# A Class of Copula-Based Bivariate Poisson Time Series Models with Applications

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. The Poisson and ZIP Distributions

#### 2.2. Copula

- $C(1,\dots ,{a}_{i},\dots ,1)={a}_{i},\phantom{\rule{0.277778em}{0ex}}\forall i=1,2,\dots ,p\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{0.277778em}{0ex}}{a}_{i}\in [0,1]$;
- $C({a}_{1},{a}_{2},\dots ,{a}_{p})=0\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}at\phantom{\rule{0.277778em}{0ex}}least\phantom{\rule{0.277778em}{0ex}}one\phantom{\rule{0.277778em}{0ex}}{a}_{i}=0\phantom{\rule{0.277778em}{0ex}}for\phantom{\rule{0.277778em}{0ex}}i=1,2,\dots ,p$;
- For any ${a}_{{i}_{1}},{a}_{{i}_{2}}\in [0,1]$ with ${a}_{{i}_{1}}\le {a}_{{i}_{2}},$ for $i=1,2,\dots ,p,$$$\begin{array}{c}\hfill \sum _{{j}_{1}=1}^{2}\sum _{{j}_{2}=1}^{2}\dots \sum _{{j}_{p}=1}^{2}{(-1)}^{{j}_{1}+{j}_{2}+\dots +{j}_{p}}C({a}_{1{j}_{1}},{a}_{2{j}_{2}}\dots ,{a}_{n{j}_{p}})\ge 0.\end{array}$$

- There exists a p-dimensional copula C such that for all ${x}_{1},{x}_{2},\dots ,{x}_{p}\in \mathbb{R}$:$$\begin{array}{c}\hfill F({x}_{1},{x}_{2},\dots ,{x}_{p})=C({F}_{1}\left({x}_{1}\right),{F}_{2}\left({x}_{2}\right),\dots ,{F}_{p}\left({x}_{p}\right));\end{array}$$
- If ${X}_{1},{X}_{2}\dots ,{X}_{p}$ are continuous, then the copula C is unique. Otherwise, C can be uniquely determined on p- dimensional rectangle $Range\left({F}_{1}\right)\times Range\left({F}_{2}\right)\times \dots \times Range\left({F}_{p}\right).$

## 3. Constructing the Bivariate Models

## 4. Estimation Method

## 5. Simulation Studies

## 6. Applications

#### 6.1. Application to Forgery and Fraud Data

#### 6.2. Application to Sandstorm Data

## 7. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Copula | Copula Function |
---|---|

Gaussian | $C({u}_{1},{u}_{2};\delta )={\mathsf{\Phi}}_{\delta}({\mathsf{\Phi}}^{-1}\left({u}_{1}\right),{\mathsf{\Phi}}^{-1}\left({u}_{2}\right)),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\delta \in [-1,1]$ |

Frank | $C({u}_{1},{u}_{2};\delta )=-\frac{1}{\delta}log\left[1+\frac{({e}^{-\delta {u}_{1}}-1)({e}^{-\delta {u}_{2}}-1)}{{e}^{-\delta -1}}\right],\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\delta \in R\phantom{\rule{4pt}{0ex}}\left\{0\right\}$ |

Gumbel | $C({u}_{1},{u}_{2};\delta )=exp\left[-{\left({(-log\left({u}_{1}\right))}^{\delta}+{(-log\left({u}_{2}\right))}^{\delta}\right)}^{1/\delta}\right],\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\delta \ge 1$ |

Clayton | $C({u}_{1},{u}_{2};\delta )={({u}_{1}^{-\delta}+{u}_{2}^{-\delta}-1)}^{-1/\delta},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\delta >0$ |

Plackett | $C({u}_{1},{u}_{2};\delta )=\frac{[1+(\delta -1)({u}_{1}+{u}_{2})]-\sqrt{{[1+(\delta -1)({u}_{1}+{u}_{2})]}^{2}-4{u}_{1}{u}_{2}\delta (\delta -1)}}{2(\delta -1)},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\delta \ge 0$ |

BVT | $C({u}_{1},{u}_{2};\delta )={\tau}_{\delta}({\tau}^{-1}\left({u}_{1}\right),{\tau}^{-1}\left({u}_{2}\right)),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\delta \in [-1,1]$ |

**Table 2.**Parameter estimates using the Gaussian copula for a univariate and joint distribution for 300 and 500 replicates.

