# Higher-Order INAR Model Based on a Flexible Innovation and Application to COVID-19 and Gold Particles Data

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## Abstract

**:**

## 1. Introduction

## 2. The INAR(1) Process with the PEE Innovations

#### The PEE-INAR(1) Model

## 3. The INAR(p) Model with PEE Innovations

## 4. Estimation

#### 4.1. Conditional Maximum Likelihood

#### 4.2. Conditional Least Squares

## 5. Simulation Study

## 6. Empirical Study

- (i)
- INAR model based on the discrete Teissier innovations (DT-INAR), see [29].
- (ii)
- INAR model based on the binomial-discrete Poisson Lindley innovations (BDPL-INAR), see [30].
- (iii)
- INAR model based on the three parameter discrete-Lindley innovations (DLi3-INAR), see [31].

#### 6.1. COVID-19 Data

#### 6.2. Gold Particles Data

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

`ppois=function(x,lambda,theta){``f=1-(((lambda+1)^(-(x+2))∗(lambda+lambda^2+theta+(x+2)∗lambda∗theta)``)``/(lambda+theta))``return(f)``}``ppois(2,0.5,1)``rpois <- function(n, L,T)``{``U <- runif(n)``X <- rep(0,n)``# loop through each uniform``for(i in 1:n)``{``# first check if you are in the first interval``if(U[i] < ppois(0,L,T))``{``X[i] <- 0``} else``{``# while loop to determine which subinterval,I, you``are in``# terminated when B = TRUE``B = FALSE``I = 0``while(B == FALSE)``{``# the interval to check``int <- c( ppois(I, L,T), ppois(I+1,L,T) )``# see if the uniform is in that interval``if( (U[i] > int[1]) & (U[i] < int[2]) )``{``# if so, quit the while loop and store``the value``X[i] <- I+1``B = TRUE``} else``{``# If not, continue the while loop and``increase I by 1``I=I+1``}``}``}``}``return(X)``}``rpois(50, 1.5, 1.2)``r.inarp.sim <- function(n, order.max, alpha,lambda,theta){``x <- rep(NA, times = n)``error <- rpois(n, lambda, theta)``for (i in 1:order.max) {``x[i] <- error[i]``}``for (t in (order.max + 1):n) {``x[t] <- 0``for (j in 1:order.max) {``x[t] <- x[t] + rbinom(1, x[t - j], alpha[j])``}``x[t] <- x[t] + error[t]``}``return(x)``}``r.inarp.sim(n = 100, order.max = 2, alpha = c(0.1,0.4),lambda = 2,theta``=0.5)`

