# New One-Parameter Over-Dispersed Discrete Distribution and Its Application to the Nonnegative Integer-Valued Autoregressive Model of Order One

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## Abstract

**:**

## 1. Introduction

## 2. Poisson New X-Lindley Distribution

#### 2.1. The Poisson New X-Lindley Distribution and Its Statistical Properties

**Definition**

**1.**

#### 2.2. Moments, Skewness, and Kurtosis

## 3. Estimation of Parameters

#### 3.1. Maximum Likelihood Estimation

#### 3.2. Method of Moments

**Proposition**

**1.**

**Proof.**

#### 3.3. Least Squares and Weighted Least Squares Estimation

#### 3.4. Simulation Study

## 4. The INAR(1) Process with PNXL Innovations

#### 4.1. Estimation of INAR(1)PNXL Process

#### 4.1.1. Conditional Maximum Likelihood

#### 4.1.2. Yule–Walker

#### 4.1.3. Conditional Least Squares

#### 4.2. Simulation of INAR(1)PNXL Process

## 5. Data Analysis

#### 5.1. Corn Borer Data

#### 5.2. Weekly Number of Syphilis Cases Data

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**ACF plot, PACF plot, time series plot, and histogram of weekly number of syphilis cases data.

n | MLE | MME | LSE | WLSE | ||||
---|---|---|---|---|---|---|---|---|

Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | |

$\theta $ = 0.5 | ||||||||

50 | 0.065 | 0.004 | 0.067 | 0.004 | 0.141 | 0.020 | 0.158 | 0.025 |

100 | 0.055 | 0.003 | 0.052 | 0.003 | 0.127 | 0.016 | 0.161 | 0.026 |

200 | 0.044 | 0.002 | 0.044 | 0.002 | 0.126 | 0.016 | 0.165 | 0.027 |

250 | 0.008 | 0.000 | 0.005 | 0.000 | 0.085 | 0.007 | 0.145 | 0.021 |

500 | 0.004 | 0.000 | 0.002 | 0.000 | 0.103 | 0.011 | 0.052 | 0.020 |

$\theta $ = 0.3 | ||||||||

50 | 0.034 | 0.001 | 0.037 | 0.001 | 0.087 | 0.008 | 0.074 | 0.006 |

100 | 0.013 | 0.000 | 0.015 | 0.000 | 0.058 | 0.003 | 0.059 | 0.004 |

200 | 0.008 | 0.000 | 0.008 | 0.000 | 0.050 | 0.003 | 0.060 | 0.004 |

250 | 0.007 | 0.000 | 0.008 | 0.000 | 0.032 | 0.001 | 0.045 | 0.002 |

500 | 0.001 | 0.000 | 0.001 | 0.000 | 0.003 | 0.001 | 0.035 | 0.001 |

$\theta $ = 1.2 | ||||||||

50 | 0.107 | 0.011 | 0.108 | 0.012 | 0.511 | 0.261 | 0.635 | 0.403 |

100 | 0.048 | 0.002 | 0.046 | 0.002 | 0.485 | 0.235 | 0.611 | 0.373 |

200 | 0.046 | 0.002 | 0.046 | 0.002 | 0.485 | 0.235 | 0.642 | 0.412 |

250 | 0.007 | 0.005 | 0.006 | 0.000 | 0.471 | 0.222 | 0.645 | 0.416 |

500 | 0.004 | 0.000 | 0.006 | 0.001 | 0.483 | 0.234 | 0.560 | 0.314 |

$\theta $ = 1.5 | ||||||||

50 | 0.055 | 0.003 | 0.052 | 0.003 | 0.684 | 0.468 | 0.897 | 0.804 |

100 | 0.030 | 0.001 | 0.029 | 0.001 | 0.677 | 0.458 | 0.868 | 0.754 |

200 | 0.025 | 0.001 | 0.026 | 0.001 | 0.689 | 0.475 | 0.880 | 0.775 |

250 | 0.021 | 0.000 | 0.025 | 0.001 | 0.699 | 0.489 | 0.864 | 0.747 |

500 | 0.020 | 0.000 | 0.021 | 0.002 | 0.666 | 0.444 | 0.889 | 0.790 |

Parameter | n | $\mathit{\alpha}$ = 0.4 and $\mathit{\theta}$ = 0.8 | |||||
---|---|---|---|---|---|---|---|

