# An Extension of the Poisson Distribution: Features and Application for Medical Data Modeling

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## Abstract

**:**

## 1. Introduction

## 2. Statistical Properties

#### 2.1. Moments and Auxiliary Statistical Measures

#### 2.2. Conditional Expectation

#### 2.3. Order Statistic (OrSc)

#### 2.4. Lorenz Curve

## 3. Estimation Methods: Unbiased and Consistent Estimators

#### 3.1. Maximal Likelihood Estimation

#### 3.2. Moment Estimation

## 4. Estimator Performance: Simulation Results

## 5. Data Analysis: Kidney Dysmorphogenetics

## 6. Results and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Scheme I ($a=0.1$) | Scheme II ($a=0.4$) | |||

n | Bias | MSE | Bias | MSE |

20 | $0.65434672$ | $0.46384901$ | $0.43678491$ | $0.34579056$ |

50 | $0.55763945$ | $0.37466489$ | $0.33578994$ | $0.28456410$ |

100 | $0.37655988$ | $0.26654540$ | $0.24234667$ | $0.17746939$ |

150 | $0.21457409$ | $0.15646744$ | $0.13456397$ | $0.10099874$ |

300 | $0.14673813$ | $0.10397464$ | $0.09478516$ | $0.07756379$ |

500 | $0.07846543$ | $0.01547531$ | $0.00466599$ | $0.00298575$ |

700 | $0.00847645$ | $0.00376549$ | $0.00016859$ | $0.00008946$ |

1000 | $0.00006455$ | $0.00002135$ | $0.00000344$ | $0.00000037$ |

scheme III ($a=0.7$) | scheme IV ($a=0.9$) | |||

Bias | MSE | Bias | MSE | |

20 | $1.28766550$ | $0.74578902$ | $0.83462601$ | $0.62399457$ |

50 | $0.98464654$ | $0.56458012$ | $0.66357910$ | $0.42095752$ |

100 | $0.67351458$ | $0.31254684$ | $0.37847655$ | $0.28546781$ |

150 | $0.38734698$ | $0.17455449$ | $0.21824765$ | $0.13683602$ |

300 | $0.21985857$ | $0.07387475$ | $0.12049875$ | $0.07488585$ |

500 | $0.12857676$ | $0.00354784$ | $0.08465547$ | $0.00110487$ |

700 | $0.08957654$ | $0.00018366$ | $0.00015424$ | $0.00028475$ |

1000 | $0.00023248$ | $0.00004571$ | $0.00000154$ | $0.00000114$ |

Scheme I ($a=0.1$) | Scheme II ($a=0.4$) | |||

n | Bias | MSE | Bias | MSE |

20 | $0.68456473$ | $0.49365504$ | $0.44984619$ | $0.36204875$ |

50 | $0.57735547$ | $0.38835547$ | $0.36825548$ | $0.29344648$ |

100 | $0.39465473$ | $0.28143054$ | $0.25846510$ | $0.18834750$ |

150 | $0.22287464$ | $0.16637550$ | $0.13876450$ | $0.10274654$ |

300 | $0.15344649$ | $0.11846548$ | $0.09535378$ | $0.09547785$ |

500 | $0.08876354$ | $0.01610476$ | $0.00524367$ | $0.00217455$ |

700 | $0.00345674$ | $0.00437386$ | $0.00022467$ | $0.00002356$ |

1000 | $0.00009365$ | $0.00003765$ | $0.00000673$ | $0.00000019$ |

scheme III ($a=0.7$) | scheme IV ($a=0.9$) | |||

Bias | MSE | Bias | MSE | |

20 | $1.13664785$ | $0.68465541$ | $0.76388994$ | $0.58635531$ |