Sample Size | Parameters | 300 Replicates | 500 Replicates | ||
---|---|---|---|---|---|

Estimate | SE | Estimate | SE | ||

${\lambda}_{1}$ | 3.185 | 0.433 | 3.166 | 0.412 | |

${\lambda}_{2}$ | 5.119 | 0.395 | 5.104 | 0.382 | |

100 | ${\delta}_{1}$ | 0.501 | 0.064 | 0.501 | 0.062 |

${\delta}_{2}$ | 0.348 | 0.079 | 0.343 | 0.078 | |

$\rho $ | 0.450 | 0.072 | 0.450 | 0.071 | |

${\lambda}_{1}$ | 3.170 | 0.224 | 3.161 | 0.235 | |

${\lambda}_{2}$ | 5.091 | 0.215 | 5.090 | 0.216 | |

300 | ${\delta}_{1}$ | 0.510 | 0.035 | 0.510 | 0.035 |

${\delta}_{2}$ | 0.349 | 0.041 | 0.348 | 0.041 | |

$\rho $ | 0.439 | 0.041 | 0.442 | 0.040 | |

${\lambda}_{1}$ | 3.173 | 0.194 | 3.164 | 0.191 | |

${\lambda}_{2}$ | 5.092 | 0.166 | 5.085 | 0.167 | |

500 | ${\delta}_{1}$ | 0.512 | 0.026 | 0.512 | 0.027 |

${\delta}_{2}$ | 0.350 | 0.034 | 0.353 | 0.034 | |

$\rho $ | 0.442 | 0.033 | 0.443 | 0.033 |

**Table 3.**Parameter estimates using the Gaussian copula for univariate distributions and the t-copula for the joint distribution for 300 and 500 replicates.

Sample Size | Parameters | 300 Replicates | 500 Replicates | ||
---|---|---|---|---|---|

Estimate | SE | Estimate | SE | ||

${\lambda}_{1}$ | 3.200 | 0.434 | 3.181 | 0.416 | |

${\lambda}_{2}$ | 5.124 | 0.399 | 5.108 | 0.388 | |

100 | ${\delta}_{1}$ | 0.495 | 0.065 | 0.495 | 0.064 |

${\delta}_{2}$ | 0.346 | 0.080 | 0.342 | 0.080 | |

$\rho $ | 0.426 | 0.074 | 0.429 | 0.071 | |

${\lambda}_{1}$ | 3.190 | 0.232 | 3.184 | 0.241 | |

${\lambda}_{2}$ | 5.094 | 0.214 | 5.096 | 0.217 | |

300 | ${\delta}_{1}$ | 0.504 | 0.037 | 0.504 | 0.036 |

${\delta}_{2}$ | 0.347 | 0.042 | 0.347 | 0.042 | |

$\rho $ | 0.418 | 0.042 | 0.420 | 0.041 | |

${\lambda}_{1}$ | 3.193 | 0.196 | 3.183 | 0.192 | |

${\lambda}_{2}$ | 5.101 | 0.168 | 5.094 | 0.170 | |

500 | ${\delta}_{1}$ | 0.505 | 0.026 | 0.505 | 0.028 |

${\delta}_{2}$ | 0.349 | 0.035 | 0.352 | 0.035 | |

$\rho $ | 0.419 | 0.033 | 0.421 | 0.033 |

**Table 4.**Parameter estimates using the Gaussian copula for the univariate and joint distributions for 300 replicates with the Poisson and ZIP marginals.