CML | CLS | YW | |||||

Bias | MSE | Bias | MSE | Bias | MSE | ||

$\alpha $ | 50 | 0.063 | 0.006 | 0.109 | 0.019 | 0.110 | 0.020 |

100 | 0.044 | 0.003 | 0.080 | 0.010 | 0.081 | 0.010 | |

200 | 0.032 | 0.002 | 0.054 | 0.005 | 0.053 | 0.005 | |

250 | 0.029 | 0.001 | 0.049 | 0.004 | 0.049 | 0.004 | |

500 | 0.019 | 0.001 | 0.035 | 0.002 | 0.035 | 0.002 | |

$\theta $ | 50 | 0.130 | 0.029 | 0.164 | 0.044 | 0.162 | 0.043 |

100 | 0.094 | 0.015 | 0.122 | 0.025 | 0.122 | 0.025 | |

200 | 0.063 | 0.007 | 0.084 | 0.012 | 0.083 | 0.012 | |

250 | 0.058 | 0.005 | 0.078 | 0.010 | 0.078 | 0.010 | |

500 | 0.041 | 0.003 | 0.056 | 0.005 | 0.056 | 0.005 | |

$\alpha $ = 0.8 and $\theta $ = 3 | |||||||

$\alpha $ | 50 | 0.041 | 0.003 | 0.098 | 0.017 | 0.105 | 0.019 |

100 | 0.028 | 0.001 | 0.061 | 0.007 | 0.065 | 0.008 | |

200 | 0.022 | 0.001 | 0.047 | 0.004 | 0.049 | 0.004 | |

250 | 0.018 | 0.001 | 0.036 | 0.002 | 0.036 | 0.002 | |

500 | 0.012 | 0.000 | 0.025 | 0.001 | 0.025 | 0.001 | |

$\theta $ | 50 | 0.745 | 0.978 | 1.024 | 1.722 | 1.008 | 1.691 |

100 | 0.512 | 0.455 | 0.764 | 0.923 | 0.761 | 0.925 | |

200 | 0.391 | 0.241 | 0.648 | 0.652 | 0.655 | 0.665 | |

250 | 0.299 | 0.148 | 0.499 | 0.405 | 0.500 | 0.407 | |

500 | 0.212 | 0.070 | 0.377 | 0.223 | 0.377 | 0.222 |

Statistic | PNXL | DIW | DG | DLL | DB | DIR | DBL | DP | CMP | |
---|---|---|---|---|---|---|---|---|---|---|

$ML{E}_{\theta}$ | 1.012 | 0.345 | 3.106 | 1.943 | 2.357 | 0.320 | 0.657 | 0.329 | 0.672 | |

$S{E}_{\theta}$ | 0.111 | 0.043 | 0.367 | 0.188 | 0.366 | 0.042 | 0.019 | 0.034 | 0.090 | |

95% CI | lower | 0.794 | 0.261 | 2.388 | 1.575 | 1.641 | 0.237 | 0.620 | 0.263 | 0.496 |

upper | 1.230 | 0.429 | 3.825 | 2.311 | 3.073 | 0.402 | 0.693 | 0.395 | 0.847 | |

$ML{E}_{\beta}$ | - | 1.541 | 0.407 | 1.401 | 0.519 | - | - | - | 0.107 | |

$S{E}_{\beta}$ | - | 0.156 | 0.029 | 0.121 | 0.051 | - | - | - | 0.116 | |

95% CI | lower | - | 1.235 | 0.349 | 1.163 | 0.419 | - | - | - | 0.121 |

upper | - | 1.847 | 0.464 | 1.638 | 0.619 | - | - | - | 0.334 |

**Table 4.**Corn borer data: $logL$, ${\chi}^{2}$-value, p-value, AIC, and BIC for the competitive models.

X | Of | PNXL | DIW | DG | DLL | DB | DIR | DBL | DP | CMP |
---|---|---|---|---|---|---|---|---|---|---|

0 | 43 | 45.355 | 41.370 | 28.553 | 41.032 | 43.836 | 38.352 | 32.734 | 64.447 | 44.995 |

1 | 35 | 30.088 | 41.850 | 37.861 | 38.938 | 39.601 | 51.874 | 39.586 | 20.149 | 30.221 |

2 | 17 | 18.705 | 15.420 | 25.585 | 17.775 | 15.622 | 15.489 | 24.277 | 9.686 | 18.855 |

3 | 11 | 11.161 | 7.170 | 12.852 | 8.432 | 7.206 | 6.028 | 12.508 | 5.647 | 11.266 |

