.0678 | 0.0234 | 0.5055 | 0.0815 | 0.0278 | 0.5329 | 0.0458 | 0.0198 | 0.5047 |

$\beta $ | 0.5409 | 0.4577 | 1.5938 | 0.1248 | 0.1263 | 1.0458 | 0.1602 | 0.1887 | 1.0943 | 0.0907 | 0.0894 | 0.9739 | 0.1283 | 0.1281 | 1.0457 | 0.0877 | 0.1027 | 1.0044 | ||

$\gamma $ | 0.0102 | 0.3130 | 1.4386 | −0.0046 | 0.1202 | 1.3193 | 0.0301 | 0.1225 | 1.3294 | −0.0394 | 0.1204 | 1.3188 | −0.0009 | 0.1195 | 1.3142 | −0.0460 | 0.1307 | 1.3629 | ||

70 | $\alpha $ | 0.2480 | 0.1925 | 1.3683 | 0.0307 | 0.0066 | 0.2859 | 0.0338 | 0.0070 | 0.2897 | 0.0276 | 0.0063 | 0.2818 | 0.0317 | 0.0067 | 0.2857 | 0.0200 | 0.0059 | 0.2807 | |

$\beta $ | 0.4794 | 0.3415 | 1.3108 | 0.0487 | 0.0228 | 0.5228 | 0.0559 | 0.0253 | 0.5293 | 0.0416 | 0.0209 | 0.5132 | 0.0495 | 0.0229 | 0.5245 | 0.0398 | 0.0228 | 0.5152 | ||

$\gamma $ | −0.0924 | 0.1667 | 1.2683 | 0.0037 | 0.0243 | 0.6074 | 0.0142 | 0.0245 | 0.6077 | −0.0001 | 0.0243 | 0.6130 | 0.0078 | 0.0243 | 0.6071 | −0.0008 | 0.0247 | 0.6169 | ||

150 | $\alpha $ | 0.1905 | 0.0663 | 0.6791 | 0.0291 | 0.0032 | 0.1852 | 0.0306 | 0.0033 | 0.1865 | 0.0277 | 0.0031 | 0.1834 | 0.0296 | 0.0033 | 0.1851 | 0.0192 | 0.0029 | 0.1820 | |

$\beta $ | 0.4187 | 0.2066 | 0.6932 | 0.0471 | 0.0112 | 0.3477 | 0.0507 | 0.0114 | 0.3504 | 0.0404 | 0.0111 | 0.3436 | 0.0476 | 0.0111 | 0.3469 | 0.0384 | 0.0140 | 0.3442 | ||

$\gamma $ | −0.0678 | 0.1216 | 0.6791 | 0.0019 | 0.0104 | 0.3929 | 0.0053 | 0.0105 | 0.3931 | −0.0001 | 0.0104 | 0.3939 | 0.0022 | 0.0104 | 0.3930 | −0.0007 | 0.0105 | 0.3951 | ||

3 | 40 | $\alpha $ | 0.2534 | 0.1426 | 1.0980 | 0.0801 | 0.0258 | 0.5305 | 0.0903 | 0.0289 | 0.5458 | 0.0701 | 0.0230 | 0.5150 | 0.0830 | 0.0263 | 0.5303 | 0.0474 | 0.0212 | 0.5222 |

$\beta $ | 0.5037 | 0.3608 | 1.2831 | 0.1410 | 0.1542 | 1.1107 | 0.1825 | 0.2317 | 1.1644 | 0.1019 | 0.1087 | 1.0485 | 0.1454 | 0.1558 | 1.1143 | 0.0963 | 0.1347 | 1.0918 | ||

$\gamma $ | −0.0634 | 0.2833 | 1.0980 | −0.0079 | 0.1179 | 1.3328 | 0.0280 | 0.1183 | 1.3187 | −0.0438 | 0.1213 | 1.3426 | −0.0058 | 0.1175 | 1.3283 | −0.0311 | 0.1231 | 1.3580 | ||

70 | $\alpha $ | 0.2146 | 0.0804 | 0.7271 | 0.0320 | 0.0068 | 0.2887 | 0.0351 | 0.0072 | 0.2958 | 0.0288 | 0.0065 | 0.2875 | 0.0329 | 0.0069 | 0.2888 | 0.0211 | 0.0061 | 0.2931 | |

