# Integer-Valued Split-BREAK Process with a General Family of Innovations and Application to Accident Count Data Modeling

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## Abstract

**:**

## 1. Introduction

## 2. Definition and Structure of the INSB(1) Process

**Definition**

**1.**

**Definition**

**2.**

## 3. Main Properties of the INSB(1) Process

**Theorem**

**1.**

**Proof.**

**Remark**

**1**

**Theorem**

**2.**

**Proof.**

**Remark**

**2**

**Over-dispersion conditions**)

**.**Recall that the over-dispersion of the PS series $\left({\epsilon}_{t}\right)$ depends on the function $g\left(a\right)=lnf\left(a\right)$. Namely, Equations (2) imply that ${D}_{\epsilon}\left(a\right):={\sigma}_{\epsilon}^{2}-{\mu}_{\epsilon}>0$ if and only if ${g}^{\u2033}\left(a\right)>0$, $\forall a\in (0,R).$ On the other hand, the over-dispersion conditions of the series $\left({\xi}_{t}\right)$ are weaker, because Equations (8) imply that ${D}_{\xi}\left(a\right):={\sigma}_{\xi}^{2}-{\mu}_{\xi}>0$ holds if and only if

**Theorem**

**3.**

**Proof.**

**Remark**

**3**

**Distributional properties**)

**.**Note that the first summand in Equation (17) exists if and only if martingale means $\left({X}_{t}\right)$ pass from the state ${X}_{t-1}=i$ to the non-increasing state ${X}_{t}=j\le i$. In addition, the transition probabilities given by Equations (17) and (18) give the marginal PMFs of the INSB series $\left({X}_{t}\right)$ and $\left({Y}_{t}\right)$:

**Theorem**

**4.**

**Proof.**

**Remark**

**4**

**Zero-inflation properties**)

**.**Based on the previous results, the proportions of zeros in the INSB series can be easily computed. Indeed, using the definition of series $\left({\epsilon}_{t}\right)$ and $\left({\xi}_{t}\right)$, given by Definition 1 and Theorem 1, one obtains

## 4. Parameter Estimation Procedure

**Theorem**

**5.**

- $\left({A}_{1}\right)$
- ${\theta}_{0}\in \Theta $ and ${\widehat{\theta}}_{T}\in \Theta $, for T large enough.
- $\left({A}_{2}\right)$
- The function$${Q}_{0}^{\left(r\right)}\left(\theta \right):={\int}_{-1}^{\phantom{\rule{0.277778em}{0ex}}1}\cdots {\int}_{-1}^{\phantom{\rule{0.277778em}{0ex}}1}w\left(\mathbf{u}\right){\left({G}_{Y}^{\left(r\right)}(\mathbf{u};\theta )-{G}_{Y}^{\left(r\right)}(\mathbf{u};{\theta}_{0})\right)}^{2}\mathrm{d}\mathbf{u}$$
- $\left({A}_{3}\right)$
- $\frac{\partial {G}_{Y}^{\left(r\right)}(\mathbf{u};{\theta}_{0})}{\partial \theta}}\phantom{\rule{4pt}{0ex}}{\displaystyle \frac{\partial {G}_{Y}^{\left(r\right)}(\mathbf{u};{\theta}_{0})}{\partial {\theta}^{\prime}}$ is a non-zero matrix uniformly bounded by some positive and w-integrable function $\mathcal{W}:{\mathbb{R}}^{r}\to {\mathbb{R}}^{+}$.
- $\left({A}_{4}\right)$
- $\frac{{\partial}^{2}{Q}_{T}^{\left(r\right)}\left({\theta}_{0}\right)}{\partial \theta \phantom{\rule{4pt}{0ex}}\partial {\theta}^{\prime}}$ is a regular matrix.

