# Monitoring Parameter Change for Time Series Models of Counts Based on Minimum Density Power Divergence Estimator

^{*}

## Abstract

**:**

## 1. Introduction

## 2. MDPDE for INGARCH Model: An Overview

## 3. MDPDE-Based Monitoring Process

**Theorem**

**1.**

**(A.1)**–

**(A.11)**hold. Then, under ${H}_{0}$, as $n\to \infty $, ${\widehat{T}}_{n,0}^{min}$ and ${\widehat{T}}_{n,0}^{max}$ converge to T in distribution, and the same holds for ${\widehat{T}}_{n}^{min}$ and ${\widehat{T}}_{n}^{max}$ if $m/n\to \infty $. Moreover, ${\widehat{T}}_{n,0}^{cusum}$ converges to ${T}^{{}^{\prime}}$ in distribution as $n\to \infty $, and so does ${\widehat{T}}_{n}^{cusum}$ if $m/n\to \lambda \in (0,\infty )$.

## 4. Simulation Results

## 5. Real Data Analysis

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CUSUM | cumulative sum |

INGARCH | integer-valued generalized autoregressive conditionally heteroscedastic |

INAR | integer-valued autoregressive |

MDPDE | minimum density power divergence etimator |

MLE | maximum likelihood estimator |

SPC | statistical process control |

## Appendix A

**Proof of**

**Theorem 1.**

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**Figure 1.**Plots of the Poisson INGARCH(1,1) time series (Case 3) with ${\theta}_{0}=(2,0.3,0.3)$, $\tau =500$ and $\delta =0$ for the left panel and $\delta =0.5$ for the right panel.

**Figure 2.**Plots of the sizes and powers in Table 10 (Part 3, Case 2) for $n=1000$. The left panel is for ${\widehat{T}}_{n}^{min}$ and the right panel is for ${\widehat{T}}_{n}^{cusum}$.

**Table 1.**Empirical sizes and powers in Case 1 for the Poisson INGARCH(1,1) model when no outliers exist with ${\theta}_{0}=(2,0.1,0.2)$.

$\mathit{\alpha}$ | n | $\mathit{\tau}$ | $\mathit{\delta}$: | 0 | 0.25 | 0.5 | 0.75 | 1 | |
---|---|---|---|---|---|---|---|---|---|

${\widehat{T}}_{n,0}^{min}$ | 0 | 500 | 250 | 0.035 | 0.541 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 500 | 250 | 0.036 | 0.428 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 500 | 250 | 0.048 | 0.997 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0 | 1000 | 500 | 0.042 | 0.791 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 1000 | 500 | 0.049 | 0.682 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 1000 | 500 | 0.052 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 500 | 250 | 0.035 | 0.523 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 500 | 250 | 0.036 | 0.398 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 500 | 250 | 0.043 | 0.995 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 1000 | 500 | 0.042 | 0.78 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 1000 | 500 | 0.051 | 0.642 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 1000 | 500 | 0.056 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 500 | 250 | 0.035 | 0.493 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 500 | 250 | 0.038 | 0.361 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 500 | 250 | 0.041 | 0.994 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 1000 | 500 | 0.04 | 0.757 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 1000 | 500 | 0.048 | 0.589 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 1000 | 500 | 0.066 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 500 | 250 | 0.035 | 0.465 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 500 | 250 | 0.042 | 0.332 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 500 | 250 | 0.036 | 0.992 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 1000 | 500 | 0.034 | 0.718 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 1000 | 500 | 0.047 | 0.551 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 1000 | 500 | 0.064 | 1 | 1 | 1 | 1 |

**Table 2.**Empirical sizes and powers in Case 2 for the Poisson INGARCH(1,1) model when no outliers exist with ${\theta}_{0}=(2,0.6,0.2)$.

