# Parameters Estimation in Non-Negative Integer-Valued Time Series: Approach Based on Probability Generating Functions

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. PGF Estimates: Definition and Asymptotic Properties

**Theorem 1.**

- (A
_{1}) - Θ is a compact set, where ${\theta}_{0}\in \mathrm{int}\left(\mathsf{\Theta}\right)$, and ${\widehat{\theta}}_{T}^{\left(r\right)}\in \mathrm{int}\left(\mathsf{\Theta}\right)$ for T large enough.
- (A
_{2}) - At the point $\theta ={\theta}_{0}$ function$${S}_{0}^{\left(r\right)}\left(\theta \right):={\int}_{-1}^{\phantom{\rule{0.277778em}{0ex}}1}\cdots {\int}_{-1}^{\phantom{\rule{0.277778em}{0ex}}1}\omega \left(\mathbf{u}\right){\left({\mathsf{\Psi}}_{X}^{\left(r\right)}(\mathbf{u};\theta )-{\mathsf{\Psi}}_{X}^{\left(r\right)}(\mathbf{u};{\theta}_{0})\right)}^{2}\mathrm{d}\mathbf{u}$$
- (A
_{3}) - $\frac{{\partial}^{2}{S}_{T}^{\left(r\right)}\left({\theta}_{0}\right)}{\partial \theta \phantom{\rule{4pt}{0ex}}\partial {\theta}^{\prime}}$ is a regular matrix.
- (A
_{4}) - $\frac{\partial {\mathsf{\Psi}}_{X}^{\left(r\right)}(\mathbf{u};{\theta}_{0})}{\partial \theta}}\phantom{\rule{4pt}{0ex}}{\displaystyle \frac{\partial {\mathsf{\Psi}}_{X}^{\left(r\right)}(\mathbf{u};{\theta}_{0})}{\partial {\theta}^{\prime}}$ is a non-zero matrix, uniformly bounded by the strictly positive ω-integrable function $\mathcal{L}:{\mathbb{R}}^{r}\to {\mathbb{R}}^{+}$.
- (A
_{5}) - The covariance function ${\gamma}_{K}(h;\theta ):=\mathrm{Cov}\left[{K}_{t}^{\left(r\right)}\left(\theta \right),\phantom{\rule{0.166667em}{0ex}}{K}_{t+h}^{\left(r\right)}\left(\theta \right)\right]$ of the series$${K}_{t}^{\left(r\right)}\left(\theta \right):={\int}_{-1}^{\phantom{\rule{0.277778em}{0ex}}1}\cdots {\int}_{-1}^{\phantom{\rule{0.277778em}{0ex}}1}\omega \left(\mathbf{u}\right)\phantom{\rule{0.277778em}{0ex}}\left[{\mathsf{\Psi}}_{X}^{\left(r\right)}(\mathbf{u};\theta )-{u}_{1}^{{X}_{t}}\cdots {u}_{r}^{{X}_{t+r-1}}\right]\frac{\partial {\mathsf{\Psi}}_{X}^{\left(r\right)}(\mathbf{u};\theta )}{\partial \theta}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\mathbf{u}.$$$$\sum _{h=-\infty}^{\infty}\left|{\gamma}_{K}(h;{\theta}_{0})\right|<+\infty .$$

**Proof.**

## 3. PGF Estimations of IID and INAR Time Series

#### 3.1. Estimation of IID Time Series

- $\left(i\right)$
- $a\left(x\right)\ge 0$ is a function that depends (only) on $x\in \mathcal{S}$;
- $\left(ii\right)$
- $\theta >0$ is a (one-dimensional) unknown parameter;
- $\left(iii\right)$
- $f\left(\theta \right):={\sum}_{x\in \mathcal{S}}a\left(x\right)\phantom{\rule{0.166667em}{0ex}}{\theta}^{x}$ is a function on $\theta $, so that $0<f\left(\theta \right)<+\infty $ when $\theta \in (0,R)$, $R>0$.

- $\left(i\right)$
- Chebyshev polynomials (of the first kind): ${\omega}_{0}\left(u\right)={(1-{u}^{2})}^{-1/2}$;
- $\left(ii\right)$
- Legendre polynomials: ${\omega}_{1}\left(u\right)\equiv 1$;
- $\left(iii\right)$
- Chebyshev polynomials (of the second kind): ${\omega}_{2}\left(u\right)={(1-{u}^{2})}^{1/2}$.