**Proof.**

**Remark**

**5.**

## 5. Numerical Simulations

## 6. Application of the Model

**o**test of predictive accuracy [41]. More precisely, the null hypothesis was that the INAR(1) and INSB(1) models have the same predictive accuracy, while the alternative hypothesis was that the INSB(1) model has better accuracy. The test statistic, labeled DM, as well as the corresponding p-values, were calculated within the package “forecast” [42] in the statistical programming language R and are presented in the lower part of Table 4. Based on this, it can be seen that the INSB(1) model has better forecast accuracy, which is in accordance with the previous results obtained. As an illustration, the lower diagrams in Figure 6 show the frequency distribution of both models, based on forecast data.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Realizations of the INSB(1) time series. (

**b**) Empirical frequency distributions of the INSB(1) time series (parameters are $\alpha =0.5,\phantom{\rule{0.277778em}{0ex}}a=2,\phantom{\rule{0.277778em}{0ex}}c=1$).

**Figure 2.**Two-dimensional (

**a**) theoretical PGFs; (

**b**) empirical PGF of the series $\left({Y}_{t}\right)$. Innovations are PS-distributed RVs with geometric distribution and parameters $a=\alpha ={\mu}_{q}=0.5$$(c=1)$.

**Figure 3.**Daily dynamics of the number of traffic accidents with fatalities in the Republic of Serbia (Series A) and the number of wild fires in the Evros region of the Republic of Greece (Series B).

**Figure 5.**Tests for i.i.d. properties of the considered time-series data: the testing of individual data ((

**left**) diagrams); the cumulative testing of data series ((

**right**) diagrams).

**Figure 6.**Plots above: Frequency distributions of the observed data fitted by the INAR(1) and INSB(1) processes. Plots below: Prediction features of INAR(1) and INSB(1) processes.

Distributions | $\mathit{S}$ | $\mathit{m}\left(\mathit{x}\right)$ | $(0,\mathit{R})$ | $\mathit{f}\left(\mathit{a}\right)$ | ${\mathit{\mu}}_{\mathit{\epsilon}}$ | ${\mathit{D}}_{\mathit{\epsilon}}\left(\mathit{a}\right)$ | ${\mathit{G}}_{\mathit{\epsilon}}(\mathit{u};\mathit{a})$ | $\mathit{R}/\mathit{a}$ |
---|---|---|---|---|---|---|---|---|

1. Bernoulli | $\left\{0,1\right\}$ | 1 | $(0,\infty )$ | $1+a$ | $\frac{a}{1+a}$ | $-\frac{{a}^{2}}{{(1+a)}^{2}}$ | $\frac{1+au}{1+a}$ | ∞ |

2. Binomial | $\left\{0,\dots ,n\right\}$ | $\left(\genfrac{}{}{0pt}{}{n}{x}\right)$ | $(0,\infty )$ | ${(1+a)}^{n}$ | $\frac{na}{1+a}$ | $-\frac{n{a}^{2}}{{(1+a)}^{2}}$ | $\left(\frac{1+au}{1+a}\right)}^{n$ | ∞ |

3. Poisson | $\left\{0,\dots ,\infty \right\}$ | $\frac{1}{x!}$ | $(0,\infty )$ | $exp\left(a\right)$ | a | 0 | $exp\left(a(u-1)\right)$ | ∞ |

4. Geometric | $\left\{0,\dots ,\infty \right\}$ | 1 | $(0,1)$ | $\frac{1}{1-a}$ | $\frac{a}{1-a}$ | $\frac{{a}^{2}}{{(1-a)}^{2}}$ | $\frac{1-a}{1-au}$ | $1/a$ |

5. Negative binomial | $\left\{0,\dots ,\infty \right\}$ | $\frac{\Gamma (x+n)}{x!\phantom{\rule{0.166667em}{0ex}}\Gamma \left(n\right)}$ | $(0,1)$ | $\frac{1}{{(1-a)}^{n}}$ | $\frac{na}{{(1-a)}^{n}}$ | $\frac{n{a}^{2}}{{(1-a)}^{2}}$ | $\left(\frac{1-a}{1-au}\right)}^{n$ | $1/a$ |

**Table 2.**The summary statistics, estimation errors and AN testing of parameter estimates of INSB$\left(1\right)$ process (true parameters are $a=\alpha =0.5$, $c=1$).

