# A Novel Flexible Class of Intervened Poisson Distribution by Lagrangian Approach

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Some Preliminaries

#### 2.1. Intervened Poisson Distribution

**Definition**

**1.**

#### 2.2. Lagrange Expansion

## 3. Some Important Members of Discrete Lagrangian Family

#### 3.1. Sudha Lagrangian Distribution

#### 3.2. Weighted Delta Poisson Distribution

#### 3.3. Linear Function Poisson Distribution

#### 3.4. Logarithmic Poisson Distribution

## 4. Lagrangian Intervened Poisson Distribution

**Definition**

**2.**

- For $\mu \to 0$, the LIPD ($\lambda ,\mu ,\rho $) reduces to the IPD.
- For $\mu \to 0$ and $\rho =0$, the LIPD ($\lambda ,\mu ,\rho $) reduces to the ZTPD.

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

## 5. Estimation

## 6. Generalized Likelihood Ratio Test

## 7. Simulation Study

- Step 1:
- Generate a random number from uniform distribution $U(0,1)$.
- Step 2:
- $i=1$, $p=\frac{(1-\mu ){e}^{-(\lambda \rho +\mu )}\lambda}{({e}^{\lambda}-1)}$, and $F=p$, where p is the probability that $X=i$ and F is the probability that X is less than or equal to i.
- Step 3:
- If $U<F$, set $X=i$, and stop.
- Step 4:
- $p=\frac{{e}^{-\mu}p}{(i+1)}$×$\frac{{\left(\mu (i+1)+\lambda (1+\rho )\right)}^{i+1}-{\left(\mu (i+1)+\lambda \rho \right)}^{i+1}}{{\left(\mu i+\lambda (1+\rho )\right)}^{i}-{\left(\mu i+\lambda \rho \right)}^{i}}$, $F=F+p$, $i=i+1$.
- Step 5:
- Go to Step 3.

- (i)
- $\lambda =0.73$, $\mu =0.18$, $\rho =0.32$;
- (ii)
- $\lambda =0.121$, $\mu =0.065$, $\rho =0.075$.

## 8. Lagrangian Intervened Poisson Regression Model

`optim`function in the R programming language under the L-BFGS-B algorithm to determine the MLEs of the parameters, just as we did in Section 5.

## 9. Applications and Empirical Study

#### 9.1. Student Enrollment Data

#### 9.2. COVID-19 Dataset

#### 9.3. National Health Insurance Scheme

## 10. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Table 1.**Mean, variance, CoV, InD, skewness, and kurtosis coefficients of the LIPD distribution for different values of the parameters.