50 | $0.90354789$ | $0.51873654$ | $0.61048655$ | $0.39746571$ |

100 | $0.61546489$ | $0.28465483$ | $0.32286556$ | $0.27454547$ |

150 | $0.35478436$ | $0.14438761$ | $0.18455675$ | $0.12445678$ |

300 | $0.19354465$ | $0.05645831$ | $0.11957650$ | $0.09877577$ |

500 | $0.10456738$ | $0.00273568$ | $0.07754891$ | $0.00465587$ |

700 | $0.07745831$ | $0.00004654$ | $0.00003456$ | $0.00065972$ |

1000 | $0.00003865$ | $0.00000314$ | $0.00000064$ | $0.00000946$ |

Observed | Expected Frequencies | |||||||
---|---|---|---|---|---|---|---|---|

X | Frequencies | DWPLT | Geo | DR | DIR | DBL | Poi | DPa |

0 | 65 | $63.34$ | $45.98$ | $10.89$ | $60.89$ | $32.08$ | $27.39$ | $65.84$ |

1 | 14 | $19.78$ | $26.76$ | $26.62$ | $33.99$ | $37.10$ | $38.08$ | $18.27$ |

2 | 10 | $9.26$ | $15.58$ | $29.45$ | $8.12$ | $21.66$ | $26.47$ | $8.16$ |

3 | 6 | $5.20$ | $9.06$ | $22.29$ | $3.00$ | $10.63$ | $12.26$ | $4.51$ |

4 | 4 | $3.25$ | $5.28$ | $12.63$ | $1.42$ | $4.84$ | $4.26$ | $2.82$ |

5 | 2 | $2.17$ | $3.07$ | $5.54$ | $0.78$ | $2.12$ | $1.19$ | $1.91$ |

6 | 2 | $1.53$ | $1.79$ | $1.91$ | $0.47$ | $0.91$ | $0.27$ | $1.37$ |

7 | 2 | $1.11$ | $1.04$ | $0.53$ | $0.31$ | $0.38$ | $0.05$ | $1.02$ |

8 | 1 | $0.83$ | $0.61$ | $0.12$ | $0.21$ | $0.16$ | $0.01$ | $0.79$ |

9 | 1 | $0.64$ | $0.35$ | $0.02$ | $0.15$ | $0.07$ | $0.00$ | $0.63$ |

10 | 1 | $0.49$ | $0.21$ | $0.00$ | $0.11$ | $0.03$ | $0.00$ | $0.51$ |

11 | 2 | $2.40$ | $0.27$ | $0.00$ | $0.55$ | $0.02$ | $0.02$ | $4.17$ |

Total | 110 | 110 | 110 | 110 | 110 | 110 | 110 | 110 |

a | $\begin{array}{c}{\mathrm{MLE}}_{a}\\ {\mathrm{Se}}_{a}\\ {\mathrm{L}.\mathrm{C}.\mathrm{I}}_{a}\\ {\mathrm{U}.\mathrm{C}.\mathrm{I}}_{a}\end{array}$ | $\begin{array}{c}0.937\\ 0.028\\ 0.881\\ 0.992\end{array}$ | $\begin{array}{c}0.582\\ 0.030\\ 0.522\\ 0.641\end{array}$ | $\begin{array}{c}0.901\\ 0.009\\ 0.883\\ 0.919\end{array}$ | $\begin{array}{c}0.554\\ 0.049\\ 0.458\\ 0.649\end{array}$ | $\begin{array}{c}0.643\\ 0.020\\ 0.604\\ 0.682\end{array}$ | $\begin{array}{c}1.390\\ 0.112\\ 1.171\\ 1.611\end{array}$ | $\begin{array}{c}0.268\\ 0.034\\ 0.201\\ 0.336\end{array}$ |

$-L$ | $169.32$ | $178.77$ | $277.78$ | $186.55$ | $207.44$ | $246.21$ | $171.19$ | |

Ac | $340.64$ | $359.53$ | $557.56$ | $375.09$ | $416.87$ | $494.42$ | $344.38$ | |

Bc | $343.34$ | $362.23$ | $560.26$ | $377.79$ | $419.57$ | $497.12$ | $347.08$ | |

CAc | $340.68$ | $359.57$ | $557.59$ | $375.13$ | $416.91$ | $494.46$ | $344.42$ | |

Hc | $341.74$ | $360.63$ | $558.65$ | $376.19$ | $417.97$ | $495.52$ | $345.48$ | |

Chi${}^{2}$ | $2.548$ | $19.109$ | $306.515$ | $40.456$ | $61.37$ | $89.277$ | $3.430$ | |

