# Coherent Forecasting for a Mixed Integer-Valued Time Series Model

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## Abstract

**:**

## 1. Introduction

## 2. Background on Integer-Valued Time Series Models

#### 2.1. First-Order Integer-Valued Autoregressive Model

#### 2.2. Pegram’s First-Order Autoregressive Process (AR(1))

#### 2.3. First-Order Mixture of Pegram and Thinning Autoregressive (MPT(1)) Process

## 3. Likelihood-Based Estimation

#### 3.1. Expectation-Maximization Algorithm

#### 3.2. Asymptotic Distribution

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

**Proposition**

**4.**

**Proposition**

**5.**

**Theorem**

**1.**

## 4. Coherent Forecasting

#### 4.1. Descriptive Measures

- A.
- Prediction root-mean-squared error (PRMSE):

- B.
- Prediction mean absolute deviation (PMAD):

- C.
- Percentage of true prediction (PTP):

#### 4.2. Confidence Interval

**Theorem**

**2.**

## 5. Simulation Study

## 6. Real Applications

#### 6.1. Burn Claims Data

#### 6.2. Burglary Data

## 7. Final Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Khoo, W.C.; Ong, S.H.; Biswas, A. Modeling time series of counts with a new class of INAR(1) model. Stat. Pap.
**2017**, 58, 393–416. [Google Scholar] [CrossRef] - Shirozhan, M.; Mohammadpour, M. An INAR(1) model based on the Pegram and thinning operators with serially dependent innovation. Commun. Stat. Simul. Comput.
**2020**, 49, 2617–2638. [Google Scholar] [CrossRef] - Kang, Y.; Wang, D.; Yang, K. A new INAR(1) process with bounded support for counts showing equidispersion, underdispersion and overdispersion. Stat. Pap.
**2021**, 62, 745–767. [Google Scholar] [CrossRef] - Yan, H.; Wang, D.H.; Li, C. A study for the NMBAR(1) processes. Commun. Stat. Simul. Comput.
**2022**, 1–22. [Google Scholar] [CrossRef] - McKenzie, E. Some simple models for discrete variate time series. Water Resour. Bull.
**1985**, 21, 645–650. [Google Scholar] [CrossRef] - Freeland, R.K. Statistical Analysis of Discrete Time Series with Application to the Analysis of Workers’ Compensation Claims Data. Ph.D. Thesis, The University of British Columbia, Vancouver, BC, Canada, 1998. [Google Scholar]
- Freeland, R.K.; McCabe, B.P.M. Forecasting discrete valued low count time series. Int. J. Forecast.
**2004**, 20, 427–434. [Google Scholar] [CrossRef] - Bu, R.B.; McCabe, B.; Hadri, K. Maximum likelihood estimation of higher-order integer-valued autoregressive process. J. Time Ser. Anal.
**2009**, 29, 973–994. [Google Scholar] [CrossRef] - McCabe, B.P.M.; Martin, G.M. Bayesian predictions of low count time series. Int. J. Forecast.
**2005**, 21, 315–330. [Google Scholar] [CrossRef] - Jung, R.C.; Tremayne, A.R. Coherent forecasting in integer time series models. Int. J. Forecast.
**2006**, 22, 223–238. [Google Scholar] [CrossRef] - Kim, H.Y.; Park, Y. Markov chain approach to forecast in the binomial autoregressive models. Commun. Korean Stat. Soc.
**2010**, 17, 441–450. [Google Scholar] [CrossRef] - Maiti, R.; Biswas, A.; Das, S. Coherent forecasting for count time series using Box-Jenkins’s AR(p) model. Stat. Neerl.
**2016**, 70, 123–145. [Google Scholar] [CrossRef] - Maiti, R.; Biswas, A.; Das, S. Time series of zero-inflated counts and their coherent forecasting. J. Forecast.
**2015**, 34, 694–707. [Google Scholar] [CrossRef] - Awale, M.; Ramanathan, T.V.; Kale, M. Coherent forecasting in integer-valued AR(1) models with geometric marginals. J. Data Sci.
**2017**, 15, 95–114. [Google Scholar] [CrossRef] - Nik, S.; Weiss, C. CLAR(1) point forecasting under estimation uncertainty. Stat. Neerl.
**2020**, 74, 489–526. [Google Scholar] [CrossRef] - Weiss, C. Thinning operations for modelling time series of counts—A survey. AStA Adv. Stat. Anal.
**2008**, 92, 319. [Google Scholar] [CrossRef] - Pegram, G.G.S. An autoregressive model for multilag Markov chain. J. Appl. Probab.
**1980**, 17, 350–362. [Google Scholar] [CrossRef] - Biswas, A.; Song, X.-K. Peter. Discrete-valued ARMA processes. Stat. Probab. Lett.
**2009**, 79, 1884–1889. [Google Scholar] [CrossRef] - Jacobs, P.A.; Lewis, A.W. Discrete Time Series Generated by Mixtures III: Autoregressive Processes (DAR(p)); Naval Postgraduate School: Monterey, CA, USA, 1978. [Google Scholar]
- Grunwald, G.K.; Hyndman, R.J.; Tedesco, L.; Tweedie, R.L. Non-Gaussian conditional linear AR(1) models. Aust. N. Z. J. Stat.
**2000**, 42, 479–495. [Google Scholar] [CrossRef] - Dempster, A.; Laird, N.; Rubin, D. Maximum Likelihood from Incomplete Data via the EM Algorithm. J. R. Stat. Soc. B
**1977**, 39, 1–38. [Google Scholar] - Karlis, D.; Xekalaki, E. Improving the EM algorithm for mixtures. Stat. Comput.
**1999**, 9, 303–307. [Google Scholar] [CrossRef] - Marcellino, M.; Stock, J.H.; Watson, M.W. A comparison of direct and iterated multistep AR methods for forecasting macroeconomic time series. J. Econ.
**2006**, 135, 499–526. [Google Scholar] [CrossRef]