$\mathit{\mu}$ | |||||||
---|---|---|---|---|---|---|---|

0.01 | 0.02 | 0.03 | 0.04 | 0.05 | |||

$\lambda =2$ | $\rho =0.1$ | Mean | 1.9590 | 1.9895 | 2.0208 | 2.0531 | 2.0862 |

Variance | 1.8617 | 1.9382 | 2.0188 | 2.1037 | 2.1932 | ||

CoV | 0.6964 | 0.6997 | 0.7030 | 0.7064 | 0.7098 | ||

InD | 0.9503 | 0.9742 | 0.9989 | 1.0246 | 1.0512 | ||

Skewness | 3.2757 | 3.2447 | 3.2135 | 3.1823 | 3.1509 | ||

Kurtosis | 9.6539 | 9.4461 | 9.2392 | 9.0332 | 8.8281 | ||

$\lambda =3$ | $\rho =0.1$ | Mean | 3.1926 | 3.2357 | 3.2799 | 3.3253 | 3.3718 |

Variance | 3.0671 | 3.1785 | 3.2954 | 3.4180 | 3.5468 | ||

CoV | 0.5485 | 0.5509 | 0.5534 | 0.5559 | 0.5585 | ||

InD | 0.9606 | 0.9823 | 1.0047 | 1.0278 | 1.0519 | ||

Skewness | 2.7212 | 2.6950 | 2.6686 | 2.6421 | 2.6155 | ||

Kurtosis | 6.2383 | 6.0874 | 5.9371 | 5.7874 | 5.6383 | ||

$\lambda =2$ | $\rho =0.2$ | Mean | 2.1610 | 2.1936 | 2.2270 | 2.2614 | 2.2968 |

Variance | 2.0678 | 2.1507 | 2.2379 | 2.3298 | 2.4265 | ||

CoV | 0.6654 | 0.6685 | 0.6717 | 0.6749 | 0.6782 | ||

InD | 0.9568 | 0.9804 | 1.0049 | 1.0302 | 1.0564 | ||

Skewness | 3.1444 | 3.1145 | 3.0845 | 3.0544 | 3.0241 | ||

Kurtosis | 8.8581 | 8.6616 | 8.4658 | 8.2709 | 8.0769 | ||

$\lambda =3$ | $\rho =0.2$ | Mean | 3.4956 | 3.5418 | 3.5892 | 3.6378 | 3.6876 |

Variance | 3.3763 | 3.4973 | 3.6241 | 3.7571 | 3.8967 | ||

CoV | 0.5256 | 0.5280 | 0.5203 | 0.5328 | 0.5353 | ||

InD | 0.9658 | 0.9874 | 1.0097 | 1.0328 | 1.0567 | ||

Skewness | 2.6439 | 2.6176 | 2.5912 | 2.5645 | 2.5378 | ||

Kurtosis | 5.8118 | 5.6633 | 5.5161 | 5.3697 | 5.2240 |

Parameter Set | Sample Size | Parameters | Estimates | Absolute Bias | MSE |
---|---|---|---|---|---|

$\lambda =0.73$, $\mu =0.18$, $\rho =0.32$ | $n=$ 50 | $\lambda $ | 0.6164 | 0.1135 | 0.0910 |

$\mu $ | 0.2225 | 0.0425 | 0.0067 | ||

$\rho $ | 0.1876 | 0.1323 | 0.1579 | ||

$n=$ 125 | $\lambda $ | 0.6928 | 0.0371 | 0.0625 | |

$\mu $ | 0.2131 | 0.0331 | 0.0031 | ||

$\rho $ | 0.3353 | 0.0846 | 0.1439 | ||

$n=$ 500 | $\lambda $ | 0.7586 | 0.0286 | 0.0425 | |

$\mu $ | 0.1894 | 0.0094 | 0.0007 | ||

$\rho $ | 0.3032 | 0.0167 | 0.0986 | ||

$n=$ 1000 | $\lambda $ | 0.7213 | 0.0086 | 0.0390 | |

$\mu $ | 0.1829 | 0.0029 | 0.0003 | ||

$\rho $ | 0.3240 | 0.0040 | 0.0901 | ||

$\lambda =0.121$, $\mu =0.065$, $\rho =0.075$ | $n=$ 100 | $\lambda $ | 0.1825 | 0.1315 | 0.0317 |

$\mu $ | 0.0347 | 0.0303 | 0.0018 | ||

$\rho $ | 0.1185 | 0.0935 | 0.0829 | ||

$n=$ 250 | $\lambda $ | 0.1496 | 0.0986 | 0.0203 | |

$\mu $ | 0.0384 | 0.0265 | 0.0014 | ||

$\rho $ | 0.1006 | 0.0756 | 0.0527 | ||

$n=$ 500 | $\lambda $ | 0.1339 | 0.0829 | 0.0155 | |

$\mu $ | 0.0433 | 0.0216 | 0.0010 | ||

$\rho $ | 0.0898 | 0.0648 | 0.0473 | ||

$n=$ 1000 | $\lambda $ | 0.1239 | 0.0029 | 0.0005 | |

$\mu $ | 0.0643 | 0.0007 | 0.0001 | ||

$\rho $ | 0.0798 | 0.0048 | 0.0003 |

Statistic | n | min | ${\mathit{Q}}_{1}$ | Md | ${\mathit{Q}}_{3}$ | max | IQR |
---|---|---|---|---|---|---|---|

Values | 56 | 1 | 4 | 7 | 17 | 37 | 13 |

Distributions | Abbreviation | Reference |
---|---|---|

zero-truncated Poisson distribution | ZTPD | - |

intervened Poisson distribution | IPD | [2] |

zero-truncated generalized binomial distribution | ZTGBD | - |

zero-truncated generalized negative binomial distribution | ZTGNBD | [23] |

intervened generalized Poisson distribution | IGPD | [8] |

Model | MLEs | $-log\mathbf{L}$ | ${\mathit{\chi}}^{2}$ | $\mathit{df}$ | AIC | BIC |
---|---|---|---|---|---|---|

ZTPD | $\lambda =10.9646$ | 311.4949 | 3924.5470 | 5 | 624.9899 | 627.6503 |

ZTGBD | $\lambda =19.7818$ | 192.1601 | 185.4118 | 3 | 390.3202 | 396.5015 |

$\mu =0.1689$ | ||||||

$\rho =4.0671$ | ||||||

ZTGNBD | $\lambda =19.8175$ | 192.1601 | 185.3881 | 3 | 390.3201 | 396.5015 |

$\mu =0.1687$ | ||||||

$\rho =3.0725$ | ||||||

IPD | $\rho =10.965$ | 311.4949 | 2998.871 | 4 | 626.99 | 631.1108 |

$\lambda =0.0000012$ | ||||||

IGPD | $\lambda =3.3266$ | 192.06 | 187.4251 | 3 | 390.12 | 395.79 |

$\mu =0.6607$ | ||||||

$\rho =0.0647$ | ||||||

LIPD | $\lambda =0.0781$ | 191.5816 | 181.3314 | 3 | 389.1631 | 395.3444 |

$\mu =0.6159$ | ||||||

$\rho =20.0777$ |

Statistic | n | min | ${\mathit{Q}}_{1}$ | Md | ${\mathit{Q}}_{3}$ | max | IQR |
---|---|---|---|---|---|---|---|