Dm | 4 | 4 | 4 | 2 | 3 | 3 | 4 | |

Pv | $0.636$ | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | $0.489$ |

Observed | Expected Frequencies | |||||||
---|---|---|---|---|---|---|---|---|

X | Frequencies | DWPLT | DF-I | DLogL | DIW | DLo | Bin | DB-II |

0 | 65 | $63.34$ | $45.26$ | $63.19$ | $63.91$ | $61.62$ | $27.94$ | $64.74$ |

1 | 14 | $19.78$ | $29.09$ | $20.10$ | $20.69$ | $21.02$ | $38.44$ | $19.18$ |

2 | 10 | $9.26$ | $16.51$ | $8.64$ | $8.05$ | $9.69$ | $26.29$ | $8.48$ |

3 | 6 | $5.20$ | $8.89$ | $4.66$ | $4.23$ | $5.28$ | $11.92$ | $4.63$ |

4 | 4 | $3.25$ | $4.70$ | $2.86$ | $2.59$ | $3.19$ | $4.03$ | $2.86$ |

5 | 2 | $2.17$ | $2.49$ | $1.92$ | $1.75$ | $2.09$ | $1.08$ | $1.92$ |

6 | 2 | $1.53$ | $1.34$ | $1.39$ | $1.26$ | $1.44$ | $0.24$ | $1.37$ |

7 | 2 | $1.11$ | $0.73$ | $1.02$ | $0.95$ | $1.04$ | $0.05$ | $1.01$ |

8 | 1 | $0.83$ | $0.41$ | $0.79$ | $0.74$ | $0.77$ | $0.00$ | $0.78$ |

9 | 1 | $0.64$ | $0.23$ | $0.62$ | $0.59$ | $0.59$ | $0.00$ | $0.61$ |

10 | 1 | $0.49$ | $0.14$ | $0.50$ | $0.49$ | $0.46$ | $0.00$ | $0.49$ |

11 | 2 | $2.40$ | $0.21$ | $4.31$ | $4.75$ | $2.81$ | $0.01$ | $3.93$ |

Total | 110 | 110 | 110 | 110 | 110 | 110 | 110 | 110 |

a | $\begin{array}{c}{\mathrm{MLE}}_{a}\\ {\mathrm{Se}}_{a}\\ \mathrm{L}.{\mathrm{C}.\mathrm{I}}_{a}\\ \mathrm{U}.{\mathrm{C}.\mathrm{I}}_{a}\end{array}$ | $\begin{array}{c}0.937\\ 0.028\\ 0.881\\ 0.992\end{array}$ | $\begin{array}{c}0.623\\ 0.031\\ 0.563\\ 0.684\end{array}$ | $\begin{array}{c}0.780\\ 0.136\\ 0.514\\ 1.046\end{array}$ | $\begin{array}{c}0.581\\ 0.048\\ 0.488\\ 0.675\end{array}$ | $\begin{array}{c}0.152\\ 0.089\\ 0\\ 0.345\end{array}$ | $\begin{array}{c}170.608\\ 0.831\\ 168.979\\ 172.237\end{array}$ | $\begin{array}{c}0.278\\ 0.045\\ 0.189\\ 0.366\end{array}$ |

X | Frequencies | DWPLT | DF-I | DLogL | DIW | DLo | Bin | DB-II |

b | $\begin{array}{c}{\mathrm{MLE}}_{b}\\ {\mathrm{Se}}_{b}\\ \mathrm{L}.{\mathrm{C}.\mathrm{I}}_{b}\\ \mathrm{U}.{\mathrm{C}.\mathrm{I}}_{b}\end{array}$ | $\begin{array}{c}-\\ -\\ -\\ -\end{array}$ | $\begin{array}{c}-\\ -\\ -\\ -\end{array}$ | $\begin{array}{c}1.208\\ 0.159\\ 0.895\\ 1.520\end{array}$ | $\begin{array}{c}1.049\\ 0.146\\ 0.763\\ 1.335\end{array}$ | $\begin{array}{c}1.830\\ 0.952\\ 0\\ 3.698\end{array}$ | $\begin{array}{c}0.008\\ 0.012\\ 0\\ 0.032\end{array}$ | $\begin{array}{c}1.053\\ 0.167\\ 0.725\\ 1.381\end{array}$ |