#### 3.2. Estimation of INAR Time Series

**Lemma 1.**

**Proof.**

**Theorem 2.**

**Proof.**

**Remark 1.**

## 4. Application of the PGF Estimates

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ACF | Auto Correlation Function |

AD test | Anderson–Darling test |

AIC | Akaike Information Criterion |

AN | Asymptotic Normality |

CF | Characteristic Function |

ECF | Empirical Characteristic Function |

IID series | Independent Identical Distributed series |

INAR process | Integer-Valued Autoregressive process |

INMA | Integer Moving Average |

MSEE | Mean Squared Estimating Error |

NIINAR process | Noise-Indicator Integer-Valued Autoregressive process |

NNIV series | Non-Negative Integer-Valued series |

PGF | Probability Generating Function |

PMF | Probability Mass Distribution |

PS | Power Series |

RV | Random Variable |

SLLN | Strong Law of Large Numbers |

YW | Yule–Walker |

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**Figure 1.**Graphs of PGFs (lines) and empirical PGFs (dots) of the IID series: (

**a**) Poisson distribution; (

**b**) geometric distribution. (True parameter value is $\theta =1/2$).

**Figure 2.**Plots of the sample frequency distributions of INAR(1) processes (

**a**) and their innovations (

**b**) with Poisson distribution (plots above) and geometric distribution (plots below). True parameters’ values are $\theta =\alpha =1/2$.

**Figure 3.**Three-dimensional plots of the two-dimensional PGFs (

**a**) and empirical PGFs (

**b**) of the INAR(1) series with Poisson innovations (panels above) and geometric innovations (panels below). True parameters’ values are $\theta =\alpha =1/2$.

**Figure 4.**(

**a**) Dynamics of the number of deaths before and after the onset of the disease COVID-19 in Serbia. (

**b**) The corresponding ACFs of the observed time series.

**Figure 5.**Frequency distribution of the actual and fitted data, obtained by (

**a**) IID Poisson process; (

**b**) INAR(1) process with geometric innovations.

Distributions | $\mathit{S}$ | $\mathit{a}\left(\mathit{x}\right)$ | $(0,\mathit{R})$ | $\mathit{f}\left(\mathit{\theta}\right)$ | ${\mathbf{\Psi}}_{\mathit{\epsilon}}(\mathit{u};\mathit{\theta})$ | $\mathit{R}/\mathit{\theta}$ |
---|---|---|---|---|---|---|

1. Bernoulli | $\left\{0,1\right\}$ | 1 | $(0,\infty )$ | $1+\theta $ | $\frac{1+\theta u}{1+\theta}$ | ∞ |

2. Binomial | $\left\{0,\cdots ,n\right\}$ | $\left(\genfrac{}{}{0pt}{}{n}{x}\right)$ | $(0,\infty )$ | ${(1+\theta )}^{n}$ | $\left(\frac{1+\theta u}{1+\theta}\right)}^{n$ | ∞ |

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

4. Geometric | $\left\{0,\cdots ,\infty \right\}$ | 1 | $(0,1)$ | $\frac{1}{1-\theta}$ | $\frac{1-\theta}{1-\theta u}$ | $1/\theta $ |

5. Neg. Binomial | $\left\{0,\cdots ,\infty \right\}$ | $\frac{\Gamma (x+n)}{x!\phantom{\rule{0.166667em}{0ex}}\Gamma \left(n\right)}$ | $(0,1)$ | $\frac{1}{{(1-\theta )}^{n}}$ | $\left(\frac{1-\theta}{1-\theta u}\right)}^{n$ | $1/\theta $ |

6. Pascal | $\left\{n,\cdots ,\infty \right\}$ | $\left(\genfrac{}{}{0pt}{}{x-1}{n-1}\right)$ | $(0,1)$ | $\left(\frac{\theta}{1-\theta}\right)}^{n$ | $\left(\frac{(1-\theta )u}{1-\theta u}\right)}^{n$ | $1/\theta $ |

**Table 2.**Estimated parameters’ values of IID series $\left({\epsilon}_{t}\right)$ series with Poisson and geometric distribution. (True parameter value is $\theta =0.5$.)

