The zero-and-plus/minus-one inflated extended-Poisson distribution

Abstract

In this paper, we introduce a new distribution defined on $Z$ , called the $\mathrm{ZPMOIEP}$ distribution, which can be viewed as a natural extension of the zero-and-one-inflated Poisson ( $\mathrm{ZOIP}$ ) distribution. It is designed to fit the count data with potentially excess zeros and/or ones, and/or minus ones. We explore its various properties and investigate the estimation of the unknown parameters. Moreover, simulation experiments are carried out to attest to the performance of the estimation. Through the use of a useful data set on football scores, the applicability of the proposed distribution is examined.

Keywords: Zero-and-one-inflated Poisson distribution; count data analysis; discrete distribution defined on Z ; extended Poisson distribution; simulation.

Figure 1.

Comparison between theoretical probabilities and empirical frequencies of n = 1000 observations of the $\mathrm{ZPMOIEP}(0.05,0.13,0.08,0.6,2)$ distribution generated via Process 1.

Figure 2.

Comparison between theoretical probabilities and empirical frequencies of n = 1000 observations of the $\mathrm{ZPMOIEP}(0.05,0.13,0.08,0.6,2)$ distribution generated via Process 2.

Figure 3.

Comparison between theoretical probabilities and empirical frequencies of n = 1000 observations of the $\mathrm{ZPMOIEP}(0.05,0.13,0.08,0.6,2)$ distribution generated by using the third stochastic representation defined by Equation (4).

Figure 4.

The boxplots estimates (from the first method, see Section 4.1) for parameters set (a), where the actual values are $({\alpha }_{-1},{\alpha }_0,{\alpha }_1,p,\lambda )=(0.05,0.13,0.08,0.6,2)$.

Figure 5.

From left to right and top to bottom, empirical biases (from the first method, see Section 4.1) of ${\hat{\alpha }}_{-1}$, ${\hat{\alpha }}_0$, ${\hat{\alpha }}_1$, $\hat{p}$ and $\hat{\lambda }$, for $n=100,\dots ,\mathrm{10,000}$.

Figure 6.

From left to right and top to bottom, empirical MSEs (from the first method, see Section 4.1) of ${\hat{\alpha }}_{-1}$, ${\hat{\alpha }}_0$, ${\hat{\alpha }}_1$, $\hat{p}$ and $\hat{\lambda }$, for $n=100,\dots ,\mathrm{10,000}$.

Figure 7.

From left to right and top to bottom, normal Q-Q plots for errors (from the first method, see Section 4.1), $({\hat{\alpha }}_{-1}-0.05)$, $({\hat{\alpha }}_0-0.13)$, $({\hat{\alpha }}_1-0.08)$, $(\hat{p}-0.6)$, and $(\hat{\lambda }-2)$, for $n=\mathrm{10,000}$.

Figure 8.

The boxplots estimates (from the second method, see Section 4.2) for parameters set (a), where the actual values are $({\alpha }_{-1},{\alpha }_0,{\alpha }_1,p,\lambda )=(0.05,0.13,0.08,0.6,2)$.

Figure 9.

From left to right and top to bottom, normal Q-Q plots for errors (from the second method, see Section 4.2), $({\tilde{\alpha }}_{-1}-0.05)$, $({\tilde{\alpha }}_0-0.13)$, $({\tilde{\alpha }}_1-0.08)$, $(\tilde{p}-0.6)$, and $(\tilde{\lambda }-2)$, for $n=\mathrm{10,000}$.

Figure 10.

Plot of observed data of difference goals of ‘Le stade de Reims’ (for all official matches played from 23.08.2020 to 23.10.2022)

Figure 11.

From left to right: ACF and PACF of the observed data associated with ‘Le stade de Reims’ results.

Figure 12.

Plot of observed data of difference goals of ‘Olympique Lyonnais’ (for all official matches played from 23.08.2020 to 23.10.2022).

Figure 13.

From left to right: ACF and PACF of the observed data associated with ‘Olympique Lyonnais’ results.