$-L$ | $169.32$ | $182.29$ | $171.72$ | $172.94$ | $170.48$ | $247.74$ | $171.14$ | |

Ac | $340.64$ | $366.58$ | $347.43$ | $349.87$ | $344.96$ | $499.48$ | $346.28$ | |

Bc | $343.34$ | $369.28$ | $352.84$ | $355.27$ | $350.36$ | $504.88$ | $351.68$ | |

CAc | $340.68$ | $366.61$ | $347.55$ | $349.99$ | $345.07$ | $499.59$ | $346.39$ | |

Hc | $341.74$ | $367.67$ | $349.62$ | $352.07$ | $347.15$ | $501.67$ | $348.47$ | |

Chi${}^{2}$ | $2.548$ | $31.702$ | $4.033$ | $6.445$ | $3.238$ | $94.729$ | $2.587$ | |

Dm | 4 | 4 | 3 | 3 | 3 | 2 | 2 | |

Pv | $0.636$ | <0.001 | $0.258$ | $0.092$ | $0.356$ | <0.001 | $0.274$ |

Observed | Expected Frequencies | ||||||||
---|---|---|---|---|---|---|---|---|---|

X | Frequencies | DWPLT | DL-I | DL-II | DL-III | NDL | PoiL | DITL | DGL |

0 | 65 | $63.34$ | $40.29$ | $46.03$ | $46.01$ | $41.96$ | $44.14$ | $52.94$ | $46.01$ |

1 | 14 | $19.78$ | $29.83$ | $26.77$ | $26.77$ | $28.80$ | $28.00$ | $28.29$ | $26.76$ |

2 | 10 | $9.26$ | $18.36$ | $15.57$ | $15.58$ | $17.57$ | $16.7$ | $12.09$ | $15.57$ |

3 | 6 | $5.20$ | $10.34$ | $9.06$ | $9.06$ | $10.05$ | $9.57$ | $5.99$ | $9.06$ |

4 | 4 | $3.25$ | $5.52$ | $5.27$ | $5.27$ | $5.52$ | $5.34$ | $3.34$ | $5.27$ |

5 | 2 | $2.17$ | $2.85$ | $3.07$ | $3.07$ | $2.95$ | $2.92$ | $2.03$ | $3.07$ |

6 | 2 | $1.53$ | $1.44$ | $1.79$ | $1.78$ | $1.54$ | $1.57$ | $1.31$ | $1.78$ |

7 | 2 | $1.11$ | $0.71$ | $1.04$ | $1.04$ | $0.79$ | $0.84$ | $0.89$ | $1.04$ |

8 | 1 | $0.83$ | $0.35$ | $0.60$ | $0.60$ | $0.40$ | $0.44$ | $0.63$ | $0.60$ |

9 | 1 | $0.64$ | $0.17$ | $0.35$ | $0.35$ | $0.20$ | $0.23$ | $0.46$ | $0.35$ |

10 | 1 | $0.49$ | $0.08$ | $0.20$ | $0.20$ | $0.10$ | $0.12$ | $0.35$ | $0.20$ |

11 | 2 | $2.40$ | $0.06$ | $0.25$ | $0.27$ | $0.12$ | $0.13$ | $1.68$ | $0.29$ |

Total | 110 | 110 | 110 | 110 | 110 | 110 | 110 | 110 | 110 |

a | $\begin{array}{c}{\mathrm{MLE}}_{a}\\ {\mathrm{Se}}_{a}\\ \mathrm{L}.{\mathrm{C}.\mathrm{I}}_{a}\\ \mathrm{U}.{\mathrm{C}.\mathrm{I}}_{a}\end{array}$ | $\begin{array}{c}0.937\\ 0.028\\ 0.881\\ 0.992\end{array}$ | $\begin{array}{c}0.436\\ 0.026\\ 0.385\\ 0.488\end{array}$ | $\begin{array}{c}0.581\\ 0.045\\ 0.492\\ 0.670\end{array}$ | $\begin{array}{c}0.582\\ 0.005\\ 0.493\\ 0.671\end{array}$ | $\begin{array}{c}0.542\\ 0.026\\ 0.491\\ 0.594\end{array}$ | $\begin{array}{c}1.087\\ 0.109\\ 0.873\\ 1.301\end{array}$ | $\begin{array}{c}2.281\\ 0.221\\ 1.849\\ 2.714\end{array}$ | $\begin{array}{c}0.582\\ 0.045\\ 0.493\\ 0.670\end{array}$ |