Statistics | PGF Estimates | ||||||||
---|---|---|---|---|---|---|---|---|---|

${\widehat{\mathit{\theta}}}_{0}$ | ${\widehat{\mathit{S}}}_{{\mathit{T}}_{0}}$ | ${\widehat{\mathit{\theta}}}_{1}$ | ${\widehat{\mathit{S}}}_{{\mathit{T}}_{1}}$ | ${\widehat{\mathit{\theta}}}_{2}$ | ${\widehat{\mathit{S}}}_{{\mathit{T}}_{2}}$ | ||||

Poisson | Min. | 0.4222 | $4.63\times {10}^{-8}$ | 0.4293 | $3.20\times {10}^{-9}$ | 0.4267 | $3.83\times {10}^{-8}$ | ||

Mean | 0.5006 | $9.19\times {10}^{-5}$ | 0.4994 | $4.47\times {10}^{-5}$ | 0.5005 | $2.80\times {10}^{-5}$ | |||

Max. | 0.5967 | $7.17\times {10}^{-4}$ | 0.6098 | $4.00\times {10}^{-4}$ | 0.5938 | $2.38\times {10}^{-4}$ | |||

MSEE | 0.0220 | – | 0.0200 | – | 0.0196 | – | |||

$AD$ | 0.3849 | – | 0.1965 | – | 0.2911 | – | |||

p-value | 0.3919 | – | 0.8889 | – | 0.6076 | – | |||

Geometric | Min. | 0.4491 | $1.09\times {10}^{-6}$ | 0.4520 | $3.27\times {10}^{-7}$ | 0.4467 | $2.40\times {10}^{-7}$ | ||

Mean | 0.5004 | $2.30\times {10}^{-4}$ | 0.5001 | $1.06\times {10}^{-4}$ | 0.4996 | $6.17\times {10}^{-5}$ | |||

Max. | 0.5535 | $2.71\times {10}^{-3}$ | 0.5353 | $3.62\times {10}^{-3}$ | 0.5580 | $5.60\times {10}^{-4}$ | |||

MSEE | 0.0129 | – | 0.0121 | – | 0.0117 | – | |||

$AD$ | 0.4583 | – | 0.2214 | – | 0.2490 | – | |||

p-value | 0.2627 | – | 0.8307 | – | 0.7467 | – |

**Table 3.**Estimated parameter values of the INAR(1) process $\left({X}_{t}\right)$ with Poisson and geometric innovations. (True parameters’ values are $\theta =\alpha =1/2$).

Statistics | PGF Estimates | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\widehat{\mathit{\theta}}}_{0}$ | ${\widehat{\mathit{\alpha}}}_{0}$ | ${\mathit{S}}_{{\mathit{T}}_{0}}^{\left(2\right)}$ | ${\widehat{\mathit{\theta}}}_{1}$ | ${\widehat{\mathit{\alpha}}}_{1}$ | ${\mathit{S}}_{{\mathit{T}}_{1}}^{\left(2\right)}$ | ${\widehat{\mathit{\theta}}}_{2}$ | ${\widehat{\mathit{\alpha}}}_{2}$ | ${\mathit{S}}_{{\mathit{T}}_{2}}^{\left(2\right)}$ | ||||

Poisson | Min. | 0.3845 | 0.3606 | $2.12\times {10}^{-5}$ | 0.3653 | 0.3352 | $5.48\times {10}^{-6}$ | 0.3710 | 0.3365 | $1.65\times {10}^{-6}$ | ||

Mean | 0.5003 | 0.4986 | $8.08\times {10}^{-4}$ | 0.5002 | 0.5006 | $1.73\times {10}^{-4}$ | 0.5015 | 0.4951 | $7.36\times {10}^{-5}$ | |||

Max. | 0.6711 | 0.5971 | $7.12\times {10}^{-3}$ | 0.6948 | 0.6292 | $1.32\times {10}^{-3}$ | 0.7049 | 0.6251 | $5.13\times {10}^{-4}$ | |||

MSEE | 0.0329 | 0.0350 | – | 0.0364 | 0.0337 | – | 0.0381 | 0.0321 | – | |||

$AD$ | 0.7468 | 0.3089 | – | 0.7272 | 0.4153 | – | 0.8609 | 0.4724 | – | |||

p-value | 0.0515 | 0.5574 | – | 0.0576 | 0.3326 | – | 0.0269 * | 0.2427 | – | |||

Geometric | Min. | 0.3962 | 0.3592 | $3.96\times {10}^{-5}$ | 0.3673 | 0.3129 | $7.98\times {10}^{-6}$ | 0.3965 | 0.3199 | $2.75\times {10}^{-6}$ | ||

Mean | 0.5002 | 0.4974 | $7.31\times {10}^{-4}$ | 0.4981 | 0.4977 | $2.23\times {10}^{-4}$ | 0.4986 | 0.5005 | $8.98\times {10}^{-5}$ | |||