## References

- McKenzie, E. Some simple models for discrete variate time series 1. Jawra J. Am. Water Resour. Assoc.
**1985**, 21, 645–650. [Google Scholar] [CrossRef] - Al-Osh, M.A.; Alzaid, A.A. First-order integer-valued autoregressive (INAR(1)) process. J. Time Ser. Anal.
**1987**, 8, 261–275. [Google Scholar] [CrossRef] - McKenzie, E. Some ARMA models for dependent sequences of Poisson counts. Adv. Appl. Probab.
**1988**, 20, 822–835. [Google Scholar] [CrossRef] - Ristić, M.M.; Bakouch, H.S.; Nastić, A.S. A new geometric first-order integer-valued autoregressive (NGINAR(1)) process. J. Stat. Plan. Inference
**2009**, 139, 2218–2226. [Google Scholar] [CrossRef] - Bakouch, H.S.; Ristić, M.M. Zero truncated Poisson integer-valued AR(1) model. Metrika
**2010**, 72, 265–280. [Google Scholar] [CrossRef] - Schweer, S.; Weiß, C.H. Compound Poisson INAR(1) processes: Stochastic properties and testing for overdispersion. Comput. Stat. Data Anal.
**2014**, 77, 267–284. [Google Scholar] [CrossRef] - Bourguignon, M.; Vasconcellos, K.L. First order non-negative integer valued autoregressive processes with power series innovations. Braz. J. Probab. Stat.
**2015**, 29, 71–93. [Google Scholar] [CrossRef] - Lívio, T.; Khan, N.M.; Bourguignon, M.; Bakouch, H.S. An INAR(1) model with Poisson–Lindley innovations. Econ. Bull.
**2018**, A, 1505–1513. [Google Scholar] - Aghababaei Jazi, M.; Jones, G.; Lai, C.D. Integer valued AR(1) with geometric innovations. J. Iran. Stat. Soc.
**2012**, 11, 173–190. [Google Scholar] - Altun, E.; Bhati, D.; Khan, N.M. A new approach to model the counts of earthquakes: INARPQX(1) process. Appl. Sci.
**2021**, 3, 274. [Google Scholar] [CrossRef] - Altun, E. A new one-parameter discrete distribution with associated regression and integer-valued autoregressive models. Math. Slovaca
**2020**, 70, 979–994. [Google Scholar] [CrossRef] - Popović, P.M. A bivariate INAR(1) model with different thinning parameters. Stat. Pap.
**2016**, 57, 517–538. [Google Scholar] [CrossRef] - Mohammadpour, M.; Bakouch, H.S.; Shirozhan, M. Poisson–Lindley INAR(1) model with applications. Braz. J. Probab. Stat.
**2018**, 32, 262–280. [Google Scholar] [CrossRef] - Alzaid, A.A.; Al-Osh, M. An integer-valued pth-order autoregressive structure (INAR(p)) process. J. Appl. Probab.
**1990**, 27, 314–324. [Google Scholar] [CrossRef] - Du, J.-G.; Li, Y. The integer-valued autoregressive (INAR(p)) model. J. Time Ser. Anal.
**1991**, 12, 129–142. [Google Scholar] - Drost, F.C.; Van Den Akker, R.; Werker, B.J. Local asymptotic normality and efficient estimation for INAR(p) models. J. Time Ser. Anal.
**2008**, 29, 783–801. [Google Scholar] [CrossRef] - Drost, F.C.; Van den Akker, R.; Werker, B.J. Efficient estimation of auto-regression parameters and innovation distributions for semiparametric integer-valued AR(p) models. J. R. Stat. Soc. Ser. Stat. Methodol.
**2009**, 71, 467–485. [Google Scholar] [CrossRef] - Zhang, H.; Wang, D.; Zhu, F. Inference for INAR(p) processes with signed generalized power series thinning operator. J. Stat. Plan. Inference
**2010**, 140, 667–683. [Google Scholar] [CrossRef] - Gladyshev, E.G. Periodically correlated random sequence. Soviet. Math.
**1961**, 2, 385–388. [Google Scholar] - Monteiro, M.; Scotto, M.G.; Pereira, I. Integer-valued autoregressive processes with periodic structure. J. Stat. Plan. Inference
**2010**, 140, 1529–1541. [Google Scholar] [CrossRef] - Buteikis, A.; Leipus, R. An integer-valued autoregressive process for seasonality. J. Stat. Comput. Simul.
**2020**, 90, 391–411. [Google Scholar] [CrossRef] - Liu, Z.; Li, Q.; Zhu, F. Random environment binomial thinning integer-valued autoregressive process with Poisson or geometric marginal. Braz. J. Probab. Stat.
**2020**, 34, 251–272. [Google Scholar] [CrossRef] - Prezotti Filho, P.R.; Reisen, V.A.; Bondon, P.; Ispány, M.; Melo, M.M.; Serpa, F.S. A periodic and seasonal statistical model for non-negative integer-valued time series with an application to dispensed medications in respiratory diseases. Appl. Math. Model.
**2021**, 96, 545–558. [Google Scholar] [CrossRef] - Maya, R.; Chesneau, C.; Krishna, A.; Irshad, M.R. Poisson extended exponential distribution with associated INAR(1) process and applications. Stats
**2022**, 5, 755–772. [Google Scholar] [CrossRef] - Weiß, C.H. An Introduction to Discrete-Valued Time Series; John Wiley & Sons: Hoboken, NJ, USA, 2018. [Google Scholar]
- Al-Osh, M.; Alzaid, A.A. Integer-valued moving average (INMA) process. Stat. Pap.
**1988**, 29, 281–300. [Google Scholar] [CrossRef] - Bu, R.; McCabe, B.; Hadri, K. Maximum likelihood estimation of higher-order integer-valued autoregressive processes. J. Time Ser. Anal.
**2008**, 29, 973–994. [Google Scholar] [CrossRef] - Joe, H. Likelihood inference for generalized integer autoregressive time series models. Econometrics
**2019**, 7, 43. [Google Scholar] [CrossRef] - Irshad, M.R.; Jodrá, P.; Krishna, A.; Maya, R. On the discrete analogue of the Teissier distribution and its associated INAR(1) process. Math. Comput. Simul.
**2023**, 214, 227–245. [Google Scholar] [CrossRef] - Shirozhan, M.; Okereke, E.W.; Bakouch, H.S.; Chesneau, C. A flexible integer-valued AR(1) process: Estimation, forecasting and modeling COVID-19 data. J. Stat. Comput. Simul.
**2023**, 93, 1461–1477. [Google Scholar] [CrossRef] - Eliwa, M.S.; Altun, E.; El-Dawoody, M.; El-Morshedy, M. A new three-parameter discrete distribution with associated INAR(1) process and applications. IEEE Access
**2020**, 8, 91150–91162. [Google Scholar] [CrossRef]