**Table 1.**Estimated PRMSE, PMAD and PTP for Pegram’s AR(1), INAR(1) and MPT(1), with Poisson process.

Model | Parameters | PRMSE | PMAD | PTP (%) |
---|---|---|---|---|

Pegram’s AR(1) | (0.5,0.4) | 0.0867 | 1.6135 | 22.3474 |

(0.3,0.8) | 0.0335 | 0.4000 | 66.7706 | |

INAR(1) | (0.5,0.4) | 0.9952 | 2.0921 | 14.8930 |

(0.3,0.8) | 0.0341 | 0.3997 | 65.0158 | |

MPT(1) | (0.5,0.4) | 0.1482 | 1.4890 | 23.6388 |

(0.3,0.8) | 0.02446 | 0.4528 | 59.4330 |

Model | Conditional Mean | Conditional Median | |
---|---|---|---|

MPT(1) | PRMSE | 0.5492 | 1.3784 |

PMAD | 0.3152 | 1.3 | |

PTP (%) | 50 | 0 | |

Pegram’s AR(1) | PRMSE | 1.0585 | 1.3784 |

PMAD | 0.9511 | 1.3 | |

PTP (%) | 0 | 0 | |

INAR(1) | PRMSE | 0.9037 | 1.3784 |

PMAD | 0.7359 | 1.3 | |

PTP (%) | 0 | 0 |

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**MDPI and ACS Style**

Khoo, W.C.; Ong, S.H.; Atanu, B.
Coherent Forecasting for a Mixed Integer-Valued Time Series Model. *Mathematics* **2022**, *10*, 2961.
https://doi.org/10.3390/math10162961

**AMA Style**

Khoo WC, Ong SH, Atanu B.
Coherent Forecasting for a Mixed Integer-Valued Time Series Model. *Mathematics*. 2022; 10(16):2961.
https://doi.org/10.3390/math10162961

**Chicago/Turabian Style**

Khoo, Wooi Chen, Seng Huat Ong, and Biswas Atanu.
2022. "Coherent Forecasting for a Mixed Integer-Valued Time Series Model" *Mathematics* 10, no. 16: 2961.
https://doi.org/10.3390/math10162961