Values | 66 | 5 | 15 | 28.50 | 43.75 | 98 | 28.75 |

Model | MLEs | $-log\mathbf{L}$ | ${\mathit{\chi}}^{2}$ | $\mathit{df}$ | AIC | BIC |
---|---|---|---|---|---|---|

ZTPD | $\lambda =32.7871$ | 636.744 | 439.98 | 5 | 1275.488 | 1277.677 |

ZTGBD | $\lambda =10.9120$ | 286.645 | 350.86 | 3 | 579.291 | 585.860 |

$\mu =0.5095$ | ||||||

$\rho =1.6295$ | ||||||

ZTGNBD | $\lambda =10.9112$ | 286.6457 | 350.8655 | 3 | 579.2915 | 585.8604 |

$\mu =0.5095$ | ||||||

$\rho =0.6295$ | ||||||

IPD | $\rho =0.1646$ | 636.744 | 439.43 | 4 | 1277.488 | 1281.867 |

$\lambda =28.1515$ | ||||||

IGPD | $\lambda =2.0806$ | 286.5185 | 353.4297 | 3 | 579.037 | 585.606 |

$\mu =0.7644$ | ||||||

$\rho =2.5693$ | ||||||

LIPD | $\lambda =0.8959$ | 286.353 | 348.9881 | 3 | 578.7059 | 585.2749 |

$\mu =0.7324$ | ||||||

$\rho =5.0437$ |

Covariates | ZTPRM | ZTNBRM | IPRM | LIPRM | ||||
---|---|---|---|---|---|---|---|---|

Estimate | p-Value | Estimate | p-Value | Estimate | p-Value | Estimate | p-Value | |

${\alpha}_{0}$ | 0.0673 | <0.001 | −0.1446 | <0.001 | 1.1981 | <0.001 | 0.1787 | <0.001 |

(0.0379) | (0.0541) | (0.0019) | (0.0327) | |||||

${\alpha}_{1}$ | 0.1028 | <0.001 | −0.0947 | <0.001 | 0.0154 | <0.001 | 0.0154 | <0.001 |

(0.0297) | (0.0409) | (0.0345) | (0.0253) | |||||

${\alpha}_{2}$ | 0.0018 | <0.001 | 0.0045 | <0.001 | 0.0017 | <0.001 | 0.0017 | <0.001 |

(0.0007) | (0.0011) | (0.0019) | (0.0006) | |||||

${\alpha}_{3}$ | −0.1982 | <0.001 | −0.07849 | <0.001 | −0.1406 | <0.001 | −0.1406 | <0.001 |

(0.0408) | (0.0560) | (0.0217) | (0.0329) | |||||

${\alpha}_{4}$ | 0.8705 | <0.001 | −1.2214 | <0.001 | 0.1352 | <0.001 | 0.1826 | <0.001 |

(0.0827) | (0.1937) | (0.0109) | (0.1046) | |||||

${\alpha}_{5}$ | 0.2919 | <0.001 | 0.3802 | <0.001 | 0.1352 | <0.001 | 0.2866 | <0.001 |

(0.0045) | (0.0106) | (0.0109) | (0.0043) | |||||

$-\mathrm{logL}$ | 8379.31 | 3043.18 | 6579.37 | 2455.76 | ||||

AIC | 16,770.63 | 6102.36 | 13,172.74 | 4927.51 | ||||

BIC | 16,778.56 | 6112.09 | 13,181.25 | 4937.25 |

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## Share and Cite

**MDPI and ACS Style**

Irshad, M.R.; Monisha, M.; Chesneau, C.; Maya, R.; Shibu, D.S.
A Novel Flexible Class of Intervened Poisson Distribution by Lagrangian Approach. *Stats* **2023**, *6*, 150-168.
https://doi.org/10.3390/stats6010010

**AMA Style**

Irshad MR, Monisha M, Chesneau C, Maya R, Shibu DS.
A Novel Flexible Class of Intervened Poisson Distribution by Lagrangian Approach. *Stats*. 2023; 6(1):150-168.
https://doi.org/10.3390/stats6010010

**Chicago/Turabian Style**

Irshad, Muhammed Rasheed, Mohanan Monisha, Christophe Chesneau, Radhakumari Maya, and Damodaran Santhamani Shibu.
2023. "A Novel Flexible Class of Intervened Poisson Distribution by Lagrangian Approach" *Stats* 6, no. 1: 150-168.
https://doi.org/10.3390/stats6010010