$\beta $ | 0.4584 | 0.2673 | 0.9372 | 0.0639 | 0.0290 | 0.5607 | 0.0724 | 0.0334 | 0.5711 | 0.0554 | 0.0258 | 0.5389 | 0.0648 | 0.0291 | 0.5617 | 0.0535 | 0.0289 | 0.5414 | ||

$\gamma $ | −0.0681 | 0.1243 | 0.7271 | −0.0044 | 0.0259 | 0.6245 | 0.0033 | 0.0257 | 0.6209 | −0.0120 | 0.0262 | 0.6263 | −0.0039 | 0.0258 | 0.6247 | −0.0091 | 0.0262 | 0.6267 | ||

150 | $\alpha $ | 0.2083 | 0.0595 | 0.4978 | 0.0271 | 0.0032 | 0.1900 | 0.0285 | 0.0033 | 0.1920 | 0.0257 | 0.0031 | 0.1886 | 0.0276 | 0.0032 | 0.1901 | 0.0202 | 0.0029 | 0.1864 | |

$\beta $ | 0.4220 | 0.2014 | 0.5992 | 0.0541 | 0.0118 | 0.3386 | 0.0577 | 0.0131 | 0.3429 | 0.0506 | 0.0107 | 0.3320 | 0.0545 | 0.0119 | 0.3394 | 0.0501 | 0.0108 | 0.3319 | ||

$\gamma $ | −0.0513 | 0.1173 | 0.4978 | 0.0022 | 0.0112 | 0.4100 | 0.0033 | 0.0112 | 0.4079 | −0.0013 | 0.0112 | 0.4116 | 0.0024 | 0.0112 | 0.4098 | 0.0001 | 0.0112 | 0.4120 |

$\mathit{\alpha}=2,\mathit{\beta}=1.5$ | MLE | SELF | LINEX c = −1.2 | LINEX c = 1.2 | ELF c = −1.2 | ELF c = 1.2 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\gamma $ | n | Bias | MSE | LACI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | |

0.5 | 40 | $\alpha $ | 0.3204 | 0.8668 | 3.4284 | −0.0230 | 0.0998 | 1.2106 | 0.0099 | 0.1009 | 1.2289 | −0.0558 | 0.1015 | 1.1995 | −0.0201 | 0.0994 | 1.2044 | −0.0553 | 0.1064 | 1.2332 |

$\beta $ | 0.7350 | 0.7654 | 1.8610 | 0.1198 | 0.1096 | 1.0823 | 0.1537 | 0.1407 | 1.1266 | 0.0862 | 0.0887 | 1.0433 | 0.1233 | 0.1108 | 1.0850 | 0.0818 | 0.0954 | 1.0746 | ||

$\gamma $ | 0.1720 | 0.1555 | 3.4284 | 0.0810 | 0.0289 | 0.5330 | 0.0946 | 0.0343 | 0.5627 | 0.0677 | 0.0242 | 0.5075 | 0.0846 | 0.0299 | 0.5402 | 0.0408 | 0.0199 | 0.4891 | ||

70 | $\alpha $ | 0.2746 | 0.8403 | 3.4300 | −0.0038 | 0.0231 | 0.5906 | 0.0036 | 0.0230 | 0.5856 | −0.0112 | 0.0232 | 0.5944 | −0.0031 | 0.0230 | 0.5895 | −0.0107 | 0.0235 | 0.5982 | |

$\beta $ | 0.6511 | 0.5762 | 1.5304 | 0.0442 | 0.0240 | 0.5759 | 0.0516 | 0.0251 | 0.5832 | 0.0368 | 0.0233 | 0.5674 | 0.0450 | 0.0241 | 0.5769 | 0.0349 | 0.0248 | 0.5713 | ||

$\gamma $ | 0.2121 | 0.1267 | 3.4300 | 0.0243 | 0.0078 | 0.3137 | 0.0279 | 0.0082 | 0.3164 | 0.0207 | 0.0074 | 0.3107 | 0.0254 | 0.0078 | 0.3138 | 0.0115 | 0.0071 | 0.3114 | ||