Sample | Poisson $({\mathit{\mu}}_{\mathit{q}}\approx 0.3935)$ | ${\mathit{S}}_{\mathit{T}}^{\left(\mathbf{2}\right)}$ | Geometric $({\mathit{\mu}}_{\mathit{q}}=0.5)$ | ${\mathit{S}}_{\mathit{T}}^{\left(\mathbf{2}\right)}$ | |||||
---|---|---|---|---|---|---|---|---|---|

$\widehat{\mathit{a}}$ | $\widehat{\mathit{\alpha}}$ | ${\widehat{\mathit{\mu}}}_{\mathit{q}}$ | $\widehat{\mathit{a}}$ | $\widehat{\mathit{\alpha}}$ | ${\widehat{\mathit{\mu}}}_{\mathit{q}}$ | ||||

${w}_{0}({u}_{1},{u}_{2})$ | Min. | 0.3904 | 0.4358 | 0.3677 | $3.14\times {10}^{-4}$ | 0.3898 | 0.4397 | 0.3950 | $3.45\times {10}^{-4}$ |

Mean | 0.5105 | 0.4989 | 0.3897 | $9.69\times {10}^{-3}$ | 0.5082 | 0.5019 | 0.4997 | $3.62\times {10}^{-3}$ | |

Max. | 0.7619 | 0.5499 | 0.4361 | $3.05\times {10}^{-2}$ | 0.7142 | 0.5760 | 0.6197 | $1.33\times {10}^{-2}$ | |

MSEE | $4.54\times {10}^{-3}$ | $2.77\times {10}^{-3}$ | $1.09\times {10}^{-4}$ | – | $1.99\times {10}^{-3}$ | $2.27\times {10}^{-4}$ | $1.18\times {10}^{-4}$ | – | |

$AD$ | 0.5576 | 0.3708 | 0.8099 | – | 0.7178 | 0.2457 | 0.5065 | – | |

(p-value) | (0.1303) | (0.3892) | (0.0337 *) | – | (0.0593) | (0.7524) | (0.1967) | – | |

${w}_{1}({u}_{1},{u}_{2})$ | Min. | 0.3836 | 0.4523 | 0.3559 | $1.59\times {10}^{-4}$ | 0.3904 | 0.4255 | 0.4056 | $1.55\times {10}^{-4}$ |

Mean | 0.5091 | 0.5002 | 0.3905 | $1.41\times {10}^{-2}$ | 0.5091 | 0.5018 | 0.5016 | $1.15\times {10}^{-3}$ | |

Max. | 0.7421 | 0.5493 | 0.4234 | $4.18\times {10}^{-2}$ | 0.6832 | 0.5694 | 0.5913 | $8.01\times {10}^{-3}$ | |

MSEE | $2.27\times {10}^{-3}$ | $7.65\times {10}^{-4}$ | $1.02\times {10}^{-4}$ | – | $2.06\times {10}^{-3}$ | $2.06\times {10}^{-4}$ | $1.17\times {10}^{-4}$ | – | |

$AD$ | 0.2437 | 0.2861 | 0.3188 | – | 0.5911 | 0.2410 | 0.5262 | – | |

(p-value) | (0.7589) | (0.6178) | (0.5301) | – | (0.1191) | (0.7613) | (0.1714) | – | |

${w}_{2}({u}_{1},{u}_{2})$ | Min. | 0.3951 | 0.4765 | 0.3807 | $1.12\times {10}^{-4}$ | 0.4065 | 0.4175 | 0.4070 | $2.38\times {10}^{-4}$ |

Mean | 0.5068 | 0.5012 | 0.3914 | $8.98\times {10}^{-3}$ | 0.5070 | 0.4987 | 0.5018 | $1.12\times {10}^{-3}$ | |

Max. | 0.7581 | 0.5471 | 0.4284 | $2.88\times {10}^{-2}$ | 0.6486 | 0.5607 | 0.6179 | $6.52\times {10}^{-3}$ | |

MSEE | $2.57\times {10}^{-3}$ | $6.17\times {10}^{-4}$ | $3.08\times {10}^{-4}$ | – | $1.98\times {10}^{-3}$ | $2.09\times {10}^{-4}$ | $9.23\times {10}^{-5}$ | – | |

