# A Type I Generalized Logistic Distribution: Solving Its Estimation Problems with a Bayesian Approach and Numerical Applications Based on Simulated and Engineering Data

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## Abstract

**:**

## 1. Introduction

`R`package named

`glogis`[18] was published without further discussion about this problem [19].

`R`package named

`lmom`[25] has been implemented for the L-moment method.

## 2. The Type I Generalized Logistic Distribution and Related Distributions

## 3. Inference: Classical Approaches

## 4. Inference: A Bayesian Approach

`R`,

`Python`,

`MatLab`, and

`C++`. Using the

`R`language for our simulated scenarios here, it was not possible to achieve reasonable results.

## 5. Simulation Studies

`glogisfit`function of the

`glogis`package [36].

`mipfp`package of

`R`without achieving reliable results. We do not show these results in the paper due to restrictions of space. Table 2 reports the results obtained for the estimates with the moment method and 1000 replicates for the size samples, cases, parameter, and indicators mentioned. These results are not satisfactory and, for the ML method, we do not report the results because they are totally unsatisfactory.

- Case 1: $\mathrm{E}\left({\mu}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})=0$, $\mathrm{Var}\left({\mu}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={({\mu}^{\u2605}+{\sigma}^{\u2605}/2+{b}^{\u2605})}^{1/2}$,$\mathrm{E}\left({\sigma}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\sigma}^{\u2605}+{b}^{\u2605}$, $\mathrm{Var}\left({\sigma}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\mu}^{\u2605\phantom{\rule{0.166667em}{0ex}}2}+3{\sigma}^{\u2605}$,$\mathrm{E}\left({b}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\mu}^{\u2605\phantom{\rule{0.166667em}{0ex}}2}/5+{b}^{\u2605}$, $\mathrm{Var}\left({b}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\sigma}^{\u2605}$.
- Case 2: $\mathrm{E}\left({\mu}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})=0$, $\mathrm{Var}\left({\mu}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\left({\mu}^{\u2605}\right)}^{2}+{\sigma}^{\u2605}/4+{b}^{\u2605}/10$,$\mathrm{E}\left({\sigma}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\sigma}^{\u2605}$, $\mathrm{Var}\left({\sigma}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\mu}^{\u2605\phantom{\rule{0.166667em}{0ex}}2}+{\sigma}^{\u2605}+{b}^{\u2605}$,$\mathrm{E}\left({b}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={b}^{\u2605}$, $\mathrm{Var}\left({b}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\mu}^{\u2605\phantom{\rule{0.166667em}{0ex}}2}+{\sigma}^{\u2605}$.
- Case 3: $\mathrm{E}\left({\mu}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\mu}^{\u2605}$, $\mathrm{Var}\left({\mu}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={({\sigma}^{\u2605}+{b}^{\u2605})}^{2}$,$\mathrm{E}\left({\sigma}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\sigma}^{\u2605}$, $\mathrm{Var}\left({\sigma}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\left(\right)}^{{\mu}^{\u2605}/2}2$.
- Case 4: $\mathrm{E}\left({\mu}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})=0$, $\mathrm{Var}\left({\mu}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\mu}^{\u2605\phantom{\rule{0.166667em}{0ex}}2}+{\sigma}^{\u2605}/4+{b}^{\u2605}/10$,$\mathrm{E}\left({\sigma}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\left({\mu}^{\u2605}\right)}^{2}+{\sigma}^{\u2605}$, $\mathrm{Var}\left({\sigma}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\mu}^{\u2605\phantom{\rule{0.166667em}{0ex}}2}+{\sigma}^{\u2605}/6+{b}^{\u2605}/10$,$\mathrm{E}\left({b}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})=9+{\mu}^{\u2605\phantom{\rule{0.166667em}{0ex}}2}+{b}^{\u2605}/10$, $\mathrm{Var}\left({b}_{t}\right|{\mu}^{\u2605},{\sigma}^{\u2605},{b}^{\u2605})={\mu}^{\u2605\phantom{\rule{0.166667em}{0ex}}2}+{\sigma}^{\u2605}/2$,

## 6. Empirical Application

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**PDF of the IGL distribution (