4 | 5 | 6.474 | 3.940 | 5.700 | 4.485 | 3.910 | 2.905 | 5.970 | 3.681 | 6.529 |

5 | 4 | 3.678 | 2.420 | 2.402 | 2.630 | 2.376 | 1.610 | 2.738 | 2.580 | 3.695 |

6 | 1 | 2.057 | 1.610 | 0.991 | 1.663 | 1.563 | 0.981 | 1.227 | 1.904 | 2.051 |

7 | 2 | 1.136 | 1.130 | 0.405 | 1.115 | 1.089 | 0.641 | 0.542 | 1.461 | 1.120 |

8 | 2 | 1.347 | 5.090 | 5.651 | 3.930 | 4.798 | 2.120 | 0.420 | 10.446 | 1.271 |

Total | 120 | 120 | 120 | 120 | 120 | 120 | 120 | 120 | 120 | 120 |

$logL$ | - | - | - | - | - | - | - | - | - | |

200.432 | 204.810 | 231.191 | 202.630 | 204.293 | 208.440 | 204.675 | 220.618 | 200.415 | ||

AIC | 402.863 | 413.621 | 430.382 | 409.261 | 412.587 | 418.881 | 411.351 | 443.236 | 404.830 | |

BIC | 405.651 | 419.195 | 435.957 | 414.836 | 418.162 | 421.668 | 414.138 | 446.024 | 410.405 | |

${\chi}^{2}$ | 1.115 | 5.511 | 7.615 | 1.311 | 2.674 | 14.295 | 6.996 | 30.518 | 1.063 | |

df | 3 | 3 | 2 | 2 | 2 | 3 | 3 | 3 | 2 | |

p-value | 0.774 | 0.138 | 0.022 | 0.519 | 0.263 | 0.003 | 0.072 | 0.000 | 0.588 |

**Table 5.**Estimates and model adequacy statistics of the fitted models for the number of syphilis cases data.

Model | Parameters | Estimate | S.E. | AIC | BIC | $\mathit{\mu}$ | ${\mathit{\sigma}}^{2}$ | DI |
---|---|---|---|---|---|---|---|---|

INAR(1)PNXL | $\alpha $ | 0.316 | 0.034 | 1660.869 | 1667.554 | 23.943 | 255.917 | 10.689 |

$\theta $ | 0.092 | 0.007 | ||||||

INAR(1)P | $\alpha $ | 0.148 | 0.026 | 2016.534 | 2023.224 | 25.349 | 25.349 | 1.000 |

$\lambda $ | 21.063 | 0.709 | ||||||

INAR(1)G | $\alpha $ | 0.347 | 0.032 | 1686.428 | 1693.112 | 23.895 | 252.431 | 10.564 |

$\lambda $ | 0.058 | 0.005 | ||||||

INAR(1)PWE | $\alpha $ | 0.058 | 0.159 | 1688.428 | 1698.455 | 24.990 | 369.211 | 14.774 |

$\lambda $ | 0.060 | 2.883 | ||||||

$\beta $ | 0.347 | 0.032 | ||||||

INAR(1)ZIP | $\alpha $ | 20.552 | 0.595 | 1732.296 | 1742.323 | 25.332 | 58.543 | 2.307 |

$\lambda $ | 0.113 | 0.024 | ||||||

$\beta $ | 0.262 | 0.024 |

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**MDPI and ACS Style**

Irshad, M.R.; Aswathy, S.; Maya, R.; Nadarajah, S.
New One-Parameter Over-Dispersed Discrete Distribution and Its Application to the Nonnegative Integer-Valued Autoregressive Model of Order One. *Mathematics* **2024**, *12*, 81.
https://doi.org/10.3390/math12010081

**AMA Style**

Irshad MR, Aswathy S, Maya R, Nadarajah S.
New One-Parameter Over-Dispersed Discrete Distribution and Its Application to the Nonnegative Integer-Valued Autoregressive Model of Order One. *Mathematics*. 2024; 12(1):81.
https://doi.org/10.3390/math12010081

**Chicago/Turabian Style**

Irshad, Muhammed Rasheed, Sreedeviamma Aswathy, Radhakumari Maya, and Saralees Nadarajah.
2024. "New One-Parameter Over-Dispersed Discrete Distribution and Its Application to the Nonnegative Integer-Valued Autoregressive Model of Order One" *Mathematics* 12, no. 1: 81.
https://doi.org/10.3390/math12010081