$AD$ | 0.2694 | 0.3384 | 0.5972 | – | 0.2123 | 0.9120 | 0.3959 | – | |

(p-value) | (0.6497) | (0.4734) | (0.1182) | – | (0.8513) | (0.0195 *) | (0.3645) | – |

**Table 3.**The summary statistics, stationarity testing and the correlation structure of real-world data.

Statistics | Series A | Series B |
---|---|---|

Minimum | 0 | 0 |

Maximum | 7 | 18 |

Mode | 0 | 0 |

Median | 1 | 0 |

Mean | 1.261 | 1.116 |

St. deviation | 1.256 | 1.763 |

Variance | 1.577 | 3.109 |

Skewness | 1.062 | 2.931 |

Kurtosis | 4.099 | 14.44 |

ADF-test | −7.705 | −5.667 |

(p-value) | (<0.01) | (<0.01) |

ACF(1) | 0.150 | 0.535 |

ACF(2) | 0.119 | 0.427 |

⋯ | ⋯ | ⋯ |

ACF(10) | 0.115 | 0.255 |

⋯ | ⋯ | ⋯ |

ACF(40) | 0.083 | 0.007 |

**Table 4.**Estimated parameters of the INAR(1) and INSB(1) processes, their respective estimation errors and predictive test statistics.

Parameters/Statistics (Stand. Errors) | Series A | Series B | ||
---|---|---|---|---|

INAR(1) | INSB(1) | INAR(1) | INSB(1) | |

a | 1.0713 ($5.24\times {10}^{-2}$) | 1.0911 ($4.07\times {10}^{-2}$) | 0.3422 ($7.79\times {10}^{-3}$) | 0.3624 ($7.47\times {10}^{-3}$) |

$\alpha $ | 0.1500 ($2.94\times {10}^{-2}$) | 0.1697 ($1.96\times {10}^{-2}$) | 0.5348 ($4.26\times {10}^{-3}$) | 0.4837 ($4.06\times {10}^{-3}$) |

${\mu}_{q}$ | – | 0.6290 ($1.96\times {10}^{-2}$) | – | 0.4503 ($4.06\times {10}^{-3}$) |

${Q}_{T}^{\left(2\right)}$ | $1.39\times {10}^{-2}$ | $4.63\times {10}^{-3}$ | $7.74\times {10}^{-3}$ | $3.04\times {10}^{-3}$ |

MSEE | $1.89\times {10}^{-2}$ | $1.82\times {10}^{-2}$ | $2.48\times {10}^{-2}$ | $1.99\times {10}^{-2}$ |

AIC | −661.95 | −698.43 | −531.39 | −553.12 |

$DM$ | 2.6741 ** | 2.4683 ** | ||

(p-value) | ($3.77\times {10}^{-3}$) | ($6.86\times {10}^{-3}$) |

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## Share and Cite

**MDPI and ACS Style**

Stojanović, V.S.; Bakouch, H.S.; Gajtanović, Z.; Almuhayfith, F.E.; Kuk, K.
Integer-Valued Split-BREAK Process with a General Family of Innovations and Application to Accident Count Data Modeling. *Axioms* **2024**, *13*, 40.
https://doi.org/10.3390/axioms13010040

**AMA Style**

Stojanović VS, Bakouch HS, Gajtanović Z, Almuhayfith FE, Kuk K.
Integer-Valued Split-BREAK Process with a General Family of Innovations and Application to Accident Count Data Modeling. *Axioms*. 2024; 13(1):40.
https://doi.org/10.3390/axioms13010040

**Chicago/Turabian Style**

Stojanović, Vladica S., Hassan S. Bakouch, Zorica Gajtanović, Fatimah E. Almuhayfith, and Kristijan Kuk.
2024. "Integer-Valued Split-BREAK Process with a General Family of Innovations and Application to Accident Count Data Modeling" *Axioms* 13, no. 1: 40.
https://doi.org/10.3390/axioms13010040