${\mathit{\alpha}}_{1}$ = 0.5, ${\mathit{\alpha}}_{2}$ = 0.3, $\mathit{\eta}$ = 1.6, $\mathit{\gamma}$ = 0.7 | |||||
---|---|---|---|---|---|

Parameter | n | CML | CLS | ||

Bias | MSE | Bias | MSE | ||

${\alpha}_{1}$ | 50 | −0.00472 | 0.01616 | −0.05084 | 0.02531 |

100 | −0.00314 | 0.00689 | −0.02297 | 0.01125 | |

200 | −0.00184 | 0.00355 | −0.01150 | 0.00573 | |

300 | −0.00125 | 0.00226 | −0.00937 | 0.00369 | |

400 | −0.00010 | 0.00181 | −0.00432 | 0.00280 | |

${\alpha}_{2}$ | 50 | −0.01962 | 0.01838 | −0.05369 | 0.02129 |

100 | −0.01627 | 0.00868 | −0.03749 | 0.01076 | |

200 | −0.00625 | 0.00416 | −0.01619 | 0.00571 | |

300 | −0.00504 | 0.00268 | −0.01262 | 0.00362 | |

400 | −0.00438 | 0.00208 | −0.01032 | 0.00287 | |

$\eta $ | 50 | −0.09165 | 0.35569 | −0.20154 | 0.37458 |

100 | −0.08853 | 0.18334 | −0.14256 | 0.28982 | |

200 | −0.07156 | 0.10112 | −0.13650 | 0.14152 | |

300 | −0.06249 | 0.08055 | −0.11754 | 0.10136 | |

400 | −0.02908 | 0.06195 | −0.06508 | 0.09931 | |

$\gamma $ | 50 | 0.19165 | 0.15335 | −0.19854 | 0.16169 |

100 | 0.15675 | 0.11126 | −0.16944 | 0.15609 | |

200 | 0.07935 | 0.05058 | −0.16451 | 0.15126 | |

300 | 0.06627 | 0.05003 | −0.13749 | 0.14404 | |

400 | 0.05120 | 0.03762 | −0.08557 | 0.11326 |

${\mathit{\alpha}}_{1}$ = 0.4, ${\mathit{\alpha}}_{2}$ = 0.2, $\mathit{\eta}$ = 1.2, $\mathit{\gamma}$ = 0.9 | |||||
---|---|---|---|---|---|