**a**) and d Skew${}_{M}(\alpha ,\beta )/\mathrm{d}\alpha $ of the IVGL distribution (

**b**) for the indicated values of the parameters.

**Figure 3.**Behavior of the Jeffreys priors, here denoted as z, for $(\sigma ,b)\in $: $(0,0.5)\times (0,0.5)$ (

**a**); $(0,0.5)\times (4,6)$ (

**b**); $(4,6)\times (0,0.5)$ (

**c**); and $(5,7)\times (9,11)$ (

**d**).

**Figure 4.**Histogram and fitted PDF (

**a**) and empirical distribution function with fitted cumulative distribution function (

**b**) and parameters estimated via classical and Bayesian methods using PLS daily flow data.

**Figure 5.**ACF of the chain $\mu \phantom{\rule{0.222222em}{0ex}}|\phantom{\rule{0.222222em}{0ex}}{W}_{n}$ (

**a**), $\sigma \phantom{\rule{0.222222em}{0ex}}|\phantom{\rule{0.222222em}{0ex}}{W}_{n}$, (

**b**), and $b\phantom{\rule{0.222222em}{0ex}}|\phantom{\rule{0.222222em}{0ex}}{W}_{n}$ (

**c**) with PLS daily flow data.

$\mathit{\mu}$ | $\mathit{\sigma}$ | b | $\mathit{n}=15$ | $\mathit{n}=30$ | $\mathit{n}=50$ | $\mathit{n}=100$ |
---|---|---|---|---|---|---|

0 | 2 | 0.05 | 93.5% | 84.5% | 83.0% | 77.0% |

0 | 4 | 0.1 | 94.5% | 85.5% | 84.5% | 80.1% |

0 | 1 | 1 | 98.0% | 98.5% | 99.5% | 100.0% |

0 | 6 | 10 | 88.5% | 80.5% | 82.0% | 70.5% |

**Table 2.**Estimates with the moment method for 1000 replicates of the listed size sample, case, parameter, and indicator.

$n=15$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Mean | −19.09638 | 13.31917 | 0.49371 | −18.81575 | 13.75909 | 0.51013 | |

Bias | −19.09638 | 11.31917 | 0.44371 | −18.81575 | 9.75909 | 0.41013 | |

Relative bias | - | 5.65958 | 8.8742 | - | 2.43977 | 4.1013 | |

Standard deviation | 14.25511 | 4.4811 | 0.21426 | 14.80999 | 4.53213 | 0.22428 | |

MSE | 567.87987 | 148.20384 | 0.24278 | 573.36827 | 115.78 | 0.21851 | |

$n=15$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Mean | −0.2572 | 0.98217 | 1.39323 | 11.42986 | 4.90052 | 2.78161 | |

Bias | −0.2572 | −0.01783 | 0.39323 | 11.42986 | -1.09948 | −7.21839 | |

Relative bias | - | −0.01783 | 0.39323 | - | −0.18325 | −0.72184 | |

Standard deviation | 1.0315 | 0.26367 | 1.42171 | 5.86088 | 1.39999 | 3.27409 | |

MSE | 1.13014 | 0.06984 | 2.17588 | 164.99163 | 3.16883 | 62.82485 | |

$n=30$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Mean | −14.43447 | 11.63792 | 0.38711 | −14.29753 | 12.14067 | 0.4039 | |

Bias | −14.43447 | 9.63792 | 0.33711 | −14.29753 | 8.14067 | 0.3039 | |

Relative bias | - | 4.81896 | 6.7422 | - | 2.03517 | 3.039 | |

Standard deviation | 9.2901 | 3.64442 | 0.14531 | 9.57106 | 3.64783 | 0.15087 | |

MSE | 294.6599 | 106.17127 | 0.13476 | 296.0245 | 79.57715 | 0.11512 | |

$n=30$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Mean | −0.1717 | 0.97755 | 1.468 | 11.33066 | 4.89198 | 3.03369 | |

Bias | −0.1717 | −0.02245 | 0.468 | 11.33066 | −1.10802 | −6.96631 | |

Relative bias | - | −0.02245 | 0.468 | - | −0.18467 | −0.69663 | |

Standard deviation | 1.04284 | 0.2548 | 3.32295 | 5.87308 | 1.14218 | 4.2669 | |

MSE | 1.117 | 0.06543 | 11.26101 | 162.87692 | 2.53227 | 66.73593 | |

$n=50$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Mean | −10.79721 | 10.10306 | 0.31954 | −10.5922 | 10.65166 | 0.33576 | |