$\mathit{\alpha}$ | n | $\mathit{\tau}$ | $\mathit{\delta}$: | 0 | $-1/5$ | $-1/3$ | $-3/7$ | $-1/2$ | |
---|---|---|---|---|---|---|---|---|---|

${\widehat{T}}_{n,0}^{min}$ | 0 | 500 | 250 | 0.05 | 0.983 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 500 | 250 | 0.06 | 0.86 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 500 | 250 | 0.049 | 0.893 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0 | 1000 | 500 | 0.052 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 1000 | 500 | 0.053 | 0.98 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 1000 | 500 | 0.059 | 0.997 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 500 | 250 | 0.047 | 0.984 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 500 | 250 | 0.058 | 0.871 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 500 | 250 | 0.046 | 0.9 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 1000 | 500 | 0.048 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 1000 | 500 | 0.041 | 0.977 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 1000 | 500 | 0.051 | 0.996 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 500 | 250 | 0.045 | 0.986 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 500 | 250 | 0.05 | 0.852 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 500 | 250 | 0.043 | 0.904 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 1000 | 500 | 0.052 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 1000 | 500 | 0.04 | 0.973 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 1000 | 500 | 0.054 | 0.997 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 500 | 250 | 0.04 | 0.985 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 500 | 250 | 0.043 | 0.845 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 500 | 250 | 0.048 | 0.912 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 1000 | 500 | 0.05 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 1000 | 500 | 0.052 | 0.978 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 1000 | 500 | 0.053 | 0.996 | 1 | 1 | 1 |

**Table 3.**Empirical sizes and powers in Case 3 for the Poisson INGARCH(1,1) model when no outliers exist with ${\theta}_{0}=(2,0.3,0.3)$.

$\mathit{\alpha}$ | n | $\mathit{\tau}$ | $\mathit{\delta}$: | 0 | 0.25 | 0.5 | 0.75 | 1 | |
---|---|---|---|---|---|---|---|---|---|

${\widehat{T}}_{n,0}^{min}$ | 0 | 500 | 250 | 0.046 | 0.309 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 500 | 250 | 0.043 | 0.216 | 0.993 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 500 | 250 | 0.047 | 0.685 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0 | 1000 | 500 | 0.039 | 0.473 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 1000 | 500 | 0.041 | 0.337 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 1000 | 500 | 0.057 | 0.969 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 500 | 250 | 0.044 | 0.292 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 500 | 250 | 0.046 | 0.208 | 0.992 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 500 | 250 | 0.054 | 0.696 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 1000 | 500 | 0.046 | 0.458 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 1000 | 500 | 0.047 | 0.314 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 1000 | 500 | 0.062 | 0.965 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 500 | 250 | 0.046 | 0.266 | 0.998 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 500 | 250 | 0.05 | 0.192 | 0.99 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 500 | 250 | 0.048 | 0.696 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 1000 | 500 | 0.044 | 0.44 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 1000 | 500 | 0.042 | 0.287 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 1000 | 500 | 0.067 | 0.962 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 500 | 250 | 0.041 | 0.244 | 0.998 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 500 | 250 | 0.051 | 0.179 | 0.986 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 500 | 250 | 0.051 | 0.669 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 1000 | 500 | 0.04 | 0.412 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 1000 | 500 | 0.045 | 0.267 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 1000 | 500 | 0.055 | 0.956 | 1 | 1 | 1 |

**Table 4.**Empirical sizes and powers in Case 4 for the Poisson INGARCH(1,1) model when no outliers exist with ${\theta}_{0}=(1,0.4,0.4)$.

$\mathit{\alpha}$ | n | $\mathit{\tau}$ | $\mathit{\delta}$: | 0 | $-1/5$ | $-1/3$ | $-3/7$ | $-1/2$ | |
---|---|---|---|---|---|---|---|---|---|

${\widehat{T}}_{n,0}^{min}$ | 0 | 500 | 250 | 0.044 | 0.687 | 0.991 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 500 | 250 | 0.049 | 0.345 | 0.75 | 0.941 | 0.986 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 500 | 250 | 0.058 | 0.364 | 0.828 | 0.957 | 0.991 | |