150 | $\alpha $ | 0.0761 | 0.1278 | 1.3698 | 0.0044 | 0.0109 | 0.3858 | 0.0035 | 0.0110 | 0.3870 | 0.0011 | 0.0109 | 0.3878 | 0.0035 | 0.0109 | 0.3856 | 0.0013 | 0.0109 | 0.3885 | |

$\beta $ | 0.5633 | 0.3521 | 0.7315 | 0.0489 | 0.0113 | 0.3596 | 0.0522 | 0.0118 | 0.3639 | 0.0455 | 0.0108 | 0.3578 | 0.0492 | 0.0113 | 0.3601 | 0.0334 | 0.0107 | 0.3584 | ||

$\gamma $ | 0.1441 | 0.0702 | 1.3698 | 0.0232 | 0.0035 | 0.2105 | 0.0249 | 0.0037 | 0.2118 | 0.0214 | 0.0033 | 0.2090 | 0.0237 | 0.0035 | 0.2103 | 0.0111 | 0.0031 | 0.2043 | ||

1.7 | 40 | $\alpha $ | 0.8060 | 2.1020 | 4.7266 | 0.0428 | 0.0867 | 1.1478 | 0.0723 | 0.0933 | 1.1575 | 0.0135 | 0.0825 | 1.1326 | 0.0452 | 0.0868 | 1.1458 | 0.0156 | 0.0863 | 1.1548 |

$\beta $ | 0.5608 | 0.4956 | 1.6690 | 0.0874 | 0.0607 | 0.8378 | 0.1111 | 0.0674 | 0.8772 | 0.0642 | 0.0556 | 0.7973 | 0.0900 | 0.0606 | 0.8436 | 0.0592 | 0.0603 | 0.8156 | ||

$\gamma $ | 0.2774 | 0.8488 | 4.7266 | 0.0518 | 0.0960 | 1.1694 | 0.0821 | 0.1060 | 1.2057 | 0.0214 | 0.0885 | 1.1471 | 0.0547 | 0.0963 | 1.1679 | 0.0191 | 0.0933 | 1.1781 | ||

70 | $\alpha $ | 0.6543 | 1.1461 | 3.3232 | 0.0117 | 0.0229 | 0.5906 | 0.0187 | 0.0234 | 0.5955 | 0.0047 | 0.0224 | 0.5811 | 0.0123 | 0.0229 | 0.5905 | 0.0053 | 0.0227 | 0.5864 | |

$\beta $ | 0.4902 | 0.3146 | 1.0690 | 0.0473 | 0.0190 | 0.4825 | 0.0537 | 0.0196 | 0.4924 | 0.0409 | 0.0187 | 0.4751 | 0.0481 | 0.0190 | 0.4837 | 0.0387 | 0.0215 | 0.4757 | ||

$\gamma $ | 0.1217 | 0.2325 | 3.3232 | 0.0121 | 0.0239 | 0.5973 | 0.0193 | 0.0244 | 0.5976 | 0.0048 | 0.0235 | 0.5964 | 0.0128 | 0.0239 | 0.5969 | 0.0042 | 0.0239 | 0.6000 | ||

150 | $\alpha $ | 0.5267 | 0.8022 | 3.1668 | 0.0110 | 0.0105 | 0.4068 | 0.0167 | 0.0106 | 0.4063 | 0.0038 | 0.0103 | 0.4067 | 0.0120 | 0.0105 | 0.4070 | 0.0041 | 0.0104 | 0.4079 | |

$\beta $ | 0.3560 | 0.2364 | 0.8854 | 0.0469 | 0.0093 | 0.3196 | 0.0500 | 0.0098 | 0.3233 | 0.0378 | 0.0089 | 0.3155 | 0.0472 | 0.0094 | 0.3201 | 0.0343 | 0.0088 | 0.3155 | ||

$\gamma $ | -0.1046 | 0.2050 | 3.0668 | 0.0120 | 0.0107 | 0.4050 | 0.0156 | 0.0109 | 0.4074 | 0.0039 | 0.0106 | 0.4035 | 0.0127 | 0.0107 | 0.4044 | 0.0039 | 0.0106 | 0.4055 | ||