Max. | 0.6071 | 0.6646 | $9.02\times {10}^{-3}$ | 0.5960 | 0.6037 | $1.57\times {10}^{-3}$ | 0.6033 | 0.6747 | $7.39\times {10}^{-4}$ | |||

MSEE | 0.0291 | 0.0521 | – | 0.0265 | 0.0446 | – | 0.0271 | 0.0425 | – | |||

$AD$ | 0.6155 | 0.8198 | – | 0.3150 | 0.5174 | – | 0.4126 | 0.7898 | – | |||

p-value | 0.1088 | 0.0340 * | – | 0.5426 | 0.1882 | – | 0.3375 | 0.0403 * | – |

**Table 4.**Summary statistics of the total number of deaths per day before and after the appearance of the COVID-19 disease in the Republic of Serbia.

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

Sample size | 1095 | 990 | ||

Minimum | 0 | 0 | ||

Mode | 6 | 1 | ||

1st Quartile | 4 | 3 | ||

Median | 6 | 9 | ||

Mean | 5.758 | 17.58 | ||

3rd Quartile | 7 | 27 | ||

Maximum | 24 | 79 | ||

St. deviation | 2.428 | 18.48 | ||

Variance | 5.895 | 341.6 | ||

Skewness | 1.797 | 1.159 | ||

Kurtosis | 10.803 | 3.161 | ||

ACF(1) | 0.085 | 0.987 | ||

⋯ | ⋯ | ⋯ | ||

ACF(10) | 0.020 | 0.911 | ||

⋯ | ⋯ | ⋯ | ||

ACF(20) | $-0.024$ | 0.754 | ||

⋯ | ⋯ | ⋯ | ||

ACF(50) | 0.049 | 0.254 | ||

⋯ | ⋯ | ⋯ | ||

ACF(100) | $-0.020$ | 0.078 |

**Table 5.**Estimated values of parameters, along with error and predictive statistics of the actual data.

Parameters/Statistics | Series A | Series B | |||||
---|---|---|---|---|---|---|---|

${\mathbf{\omega}}_{\mathbf{0}}$ | ${\mathbf{\omega}}_{\mathbf{1}}$ | ${\mathbf{\omega}}_{\mathbf{2}}$ | ${\mathbf{\omega}}_{\mathbf{0}}$ | ${\mathbf{\omega}}_{\mathbf{1}}$ | ${\mathbf{\omega}}_{\mathbf{2}}$ | ||

$\theta $ | 5.8069 | 5.8218 | 5.8176 | 0.1812 | 0.1808 | 0.1810 | |

$\alpha $ | – | – | – | 0.9872 | 0.9865 | 0.9871 | |

${S}_{T}^{\left(2\right)}$ | $1.06\times {10}^{-3}$ | $1.06\times {10}^{-3}$ | $7.16\times {10}^{-5}$ | $6.48\times {10}^{-3}$ | $3.46\times {10}^{-3}$ | $1.92\times {10}^{-3}$ | |

MSEE | 0.0726 | 0.0789 | 0.0771 | 0.0221 | 0.0288 | 0.0233 | |

AIC | 4.5992 | 4.6023 | 4.6016 | 5.3035 | 5.3041 | 5.3042 | |

$DM$ | −0.3641 | 0.1039 | 0.0974 | 0.5855 | 0.7797 | 0.6456 | |

p-value | 0.3579 | 0.5414 | 0.5388 | 0.7208 | 0.7821 | 0.7406 |

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**MDPI and ACS Style**

Stojanović, V.; Ljajko, E.; Tošić, M.
Parameters Estimation in Non-Negative Integer-Valued Time Series: Approach Based on Probability Generating Functions. *Axioms* **2023**, *12*, 112.
https://doi.org/10.3390/axioms12020112

**AMA Style**

Stojanović V, Ljajko E, Tošić M.
Parameters Estimation in Non-Negative Integer-Valued Time Series: Approach Based on Probability Generating Functions. *Axioms*. 2023; 12(2):112.
https://doi.org/10.3390/axioms12020112

**Chicago/Turabian Style**

Stojanović, Vladica, Eugen Ljajko, and Marina Tošić.
2023. "Parameters Estimation in Non-Negative Integer-Valued Time Series: Approach Based on Probability Generating Functions" *Axioms* 12, no. 2: 112.
https://doi.org/10.3390/axioms12020112