Parameter | n | CML | CLS | ||

Bias | MSE | Bias | MSE | ||

${\alpha}_{1}$ | 50 | 0.00543 | 0.01301 | −0.04288 | 0.02287 |

100 | 0.00470 | 0.00724 | −0.01902 | 0.01254 | |

200 | 0.00072 | 0.00322 | −0.01243 | 0.00567 | |

300 | 0.00234 | 0.00229 | −0.00677 | 0.00410 | |

400 | 0.00113 | 0.00165 | −0.00652 | 0.00286 | |

${\alpha}_{2}$ | 50 | −0.01558 | 0.01382 | −0.04464 | 0.01667 |

100 | −0.00969 | 0.00820 | −0.02861 | 0.01087 | |

200 | −0.00204 | 0.00374 | −0.01393 | 0.00536 | |

300 | −0.00337 | 0.00251 | −0.01279 | 0.00364 | |

400 | −0.00143 | 0.00189 | −0.00873 | 0.00265 | |

$\eta $ | 50 | −0.04760 | 0.11652 | −0.08939 | 0.11574 |

100 | −0.02667 | 0.06381 | −0.08442 | 0.06985 | |

200 | −0.00988 | 0.03173 | −0.07852 | 0.04698 | |

300 | −0.00879 | 0.01873 | −0.07821 | 0.04034 | |

400 | −0.00591 | 0.01515 | −0.02922 | 0.03297 | |

$\gamma $ | 50 | −0.24843 | 0.25267 | 0.27138 | 0.37775 |

100 | −0.23901 | 0.24417 | 0.25147 | 0.30304 | |

200 | −0.23631 | 0.24380 | 0.24401 | 0.25091 | |

300 | −0.23352 | 0.22876 | 0.24300 | 0.23349 | |

400 | −0.19476 | 0.22753 | 0.23390 | 0.23187 |

Model | Parameter | Estimate | Std Error | log L | AIC | BIC |
---|---|---|---|---|---|---|

PEE-INAR(2) | ${\alpha}_{1}$ | 0.34330 | 0.08397 | −143.33799 | 294.67599 | 304.71942 |

${\alpha}_{2}$ | 0.29190 | 0.09068 | ||||

$\eta $ | 1.38030 | 1.42716 | ||||

$\gamma $ | 0.00010 | 1.39072 | ||||

PEE-INAR(1) | $\alpha $ | 0.45897 | 1.05246 | −149.01889 | 304.03779 | 311.57037 |

$\eta $ | 1.08045 | 1.50832 | ||||

$\gamma $ | 0.11301 | 0.06388 | ||||

DT-INAR(2) | ${\alpha}_{1}$ | 0.35590 | 0.06094 | −162.03214 | 330.06429 | 337.59686 |

${\alpha}_{2}$ | 0.27323 | 0.06300 | ||||

$\lambda $ | 0.48354 | 0.02498 | ||||

DT-INAR(1) | $\alpha $ | 0.40357 | 0.01799 | −181.43955 | 366.87910 | 371.90082 |

$\lambda $ | 0.59686 | 0.05366 | ||||

BDPL-INAR(2) | ${\alpha}_{1}$ | 0.34433 | 0.08350 | −143.51806 | 295.03612 | 305.07956 |

${\alpha}_{2}$ | 0.28807 | 0.09016 | ||||

$\lambda $ | 0.99990 | 1.91169 | ||||

$\beta $ | 0.48685 | 1.08893 | ||||

BDPL-INAR(1) | $\alpha $ | 0.45899 | 117.11367 | −149.01889 | 304.03779 | 311.57037 |

$\lambda $ | 9.53718 | 116.77791 | ||||

$\beta $ | 8.82505 | 0.06388 | ||||

DLi3-INAR(2) | ${\alpha}_{1}$ | 0.34330 | 0.08397 | −143.33799 | 296.67599 | 309.23028 |

${\alpha}_{2}$ | 0.29190 | 0.09068 | ||||

$\lambda $ | 6.70660 | 5.31318 | ||||

$\beta $ | 0.42013 | 0.25201 | ||||

$\theta $ | 0.00010 | 3.91858 | ||||

DLi3-INAR(1) | $\alpha $ | 0.45926 | 59.2776 | −149.02031 | 308.04062 | 320.59491 |