3 | 40 | $\alpha $ | 0.8234 | 1.4545 | 3.4559 | 0.0740 | 0.0971 | 1.1755 | 0.1059 | 0.1094 | 1.1947 | 0.0424 | 0.0878 | 1.1452 | 0.0766 | 0.0977 | 1.1757 | 0.0457 | 0.0923 | 1.1783 |

$\beta $ | 0.4873 | 0.3432 | 1.2753 | 0.0922 | 0.0654 | 0.7968 | 0.1154 | 0.0881 | 0.8305 | 0.0696 | 0.0504 | 0.7560 | 0.0946 | 0.0662 | 0.7991 | 0.0661 | 0.0572 | 0.7675 | ||

$\gamma $ | 0.2195 | 0.8011 | 3.4559 | 0.0058 | 0.1180 | 1.3609 | 0.0420 | 0.1216 | 1.3737 | −0.0302 | 0.1180 | 1.3614 | 0.0079 | 0.1178 | 1.3566 | −0.0170 | 0.1209 | 1.3838 | ||

70 | $\alpha $ | 0.6272 | 0.6506 | 1.9889 | 0.0103 | 0.0211 | 0.5549 | 0.0174 | 0.0216 | 0.5545 | 0.0031 | 0.0208 | 0.5482 | 0.0109 | 0.0212 | 0.5542 | 0.0037 | 0.0211 | 0.5534 | |

$\beta $ | 0.4289 | 0.2312 | 0.8523 | 0.0337 | 0.0225 | 0.4743 | 0.0412 | 0.0224 | 0.4810 | 0.0263 | 0.0234 | 0.4664 | 0.0347 | 0.0219 | 0.4746 | 0.0215 | 0.0329 | 0.4695 | ||

$\gamma $ | 0.2217 | 0.1809 | 1.9889 | 0.0047 | 0.0249 | 0.6148 | 0.0144 | 0.0252 | 0.6171 | −0.0012 | 0.0246 | 0.6083 | 0.0071 | 0.0249 | 0.6162 | 0.0019 | 0.0248 | 0.6135 | ||

150 | $\alpha $ | 0.5738 | 0.6187 | 1.8251 | 0.0103 | 0.0094 | 0.3702 | 0.0184 | 0.0097 | 0.3721 | 0.0029 | 0.0093 | 0.3688 | 0.0105 | 0.0095 | 0.3698 | 0.0023 | 0.0093 | 0.3696 | |

$\beta $ | 0.4170 | 0.1999 | 0.6333 | 0.0330 | 0.0075 | 0.3050 | 0.0424 | 0.0078 | 0.3067 | 0.0254 | 0.0072 | 0.3026 | 0.0340 | 0.0075 | 0.3052 | 0.0204 | 0.0072 | 0.3030 | ||

$\gamma $ | 0.0464 | 0.1263 | 1.8251 | 0.0046 | 0.0106 | 0.3980 | 0.0082 | 0.0107 | 0.3987 | 0.0012 | 0.0105 | 0.3975 | 0.0051 | 0.0106 | 0.3979 | 0.0018 | 0.0106 | 0.3994 |

$\mathit{\alpha}=2,\mathit{\beta}=0.4$ | MLE | SELF | LINEX c = −1.2 | LINEX c = 1.2 | ELF c = −1.2 | ELF c = 1.2 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\gamma $ | n | Bias | MSE | LACI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | |

0.5 | 40 | $\alpha $ | 0.0083 | 0.3381 | 2.2801 | 0.0033 | 0.1169 | 1.3099 | 0.0386 | 0.1191 | 1.3204 | −0.0320 | 0.1181 | 1.3132 | 0.0064 | 0.1163 | 1.3090 | −0.0323 | 0.1268 | 1.3544 |

$\beta $ | 0.1379 | 0.0285 | 0.3821 | 0.0967 | 0.0328 | 0.3317 | 0.1068 | 0.0570 | 0.3433 | 0.0867 | 0.0179 | 0.3178 | 0.0994 | 0.0344 | 0.3362 | 0.0647 | 0.0148 | 0.3092 | ||