Poisson | ZIP | ||||
---|---|---|---|---|---|

Sample Size | Parameters | Estimate | SE | Estimate | SE |

100 | ${\lambda}_{1}$ | 3.168 | 0.432 | 3.402 | 0.387 |

${\omega}_{1}$ | 0.332 | 0.084 | |||

${\lambda}_{2}$ | 4.993 | 0.378 | 5.205 | 0.386 | |

${\omega}_{2}$ | 0.405 | 0.067 | |||

${\delta}_{1}$ | 0.503 | 0.065 | 0.539 | 0.0841 | |

${\delta}_{2}$ | 0.341 | 0.082 | 0.354 | 0.099 | |

$\rho $ | −0.453 | 0.075 | −0.491 | 0.095 | |

300 | ${\lambda}_{1}$ | 3.146 | 0.225 | 3.402 | 0.204 |

${\omega}_{1}$ | 0.337 | 0.044 | |||

${\lambda}_{2}$ | 4.978 | 0.199 | 5.201 | 0.191 | |

${\omega}_{2}$ | 0.409 | 0.038 | |||

${\delta}_{1}$ | 0.510 | 0.035 | 0.549 | 0.044 | |

${\delta}_{2}$ | 0.345 | 0.040 | 0.362 | 0.052 | |

$\rho $ | −0.444 | 0.039 | −0.479 | 0.052 | |

500 | ${\lambda}_{1}$ | 3.147 | 0.191 | 3.401 | 0.173 |

${\omega}_{1}$ | 0.338 | 0.035 | |||

${\lambda}_{2}$ | 4.970 | 0.157 | 5.199 | 0.148 | |

${\omega}_{2}$ | 0.409 | 0.03 | |||

${\delta}_{1}$ | 0.510 | 0.026 | 0.55 | 0.033 | |

${\delta}_{2}$ | 0.350 | 0.031 | 0.367 | 0.041 | |

$\rho $ | −0.441 | 0.032 | −0.476 | 0.041 |

Univariate Copula Family | Joint Copula Family | |
---|---|---|

Gaussian | t | |

Gaussian | 1448.63 | 1462.68 |

Frank | 1461.7 | 1475.61 |

Clayton | 1433.78 | 1447.85 |

Gumbel | 1458.4 | 2315.55 |

Parameter | Estimate | SE |
---|---|---|

${\lambda}_{1}$ | 2.695 | 0.0657 |

${\lambda}_{2}$ | 8.332 | 0.0350 |

${\delta}_{1}$ | 0.261 | 0.0527 |

${\delta}_{2}$ | 0.192 | 0.0478 |

$\rho $ | 0.178 | 0.0532 |

Univariate Copula Family | Joint Copula Family | |
---|---|---|

Gaussian | t | |

Gaussian | 2232.2 | 2272.09 |

Frank | 2122.9 | 2153.99 |

Clayton | 2089.8 | 2268.88 |

Gumbel | 4574.24 | 7646.1 |

Parameter | Estimate | SE |
---|---|---|

${\lambda}_{1}$ | 1.384 | 0.0003 |

${\omega}_{1}$ | 0.502 | 0.0003 |

${\lambda}_{2}$ | 2.505 | 0.0604 |

${\omega}_{2}$ | 0.690 | 0.1501 |

${\delta}_{1}$ | 0.327 | 0.0024 |

${\delta}_{2}$ | 0.300 | 0.0454 |

$\rho $ | 0.210 | 0.0469 |

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**MDPI and ACS Style**

Alqawba, M.; Fernando, D.; Diawara, N.
A Class of Copula-Based Bivariate Poisson Time Series Models with Applications. *Computation* **2021**, *9*, 108.
https://doi.org/10.3390/computation9100108

**AMA Style**

Alqawba M, Fernando D, Diawara N.
A Class of Copula-Based Bivariate Poisson Time Series Models with Applications. *Computation*. 2021; 9(10):108.
https://doi.org/10.3390/computation9100108

**Chicago/Turabian Style**

Alqawba, Mohammed, Dimuthu Fernando, and Norou Diawara.
2021. "A Class of Copula-Based Bivariate Poisson Time Series Models with Applications" *Computation* 9, no. 10: 108.
https://doi.org/10.3390/computation9100108