Bias | −10.79721 | 8.10306 | 0.26954 | -10.5922 | 6.65166 | 0.23576 | |

Relative bias | - | 4.05153 | 5.3908 | - | 1.66291 | 2.3576 | |

Standard deviation | 8.38061 | 3.34231 | 0.13118 | 8.62776 | 3.34787 | 0.13503 | |

MSE | 186.81438 | 76.83061 | 0.08986 | 186.63291 | 55.45283 | 0.07382 | |

$n=50$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Mean | −0.12718 | 0.97281 | 1.30513 | 9.72513 | 5.09282 | 3.83445 | |

Bias | −0.12718 | −0.02719 | 0.30513 | 9.72513 | −0.90718 | −6.16555 | |

Relative bias | - | −0.02719 | 0.30513 | - | −0.1512 | −0.61656 | |

Standard deviation | 0.92368 | 0.21061 | 1.61055 | 5.55714 | 0.89339 | 5.36295 | |

MSE | 0.86936 | 0.04509 | 2.68698 | 125.45997 | 1.62112 | 66.77523 | |

$n=100$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Mean | −8.57027 | 9.05183 | 0.272 | −8.21102 | 9.56039 | 0.28589 | |

Bias | −8.57027 | 7.05183 | 0.222 | −8.21102 | 5.56039 | 0.18589 | |

Relative bias | - | 3.52591 | 4.44 | - | 1.3901 | 1.8589 | |

Standard deviation | 5.91348 | 2.60411 | 0.09776 | 6.22217 | 2.70276 | 0.10234 | |

MSE | 108.41877 | 56.50969 | 0.05884 | 106.13622 | 38.22284 | 0.04503 | |

$n=100$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Mean | −0.19489 | 1.00876 | 1.23021 | 7.13316 | 5.4028 | 5.68398 | |

Bias | −0.19489 | 0.00876 | 0.23021 | 7.13316 | −0.5972 | −4.31602 | |

Relative bias | - | 0.00876 | 0.23021 | - | −0.09953 | −0.4316 | |

Standard deviation | 0.67739 | 0.15986 | 0.68861 | 5.93529 | 0.75779 | 7.43532 | |

MSE | 0.49684 | 0.02563 | 0.52718 | 86.10964 | 0.93089 | 73.91197 |

**Table 3.**Bayesian estimate (posterior mean) for 1000 replicates of the listed size sample, case, parameter, and indicator.

$n=15$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Mean | −0.019 | 3.102 | 0.081 | 0.061 | 3.711 | 0.093 | |

Bias | −0.019 | 1.102 | 0.031 | 0.061 | −0.289 | −0.007 | |

Relative bias | - | 0.551 | 0.62 | - | −0.072 | −0.07 | |

Standard deviation | 0.08 | 0.632 | 0.024 | 0.406 | 0.157 | 0.024 | |

MSE | 0.007 | 1.614 | 0.002 | 0.169 | 0.108 | 0.001 | |

$n=15$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Mean | 0.086 | 0.962 | 1.363 | 0.084 | 5.918 | 10.517 | |

Bias | 0.086 | −0.038 | 0.363 | 0.084 | −0.082 | 0.517 | |

Relative bias | - | −0.038 | 0.363 | - | −0.014 | 0.052 | |

Standard deviation | 0.85 | 0.331 | 0.683 | 0.172 | 0.746 | 0.742 | |

MSE | 0.729 | 0.111 | 0.598 | 0.037 | 0.563 | 0.818 | |

$n=30$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Mean | −0.025 | 2.737 | 0.07 | 0.062 | 3.705 | 0.093 | |

Bias | −0.025 | 0.737 | 0.02 | 0.062 | −0.295 | −0.007 | |

Relative bias | - | 0.368 | 0.4 | - | −0.074 | −0.07 | |

Standard deviation | 0.122 | 0.719 | 0.02 | 0.558 | 0.221 | 0.018 | |

MSE | 0.015 | 1.061 | 0.001 | 0.315 | 0.136 | 0 | |

$n=30$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Mean | −0.097 | 0.996 | 1.419 | 0.088 | 5.956 | 10.429 | |