$\gamma $ | 0.1991 | 0.1262 | 2.2801 | 0.1147 | 0.0360 | 0.5585 | 0.1295 | 0.0428 | 0.5770 | 0.1000 | 0.0299 | 0.5391 | 0.1183 | 0.0373 | 0.5626 | 0.0722 | 0.0234 | 0.4975 | ||

70 | $\alpha $ | 0.0450 | 0.1868 | 1.6861 | 0.0078 | 0.0253 | 0.6177 | 0.0154 | 0.0257 | 0.6219 | 0.0002 | 0.0251 | 0.6164 | 0.0084 | 0.0253 | 0.6186 | 0.0008 | 0.0254 | 0.6197 | |

$\beta $ | 0.1253 | 0.0207 | 0.2775 | 0.0586 | 0.0078 | 0.2400 | 0.0618 | 0.0084 | 0.2440 | 0.0554 | 0.0073 | 0.2366 | 0.0597 | 0.0080 | 0.2398 | 0.0459 | 0.0061 | 0.2308 | ||

$\gamma $ | 0.1478 | 0.0545 | 1.6861 | 0.0579 | 0.0110 | 0.3304 | 0.0623 | 0.0119 | 0.3397 | 0.0535 | 0.0102 | 0.3230 | 0.0592 | 0.0112 | 0.3312 | 0.0436 | 0.0089 | 0.3162 | ||

150 | $\alpha $ | −0.0186 | 0.2928 | 2.1209 | 0.0105 | 0.0116 | 0.4142 | 0.0140 | 0.0116 | 0.4153 | 0.0071 | 0.0115 | 0.4144 | 0.0108 | 0.0116 | 0.4140 | 0.0073 | 0.0116 | 0.4153 | |

$\beta $ | 0.1209 | 0.0170 | 0.1923 | 0.0555 | 0.0049 | 0.1684 | 0.0570 | 0.0051 | 0.1699 | 0.0539 | 0.0047 | 0.1648 | 0.0560 | 0.0050 | 0.1693 | 0.0493 | 0.0041 | 0.1606 | ||

$\gamma $ | 0.1692 | 0.0742 | 2.1209 | 0.0484 | 0.0059 | 0.2218 | 0.0506 | 0.0062 | 0.2305 | 0.0462 | 0.0056 | 0.2179 | 0.0491 | 0.0060 | 0.2239 | 0.0413 | 0.0049 | 0.2118 | ||

1.7 | 40 | $\alpha $ | 0.3465 | 0.1953 | 1.0757 | 0.0181 | 0.1107 | 1.3160 | 0.0507 | 0.1143 | 1.3186 | −0.0142 | 0.1096 | 1.2998 | 0.0209 | 0.1104 | 1.3154 | −0.0133 | 0.1159 | 1.3541 |

$\beta $ | 0.1095 | 0.0159 | 0.2446 | 0.0443 | 0.0092 | 0.1889 | 0.0481 | 0.0156 | 0.1912 | 0.0407 | 0.0058 | 0.1861 | 0.0454 | 0.0097 | 0.1893 | 0.0325 | 0.0048 | 0.1839 | ||

$\gamma $ | 0.5075 | 0.4021 | 1.0757 | 0.0860 | 0.0925 | 1.1049 | 0.1165 | 0.1052 | 1.1401 | 0.0556 | 0.0823 | 1.0806 | 0.0888 | 0.0932 | 1.1016 | 0.0544 | 0.0864 | 1.1150 | ||

70 | $\alpha $ | 0.2739 | 0.1824 | 1.0594 | 0.0058 | 0.0258 | 0.5942 | 0.0131 | 0.0261 | 0.6032 | −0.0015 | 0.0256 | 0.5983 | 0.0064 | 0.0258 | 0.5922 | −0.0009 | 0.0259 | 0.6005 | |

$\beta $ | 0.1106 | 0.0145 | 0.1856 | 0.0332 | 0.0025 | 0.1311 | 0.0344 | 0.0026 | 0.1328 | 0.0319 | 0.0024 | 0.1298 | 0.0336 | 0.0025 | 0.1316 | 0.0278 | 0.0021 | 0.1288 | ||