${\widehat{T}}_{n,0}^{min}$ | 0 | 1000 | 500 | 0.038 | 0.946 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 1000 | 500 | 0.039 | 0.626 | 0.969 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 1000 | 500 | 0.058 | 0.796 | 0.998 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 500 | 250 | 0.044 | 0.688 | 0.99 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 500 | 250 | 0.054 | 0.349 | 0.752 | 0.938 | 0.985 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 500 | 250 | 0.06 | 0.376 | 0.841 | 0.964 | 0.993 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 1000 | 500 | 0.042 | 0.945 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 1000 | 500 | 0.042 | 0.616 | 0.966 | 0.999 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 1000 | 500 | 0.053 | 0.782 | 0.997 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 500 | 250 | 0.047 | 0.686 | 0.989 | 0.999 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 500 | 250 | 0.056 | 0.357 | 0.757 | 0.939 | 0.986 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 500 | 250 | 0.056 | 0.378 | 0.832 | 0.965 | 0.991 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 1000 | 500 | 0.042 | 0.94 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 1000 | 500 | 0.039 | 0.597 | 0.965 | 0.999 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 1000 | 500 | 0.059 | 0.793 | 0.997 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 500 | 250 | 0.049 | 0.677 | 0.985 | 0.999 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 500 | 250 | 0.048 | 0.321 | 0.721 | 0.917 | 0.977 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 500 | 250 | 0.054 | 0.381 | 0.831 | 0.963 | 0.991 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 1000 | 500 | 0.043 | 0.931 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 1000 | 500 | 0.047 | 0.606 | 0.962 | 0.999 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 1000 | 500 | 0.064 | 0.792 | 0.997 | 1 | 1 |

**Table 5.**Empirical sizes and powers in Case 1 for the Poisson INGARCH(1,1) model when no outliers exist with ${\theta}_{0}=(2,0.1,0.2)$.

$\mathit{\alpha}$ | n | $\mathit{\tau}$ | $\mathit{\delta}$: | 0 | 0.25 | 0.5 | 0.75 | 1 | |
---|---|---|---|---|---|---|---|---|---|

${\widehat{T}}_{n,0}^{min}$ | 0 | 500 | 125 | 0.035 | 0.759 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 500 | 125 | 0.036 | 0.636 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 500 | 125 | 0.048 | 0.983 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0 | 1000 | 250 | 0.042 | 0.936 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 1000 | 250 | 0.049 | 0.874 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 1000 | 250 | 0.052 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 500 | 125 | 0.035 | 0.739 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 500 | 125 | 0.036 | 0.606 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 500 | 125 | 0.043 | 0.981 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 1000 | 250 | 0.042 | 0.938 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 1000 | 250 | 0.051 | 0.861 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 1000 | 250 | 0.056 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 500 | 125 | 0.035 | 0.716 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 500 | 125 | 0.038 | 0.57 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 500 | 125 | 0.041 | 0.981 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 1000 | 250 | 0.04 | 0.936 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 1000 | 250 | 0.048 | 0.842 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 1000 | 250 | 0.066 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 500 | 125 | 0.035 | 0.693 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 500 | 125 | 0.042 | 0.542 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 500 | 125 | 0.036 | 0.976 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 1000 | 250 | 0.034 | 0.93 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 1000 | 250 | 0.047 | 0.828 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 1000 | 250 | 0.064 | 1 | 1 | 1 | 1 |

**Table 6.**Empirical sizes and powers Case 2 for the Poisson INGARCH(1,1) model when no outliers exist with ${\theta}_{0}=(2,0.6,0.2)$.

$\mathit{\alpha}$ | n | $\mathit{\tau}$ | $\mathit{\delta}$: | 0 | $-1/5$ | $-1/3$ | $-3/7$ | $-1/2$ | |
---|---|---|---|---|---|---|---|---|---|

${\widehat{T}}_{n,0}^{min}$ | 0 | 500 | 125 | 0.05 | 0.999 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 500 | 125 | 0.06 | 0.969 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 500 | 125 | 0.049 | 0.844 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0 | 1000 | 250 | 0.052 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 1000 | 250 | 0.053 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 1000 | 250 | 0.059 | 0.988 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 500 | 125 | 0.047 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 500 | 125 | 0.058 | 0.971 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 500 | 125 | 0.046 | 0.85 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 1000 | 250 | 0.048 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 1000 | 250 | 0.041 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 1000 | 250 | 0.051 | 0.988 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 500 | 125 | 0.045 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 500 | 125 | 0.05 | 0.967 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 500 | 125 | 0.043 | 0.845 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 1000 | 250 | 0.052 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 1000 | 250 | 0.04 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 1000 | 250 | 0.054 | 0.991 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 500 | 125 | 0.04 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 500 | 125 | 0.043 | 0.962 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 500 | 125 | 0.048 | 0.863 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 1000 | 250 | 0.05 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 1000 | 250 | 0.052 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 1000 | 250 | 0.053 | 0.986 | 1 | 1 | 1 |

**Table 7.**Empirical sizes and powers Case 3 for the Poisson INGARCH(1,1) model when no outliers exist with ${\theta}_{0}=(2,0.3,0.3)$.