Bias | −0.097 | −0.004 | 0.419 | 0.088 | −0.044 | 0.429 | |

Relative bias | - | −0.004 | 0.419 | - | −0.007 | 0.043 | |

Standard deviation | 0.836 | 0.267 | 0.761 | 0.21 | 0.581 | 0.949 | |

MSE | 0.709 | 0.072 | 0.755 | 0.052 | 0.339 | 1.085 | |

$n=50$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Mean | −0.007 | 2.502 | 0.064 | 0.086 | 3.71 | 0.093 | |

Bias | −0.007 | 0.502 | 0.014 | 0.086 | −0.29 | −0.007 | |

Relative bias | - | 0.251 | 0.28 | - | −0.072 | −0.07 | |

Standard deviation | 0.151 | 0.666 | 0.018 | 0.634 | 0.241 | 0.014 | |

MSE | 0.023 | 0.696 | 0.001 | 0.409 | 0.142 | 0 | |

$n=50$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Mean | −0.186 | 1.017 | 1.404 | 0.078 | 5.961 | 10.379 | |

Bias | −0.186 | 0.017 | 0.404 | 0.078 | −0.039 | 0.379 | |

Relative bias | - | 0.017 | 0.404 | - | −0.006 | 0.038 | |

Standard deviation | 0.754 | 0.196 | 0.754 | 0.238 | 0.47 | 1.063 | |

MSE | 0.604 | 0.039 | 0.731 | 0.063 | 0.223 | 1.273 | |

$n=100$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Mean | −0.012 | 2.383 | 0.061 | 0.05 | 3.779 | 0.096 | |

Bias | −0.012 | 0.383 | 0.011 | 0.05 | −0.221 | −0.004 | |

Relative bias | - | 0.192 | 0.22 | - | −0.055 | −0.040 | |

Standard deviation | 0.194 | 0.572 | 0.016 | 0.707 | 0.272 | 0.011 | |

MSE | 0.038 | 0.474 | 0 | 0.502 | 0.123 | 0 | |

$n=100$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Mean | −0.221 | 1.027 | 1.314 | 0.067 | 5.96 | 10.389 | |

Bias | −0.221 | 0.027 | 0.314 | 0.067 | −0.04 | 0.389 | |

Relative bias | - | 0.027 | 0.314 | - | −0.007 | 0.039 | |

Standard deviation | 0.561 | 0.151 | 0.531 | 0.187 | 0.357 | 0.879 | |

MSE | 0.364 | 0.023 | 0.38 | 0.039 | 0.129 | 0.924 |

**Table 4.**Bayesian estimate (posterior median) for 1000 replicates of the listed size sample, case, parameter, and indicator.

$n=15$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Mean | −0.026 | 2.747 | 0.069 | 0.065 | 3.638 | 0.087 | |

Bias | −0.026 | 0.747 | 0.019 | 0.065 | −0.362 | −0.013 | |

Relative bias | - | 0.374 | 0.38 | - | −0.09 | −0.13 | |

Standard deviation | 0.095 | 0.624 | 0.023 | 0.398 | 0.158 | 0.022 | |

MSE | 0.01 | 0.947 | 0.001 | 0.163 | 0.156 | 0.001 | |

$n=15$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Mean | 0.22 | 0.907 | 0.935 | 0.081 | 5.84 | 10.267 | |

Bias | 0.22 | −0.093 | −0.065 | 0.081 | −0.16 | 0.267 | |

Relative bias | - | −0.093 | −0.065 | - | −0.027 | 0.027 | |

Standard deviation | 0.924 | 0.346 | 0.513 | 0.191 | 0.724 | 0.728 | |

MSE | 0.903 | 0.128 | 0.267 | 0.043 | 0.549 | 0.602 | |

$n=30$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Mean | −0.034 | 2.472 | 0.062 | 0.064 | 3.632 | 0.089 | |