b | $\begin{array}{c}{\mathrm{MLE}}_{b}\\ {\mathrm{Se}}_{b}\\ \mathrm{L}.{\mathrm{C}.\mathrm{I}}_{b}\\ \mathrm{U}.{\mathrm{C}.\mathrm{I}}_{b}\end{array}$ | $\begin{array}{c}-\\ -\\ -\\ -\end{array}$ | $\begin{array}{c}-\\ -\\ -\\ -\end{array}$ | $\begin{array}{c}0.001\\ 0.058\\ 0\\ 0.116\end{array}$ | $\begin{array}{c}358.728\\ 11,863.370\\ 0\\ 2.3\times {10}^{4}\end{array}$ | $\begin{array}{c}-\\ -\\ -\\ -\end{array}$ | $\begin{array}{c}-\\ -\\ -\\ -\end{array}$ | $\begin{array}{c}-\\ -\\ -\\ -\end{array}$ | $\begin{array}{c}0.351\\ 0.065\\ 0.223\\ 0.479\end{array}$ |

c | $\begin{array}{c}{\mathrm{MLE}}_{c}\\ {\mathrm{Se}}_{c}\\ \mathrm{L}.{\mathrm{C}.\mathrm{I}}_{c}\\ \mathrm{U}.{\mathrm{C}.\mathrm{I}}_{c}\end{array}$ | $\begin{array}{c}-\\ -\\ -\\ -\end{array}$ | $\begin{array}{c}-\\ -\\ -\\ -\end{array}$ | $\begin{array}{c}-\\ -\\ -\\ -\end{array}$ | $\begin{array}{c}0.001\\ 20.698\\ 0\\ 22.691\end{array}$ | $\begin{array}{c}-\\ -\\ -\\ -\end{array}$ | $\begin{array}{c}-\\ -\\ -\\ -\end{array}$ | $\begin{array}{c}-\\ -\\ -\\ -\end{array}$ | $\begin{array}{c}-\\ -\\ -\\ -\end{array}$ |

$-L$ | $169.32$ | $189.11$ | $178.77$ | $178.77$ | $185.98$ | $183.11$ | $174.95$ | $178.77$ | |

Ac | $340.64$ | $380.22$ | $361.53$ | $363.53$ | $373.96$ | $368.23$ | $351.89$ | $361.53$ | |

Bc | $343.34$ | $382.92$ | $366.93$ | $371.63$ | $376.66$ | $370.93$ | $354.59$ | $366.93$ | |

CAc | $340.68$ | $380.26$ | $361.65$ | $363.76$ | $373.99$ | $368.26$ | $351.93$ | $361.65$ | |

Hc | $341.74$ | $381.32$ | $363.72$ | $366.82$ | $375.05$ | $369.32$ | $352.99$ | $363.72$ | |

Chi${}^{2}$ | $2.548$ | $34.635$ | $19.091$ | $19.096$ | $29.505$ | $24.824$ | $12.065$ | $19.092$ | |

Dm | 4 | 4 | 3 | 2 | 4 | 4 | 3 | 4 | |

Pv | $0.636$ | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | $0.007$ | <0.001 |

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**MDPI and ACS Style**

El-Dawoody, M.; Eliwa, M.S.; El-Morshedy, M.
An Extension of the Poisson Distribution: Features and Application for Medical Data Modeling. *Processes* **2023**, *11*, 1195.
https://doi.org/10.3390/pr11041195

**AMA Style**

El-Dawoody M, Eliwa MS, El-Morshedy M.
An Extension of the Poisson Distribution: Features and Application for Medical Data Modeling. *Processes*. 2023; 11(4):1195.
https://doi.org/10.3390/pr11041195

**Chicago/Turabian Style**

El-Dawoody, Mohamed, Mohamed S. Eliwa, and Mahmoud El-Morshedy.
2023. "An Extension of the Poisson Distribution: Features and Application for Medical Data Modeling" *Processes* 11, no. 4: 1195.
https://doi.org/10.3390/pr11041195