$\mathit{\alpha}$ | n | $\mathit{\tau}$ | $\mathit{\delta}$: | 0 | 0.25 | 0.5 | 0.75 | 1 | |
---|---|---|---|---|---|---|---|---|---|

${\widehat{T}}_{n,0}^{min}$ | 0 | 500 | 125 | 0.046 | 0.488 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 500 | 125 | 0.043 | 0.33 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 500 | 125 | 0.047 | 0.614 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0 | 1000 | 250 | 0.039 | 0.716 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 1000 | 250 | 0.041 | 0.554 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 1000 | 250 | 0.057 | 0.916 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 500 | 125 | 0.044 | 0.455 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 500 | 125 | 0.046 | 0.314 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 500 | 125 | 0.054 | 0.614 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 1000 | 250 | 0.046 | 0.706 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 1000 | 250 | 0.047 | 0.531 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 1000 | 250 | 0.062 | 0.914 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 500 | 125 | 0.046 | 0.434 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 500 | 125 | 0.05 | 0.295 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 500 | 125 | 0.048 | 0.601 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 1000 | 250 | 0.044 | 0.701 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 1000 | 250 | 0.042 | 0.505 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 1000 | 250 | 0.067 | 0.901 | 1 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 500 | 125 | 0.041 | 0.416 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 500 | 125 | 0.051 | 0.283 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 500 | 125 | 0.051 | 0.573 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 1000 | 250 | 0.04 | 0.684 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 1000 | 250 | 0.045 | 0.485 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 1000 | 250 | 0.055 | 0.869 | 1 | 1 | 1 |

**Table 8.**Empirical sizes and powers in Case 4 for the Poisson INGARCH(1,1) model when no outliers exist with ${\theta}_{0}=(1,0.4,0.4)$.

$\mathit{\alpha}$ | n | $\mathit{\tau}$ | $\mathit{\delta}$: | 0 | $-1/5$ | $-1/3$ | $-3/7$ | $-1/2$ | |
---|---|---|---|---|---|---|---|---|---|

${\widehat{T}}_{n,0}^{min}$ | 0 | 500 | 125 | 0.044 | 0.958 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 500 | 125 | 0.049 | 0.559 | 0.937 | 0.995 | 0.999 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 500 | 125 | 0.058 | 0.242 | 0.636 | 0.869 | 0.943 | |

${\widehat{T}}_{n,0}^{min}$ | 0 | 1000 | 250 | 0.038 | 0.998 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0 | 1000 | 250 | 0.039 | 0.887 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 1000 | 250 | 0.058 | 0.543 | 0.961 | 0.998 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 500 | 125 | 0.044 | 0.955 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 500 | 125 | 0.054 | 0.565 | 0.937 | 0.994 | 0.999 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 500 | 125 | 0.06 | 0.283 | 0.667 | 0.881 | 0.953 | |

${\widehat{T}}_{n,0}^{min}$ | 0.1 | 1000 | 250 | 0.042 | 0.999 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 1000 | 250 | 0.042 | 0.883 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 1000 | 250 | 0.053 | 0.542 | 0.96 | 0.998 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 500 | 125 | 0.047 | 0.95 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 500 | 125 | 0.056 | 0.574 | 0.941 | 0.992 | 0.999 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 500 | 125 | 0.056 | 0.291 | 0.669 | 0.88 | 0.951 | |

${\widehat{T}}_{n,0}^{min}$ | 0.2 | 1000 | 250 | 0.042 | 0.999 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 1000 | 250 | 0.039 | 0.873 | 0.997 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 1000 | 250 | 0.059 | 0.56 | 0.965 | 0.999 | 1 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 500 | 125 | 0.049 | 0.945 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 500 | 125 | 0.048 | 0.535 | 0.931 | 0.987 | 0.997 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 500 | 125 | 0.054 | 0.294 | 0.662 | 0.873 | 0.95 | |

${\widehat{T}}_{n,0}^{min}$ | 0.3 | 1000 | 250 | 0.043 | 0.999 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 1000 | 250 | 0.047 | 0.885 | 0.996 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 1000 | 250 | 0.064 | 0.569 | 0.967 | 0.998 | 1 |

**Table 9.**Empirical sizes and powers in Case 1 for the Poisson INGARCH(1,1) model when ${\theta}_{0}=(2,0.1,0.2)$, $p=0.1$ and $\lambda =10$.