$\gamma $ | 0.6235 | 0.3596 | 1.0594 | 0.0265 | 0.0218 | 0.5531 | 0.0332 | 0.0225 | 0.5546 | 0.0197 | 0.0211 | 0.5511 | 0.0271 | 0.0218 | 0.5517 | 0.0192 | 0.0214 | 0.5540 | ||

150 | $\alpha $ | 0.2996 | 0.1379 | 0.8608 | 0.0041 | 0.0115 | 0.4127 | 0.0153 | 0.0117 | 0.4174 | 0.0087 | 0.0113 | 0.4112 | 0.0123 | 0.0115 | 0.4127 | 0.0090 | 0.0114 | 0.4123 | |

$\beta $ | 0.1026 | 0.0116 | 0.1287 | 0.0323 | 0.0020 | 0.0917 | 0.0330 | 0.0023 | 0.0925 | 0.0316 | 0.0019 | 0.0913 | 0.0326 | 0.0021 | 0.0920 | 0.0296 | 0.0016 | 0.0903 | ||

$\gamma $ | 0.5119 | 0.3213 | 0.8608 | 0.0233 | 0.0111 | 0.4042 | 0.0362 | 0.0116 | 0.4083 | 0.0294 | 0.0107 | 0.3999 | 0.0331 | 0.0112 | 0.4044 | 0.0293 | 0.0108 | 0.4013 |

Models | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ | $\mathit{\rho}$ | $\mathit{\theta}$ | AIC | CAIC | BIC | HQIC | CVM | AD | KSD | PVKS |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

KMWE | 0.085 | 34770.449 | 89.070 | - | - | 103.606 | 106.593 | 105.106 | 104.189 | 0.033 | 0.246 | 0.094 | 0.994 |

SEWE | 25.458 | 5.854 | 0.097 | 0.010 | - | 105.516 | 108.183 | 109.499 | 106.294 | 0.032 | 0.232 | 0.097 | 0.991 |

EGWGP | 12.999 | 0.003 | 0.282 | 0.123 | 0.907 | 119.739 | 124.025 | 124.718 | 120.711 | 0.032 | 0.232 | 0.197 | 0.420 |

EGWGP | 272.716 | 45.047 | 1048.387 | 22.000 | 0.073 | 140.606 | 144.892 | 145.585 | 141.578 | 0.033 | 0.238 | 0.331 | 0.025 |

WL | 39.638 | 94.626 | 0.209 | 4.361 | - | 108.018 | 110.685 | 112.001 | 108.796 | 0.068 | 0.481 | 0.142 | 0.818 |

MOAPW | 8.685 | 13.482 | 14.556 | 94.164 | 108.963 | 111.629 | 112.946 | 109.740 | 0.049 | 0.370 | 0.131 | 0.880 | |

EOWL | 57.762 | 0.923 | 1.414 | - | 163.848 | 106.082 | 108.749 | 110.065 | 106.860 | 0.028 | 0.218 | 0.100 | 0.988 |

MKITL | 112.748 | 0.174 | - | - | - | 104.023 | 104.729 | 106.014 | 104.412 | 0.068 | 0.482 | 0.142 | 0.817 |

OWITL | 113.746 | 82.382 | - | - | 0.170 | 106.022 | 107.522 | 109.009 | 106.605 | 0.068 | 0.482 | 0.142 | 0.817 |

EW | 38.762 | 132.052 | - | - | 55.135 | 106.086 | 107.586 | 109.073 | 106.669 | 0.069 | 0.488 | 0.142 | 0.813 |

Methods | Estimates | SE | Lower | Upper | CV | |
---|---|---|---|---|---|---|

MLE | $\alpha $ | 0.0849 | 0.0110 | 0.0632 | 0.1065 | 13.01% |

$\beta $ | 34,770.4490 | 2973.6521 | 28,942.0909 | 40,598.8070 | 8.55% | |

$\gamma $ | 89.0704 | 40.1604 | 10.3561 | 167.7847 | 45.09% | |

Bayesian | $\alpha $ | 0.0848 | 0.0088 | 0.0674 | 0.1014 | 10.38% |

$\beta $ | 34,769.9281 | 172.2018 | 34,449.7473 | 35,119.9013 | 0.50% | |

$\gamma $ | 89.0796 | 12.2706 | 64.5373 | 113.3496 | 13.77% |

Models | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ | $\mathit{\rho}$ | $\mathit{\theta}$ | AIC | CAIC | BIC | HQIC | CVM | AD | KSD | PVKS |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