Bias | −0.034 | 0.472 | 0.012 | 0.064 | −0.368 | −0.011 | |

Relative bias | - | 0.236 | 0.24 | - | −0.092 | −0.11 | |

Standard deviation | 0.141 | 0.717 | 0.02 | 0.543 | 0.22 | 0.017 | |

MSE | 0.021 | 0.737 | 0.001 | 0.299 | 0.184 | 0 | |

$n=30$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Mean | 0.017 | 0.966 | 1.079 | 0.092 | 5.903 | 10.214 | |

Bias | 0.017 | −0.034 | 0.079 | 0.092 | −0.097 | 0.214 | |

Relative bias | - | −0.034 | 0.079 | - | −0.016 | 0.021 | |

Standard deviation | 0.845 | 0.27 | 0.571 | 0.245 | 0.568 | 0.935 | |

MSE | 0.714 | 0.074 | 0.332 | 0.068 | 0.332 | 0.92 | |

$n=50$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Mean | −0.014 | 2.304 | 0.058 | 0.09 | 3.644 | 0.09 | |

Bias | −0.014 | 0.304 | 0.008 | 0.09 | −0.356 | −0.01 | |

Relative bias | - | 0.152 | 0.16 | - | −0.089 | −0.1 | |

Standard deviation | 0.18 | 0.672 | 0.018 | 0.624 | 0.239 | 0.014 | |

MSE | 0.033 | 0.545 | 0 | 0.398 | 0.184 | 0 | |

$n=50$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Mean | −0.089 | 0.996 | 1.142 | 0.085 | 5.922 | 10.185 | |

Bias | −0.089 | −0.004 | 0.142 | 0.085 | −0.078 | 0.185 | |

Relative bias | - | −0.004 | 0.142 | - | −0.013 | 0.018 | |

Standard deviation | 0.751 | 0.197 | 0.57 | 0.282 | 0.46 | 1.045 | |

MSE | 0.572 | 0.039 | 0.346 | 0.087 | 0.217 | 1.126 | |

$n=100$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Mean | −0.022 | 2.258 | 0.057 | 0.064 | 3.72 | 0.093 | |

Bias | −0.022 | 0.258 | 0.007 | 0.064 | −0.28 | −0.007 | |

Relative bias | - | 0.129 | 0.14 | - | −0.07 | −0.07 | |

Standard deviation | 0.231 | 0.582 | 0.016 | 0.693 | 0.272 | 0.011 | |

MSE | 0.054 | 0.406 | 0 | 0.485 | 0.152 | 0 | |

$n=100$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Mean | −0.154 | 1.015 | 1.16 | 0.076 | 5.938 | 10.218 | |

Bias | −0.154 | 0.015 | 0.16 | 0.076 | −0.062 | 0.218 | |

Relative bias | - | 0.015 | 0.16 | - | −0.01 | 0.022 | |

Standard deviation | 0.554 | 0.152 | 0.43 | 0.223 | 0.349 | 0.856 | |

MSE | 0.331 | 0.023 | 0.211 | 0.055 | 0.125 | 0.781 |

**Table 5.**Results of the diagnostics for 1000 replicates of the listed size sample, case, parameter, and test.

$n=15$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Gelman-Rubin | - | 1.2081 | - | - | 1.0644 | - | |

Geweke | 0.9516 | 0.648 | 0.6535 | 0.6399 | 0.3411 | 0.1022 | |

Ljung-Box | 0.112 | 0.2358 | 0.2497 | 0.0729 | 0.2563 | 0.1289 | |

$n=15$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Gelman-Rubin | - | 1.1755 | - | - | 1.0706 | - | |

Geweke | 0.2091 | 0.438 | 0.3774 | 0.7777 | 0.5109 | 0.7443 | |

Ljung-Box | 0.525 | 0.4431 | 0.2327 | 0.6502 | 0.1333 | 0.1462 | |

$n=30$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Gelman-Rubin | - | 1.017 | - | - | 1.0218 | - | |

Geweke | 0.2203 | 0.0943 | 0.1702 | 0.9785 | 0.9788 | 0.427 | |

Ljung-Box | 0.6654 | 0.0271 | 0.0022 | 0.737 | 0.4986 | 0.7545 | |

$n=30$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Gelman-Rubin | - | 1.0758 | - | - | 1.0414 | - | |