$\lambda $ | 0.6639 | 3.2197 | ||||

$\beta $ | 0.0352 | 0.2434 | ||||

$\theta $ | 0.4805 | 0.0638 |

Model | Parameter | Estimate | Std Error | log L | AIC | BIC |
---|---|---|---|---|---|---|

PEE-INAR(2) | ${\alpha}_{1}$ | 0.49188 | 0.04530 | −522.31596 | 1052.63192 | 1068.39260 |

${\alpha}_{2}$ | 0.20424 | 0.05276 | ||||

$\eta $ | 4.21543 | 0.58439 | ||||

$\gamma $ | 9.9990 | 4.63723 | ||||

PEE-INAR(1) | $\alpha $ | 0.57037 | 0.27852 | −534.38144 | 1074.76288 | 1086.58339 |

$\eta $ | 2.64733 | 7.87235 | ||||

$\gamma $ | 9.99000 | 0.03105 | ||||

DT-INAR(2) | ${\alpha}_{1}$ | 0.34821 | 0.04483 | −535.07328 | 1076.14657 | 1087.96708 |

${\alpha}_{2}$ | 0.17003 | 0.04221 | ||||

$\lambda $ | 0.46375 | 0.01549 | ||||

DT-INAR(1) | $\alpha $ | 0.41251 | 0.01157 | −550.51787 | 1105.03574 | 1112.91608 |

$\lambda $ | 0.51266 | 0.03352 | ||||

BDPL-INAR(2) | ${\alpha}_{1}$ | 0.48773 | 0.04601 | −522.46265 | 1052.92530 | 1068.68599 |

${\alpha}_{2}$ | 0.21088 | 0.05324 | ||||

$\lambda $ | 0.06186 | 0.06766 | ||||

$\beta $ | 0.01499 | 0.01673 | ||||

BDPL-INAR(1) | $\alpha $ | 0.56887 | 0.03965 | −533.39932 | 1072.79864 | 1084.61915 |

$\lambda $ | 0.04884 | 0.01399 | ||||

$\beta $ | 0.01694 | 0.03114 | ||||

DLi3-INAR(2) | ${\alpha}_{1}$ | 0.49812 | 0.04445 | −524.88199 | 1059.76397 | 1079.46483 |

${\alpha}_{2}$ | 0.22716 | 0.05073 | ||||

$\lambda $ | 3.87526 | 2.34043 | ||||

$\beta $ | 0.29820 | 0.04151 | ||||

$\theta $ | 0.23710 | 12.46557 | ||||

DLi3-INAR(1) | $\alpha $ | 0.54084 | 3.69433 | −530.72820 | 1071.45641 | 1091.15726 |

$\lambda $ | 0.00856 | 719.40720 | ||||

$\beta $ | 1.66825 | 0.02747 | ||||

$\theta $ | 0.19226 | 0.03634 |

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## Share and Cite

**MDPI and ACS Style**

Almuhayfith, F.E.; Krishna, A.; Maya, R.; Irshad, M.R.; Bakouch, H.S.; Almulhim, M.
Higher-Order INAR Model Based on a Flexible Innovation and Application to COVID-19 and Gold Particles Data. *Axioms* **2024**, *13*, 32.
https://doi.org/10.3390/axioms13010032

**AMA Style**

Almuhayfith FE, Krishna A, Maya R, Irshad MR, Bakouch HS, Almulhim M.
Higher-Order INAR Model Based on a Flexible Innovation and Application to COVID-19 and Gold Particles Data. *Axioms*. 2024; 13(1):32.
https://doi.org/10.3390/axioms13010032

**Chicago/Turabian Style**

Almuhayfith, Fatimah E., Anuresha Krishna, Radhakumari Maya, Muhammad Rasheed Irshad, Hassan S. Bakouch, and Munirah Almulhim.
2024. "Higher-Order INAR Model Based on a Flexible Innovation and Application to COVID-19 and Gold Particles Data" *Axioms* 13, no. 1: 32.
https://doi.org/10.3390/axioms13010032