KMWE | 0.082 | 20539.668 | 18.838 | - | - | 118.941 | 121.928 | 120.441 | 119.524 | 0.027 | 0.232 | 0.092 | 0.996 |

SEWE | 27.567 | 2.619 | 0.017 | 0.020 | - | 121.234 | 123.900 | 125.217 | 122.011 | 0.029 | 0.251 | 0.094 | 0.995 |

EGWGP | 7.494 | 0.054 | 4.458 | 1.189 | 0.650 | 123.381 | 127.667 | 128.359 | 124.353 | 0.031 | 0.267 | 0.100 | 0.989 |

WL | 0.002 | 45.047 | 0.350 | 13.751 | - | 124.276 | 126.942 | 128.259 | 125.053 | 0.072 | 0.523 | 0.149 | 0.765 |

MOAPW | 378.169 | 5.184 | 449.679 | 71.020 | - | 123.167 | 125.833 | 127.149 | 123.944 | 0.037 | 0.318 | 0.106 | 0.977 |

EOWL | 46.765 | 1.246 | 1.120 | - | 122.998 | 121.761 | 124.428 | 125.744 | 122.539 | 0.029 | 0.239 | 0.100 | 0.989 |

MKITL | 76.658 | 0.173 | - | - | - | 120.276 | 120.982 | 122.268 | 120.665 | 0.072 | 0.523 | 0.149 | 0.769 |

OWITL | 77.449 | 38.926 | - | - | 0.167 | 122.275 | 123.775 | 125.262 | 122.858 | 0.072 | 0.523 | 0.149 | 0.766 |

EW | 26.184 | 153.169 | - | - | 63.485 | 122.379 | 123.879 | 125.366 | 122.962 | 0.074 | 0.532 | 0.150 | 0.757 |

Methods | Estimates | SE | Lower | Upper | CV | |
---|---|---|---|---|---|---|

MLE | $\alpha $ | 0.082 | 0.007 | 0.067 | 0.101 | 8.58% |

$\beta $ | 20,539.668 | 123.556 | 34,449.747 | 35,119.901 | 0.60% | |

$\gamma $ | 18.838 | 7.919 | 64.537 | 113.350 | 42.04% | |

Bayesian | $\alpha $ | 0.082 | 0.007 | 0.068 | 0.095 | 8.46% |

$\beta $ | 20,539.536 | 11.230 | 20,517.754 | 20,561.557 | 0.05% | |

$\gamma $ | 18.831 | 2.834 | 13.536 | 24.629 | 15.05% |

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## Share and Cite

**MDPI and ACS Style**

Eldessouky, E.A.; Hassan, O.H.M.; Elgarhy, M.; Hassan, E.A.A.; Elbatal, I.; Almetwally, E.M.
A New Extension of the Kumaraswamy Exponential Model with Modeling of Food Chain Data. *Axioms* **2023**, *12*, 379.
https://doi.org/10.3390/axioms12040379

**AMA Style**

Eldessouky EA, Hassan OHM, Elgarhy M, Hassan EAA, Elbatal I, Almetwally EM.
A New Extension of the Kumaraswamy Exponential Model with Modeling of Food Chain Data. *Axioms*. 2023; 12(4):379.
https://doi.org/10.3390/axioms12040379

**Chicago/Turabian Style**

Eldessouky, Eman A., Osama H. Mahmoud Hassan, Mohammed Elgarhy, Eid A. A. Hassan, Ibrahim Elbatal, and Ehab M. Almetwally.
2023. "A New Extension of the Kumaraswamy Exponential Model with Modeling of Food Chain Data" *Axioms* 12, no. 4: 379.
https://doi.org/10.3390/axioms12040379