Geweke | 0.5641 | 0.2097 | 0.4419 | 0.722 | 0.5419 | 0.7722 | |

Ljung-Box | 0.2461 | 0.8968 | 0.193 | 0.0779 | 0.0902 | 0.3097 | |

$n=50$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Gelman-Rubin | - | 1.0209 | - | - | 1.0305 | - | |

Geweke | 0.8796 | 0.542 | 0.6853 | 0.5699 | 0.2915 | 0.5914 | |

Ljung-Box | 0.0822 | 0.4869 | 0.2854 | 0.2587 | 0.7124 | 0.6139 | |

$n=50$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Gelman-Rubin | - | 1.1348 | - | - | 1.1341 | - | |

Geweke | 0.5635 | 0.4084 | 0.5817 | 0.9604 | 0.7169 | 0.6362 | |

Ljung-Box | 0.1675 | 0.6055 | 0.0738 | 0.4626 | 0.2712 | 0.0961 | |

$n=100$ | Cases 1–2 | $\mu =0$ | $\sigma =2$ | $b=0.05$ | $\mu =0$ | $\sigma =4$ | $b=0.1$ |

Gelman-Rubin | - | 1.1659 | - | - | 1.0252 | - | |

Geweke | 0.6261 | 0.5756 | 0.8681 | 0.8935 | 0.6845 | 0.9877 | |

Ljung-Box | 0.0754 | 0.1719 | 0.0878 | 0.0942 | 0.2024 | 0.5346 | |

$n=100$ | Cases 3–4 | $\mu =0$ | $\sigma =1$ | $b=1$ | $\mu =0$ | $\sigma =6$ | $b=10$ |

Gelman-Rubin | - | 1.0038 | - | - | 1.1379 | - | |

Geweke | 0.4396 | 0.1397 | 0.4124 | 0.0997 | 0.8876 | 0.5265 | |

Ljung-Box | 0.7793 | 0.1469 | 0.3915 | 0.1579 | 0.2634 | 0.1191 |

Mean | Median | Variance | Standard Deviation | Coefficient of Skewness |
---|---|---|---|---|

10,739.57 | 10,999.74 | 318,485.03 | 564.34 | −3.50 |

Parameter | $\mathit{\mu}$ | $\mathit{\sigma}$ | b |
---|---|---|---|

p-value | 0.60 | 0.34 | 0.04 |

**Table 8.**Classical and Bayesian estimates of $\theta ={(\mu ,\sigma ,b)}^{\top}$ with PLS daily flow data.

Parameter | |||
---|---|---|---|

Indicator | $\mathit{\mu}$ | $\mathit{\sigma}$ | b |

Estimate | 604.75 | 1184.63 | 3866.72 |

Posterior mean | 11,000.27 | 177.55 | 0.49 |

Posterior median | 11,000.24 | 177.55 | 0.49 |

Variance | 1.18 | 4.15 | 0.00 |

Standar deviation | 1.09 | 2.04 | 0.02 |

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**MDPI and ACS Style**

Lagos-Álvarez, B.; Jerez-Lillo, N.; Navarrete, J.P.; Figueroa-Zúñiga, J.; Leiva, V.
A Type I Generalized Logistic Distribution: Solving Its Estimation Problems with a Bayesian Approach and Numerical Applications Based on Simulated and Engineering Data. *Symmetry* **2022**, *14*, 655.
https://doi.org/10.3390/sym14040655

**AMA Style**

Lagos-Álvarez B, Jerez-Lillo N, Navarrete JP, Figueroa-Zúñiga J, Leiva V.
A Type I Generalized Logistic Distribution: Solving Its Estimation Problems with a Bayesian Approach and Numerical Applications Based on Simulated and Engineering Data. *Symmetry*. 2022; 14(4):655.
https://doi.org/10.3390/sym14040655

**Chicago/Turabian Style**

Lagos-Álvarez, Bernardo, Nixon Jerez-Lillo, Jean P. Navarrete, Jorge Figueroa-Zúñiga, and Víctor Leiva.
2022. "A Type I Generalized Logistic Distribution: Solving Its Estimation Problems with a Bayesian Approach and Numerical Applications Based on Simulated and Engineering Data" *Symmetry* 14, no. 4: 655.
https://doi.org/10.3390/sym14040655