$\mathit{\alpha}$ | n | $\mathit{\tau}$ | $\mathit{\delta}$: | 0 | 0.25 | 0.5 | 0.75 | 1 | |
---|---|---|---|---|---|---|---|---|---|

${\widehat{T}}_{n}^{min}$ | 0 | 500 | 250 | 0.065 | 0.058 | 0.145 | 0.8 | 0.958 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 500 | 250 | 0.048 | 0.047 | 0.066 | 0.512 | 0.997 | |

${\widehat{T}}_{n}^{min}$ | 0 | 1000 | 500 | 0.061 | 0.058 | 0.367 | 0.958 | 0.991 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 1000 | 500 | 0.053 | 0.053 | 0.095 | 0.978 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 500 | 250 | 0.042 | 0.039 | 0.23 | 0.891 | 0.962 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 500 | 250 | 0.035 | 0.037 | 0.122 | 0.897 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 1000 | 500 | 0.056 | 0.046 | 0.653 | 0.979 | 0.995 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 1000 | 500 | 0.053 | 0.054 | 0.963 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 500 | 250 | 0.036 | 0.032 | 0.162 | 0.842 | 0.951 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 500 | 250 | 0.035 | 0.036 | 0.111 | 0.804 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 1000 | 500 | 0.026 | 0.025 | 0.454 | 0.976 | 0.993 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 1000 | 500 | 0.023 | 0.023 | 0.514 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 500 | 250 | 0.032 | 0.034 | 0.201 | 0.855 | 0.95 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 500 | 250 | 0.032 | 0.032 | 0.114 | 0.771 | 0.979 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 1000 | 500 | 0.024 | 0.02 | 0.485 | 0.973 | 0.991 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 1000 | 500 | 0.021 | 0.021 | 0.284 | 0.999 | 1 |

**Table 10.**Empirical sizes and powers in Case 2 for the Poisson INGARCH(1,1) model when ${\theta}_{0}=(2,0.6,0.2)$, $p=0.1$ and $\lambda =30$.

$\mathit{\alpha}$ | n | $\mathit{\tau}$ | $\mathit{\delta}$: | 0 | 0.25 | 0.5 | 0.75 | 1 | |
---|---|---|---|---|---|---|---|---|---|

${\widehat{T}}_{n}^{min}$ | 0 | 500 | 250 | 0.08 | 0.975 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 500 | 250 | 0.065 | 0.11 | 0.194 | 0.329 | 0.456 | |

${\widehat{T}}_{n}^{min}$ | 0 | 1000 | 500 | 0.055 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 1000 | 500 | 0.05 | 0.203 | 0.594 | 0.795 | 0.902 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 500 | 250 | 0.057 | 0.935 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 500 | 250 | 0.062 | 0.169 | 0.666 | 0.927 | 0.993 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 1000 | 500 | 0.091 | 0.999 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 1000 | 500 | 0.052 | 0.615 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 500 | 250 | 0.054 | 0.875 | 0.998 | 0.999 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 500 | 250 | 0.043 | 0.069 | 0.309 | 0.663 | 0.784 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 1000 | 500 | 0.135 | 0.993 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 1000 | 500 | 0.046 | 0.569 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 500 | 250 | 0.063 | 0.896 | 0.998 | 0.999 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 500 | 250 | 0.046 | 0.086 | 0.455 | 0.763 | 0.853 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 1000 | 500 | 0.159 | 0.992 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 1000 | 500 | 0.047 | 0.675 | 0.999 | 1 | 1 |

**Table 11.**Empirical sizes and powers in Case 3 for the Poisson INGARCH(1,1) model when ${\theta}_{0}=(2,0.3,0.3)$, $p=0.1$ and $\lambda =30$.

$\mathit{\alpha}$ | n | $\mathit{\tau}$ | $\mathit{\delta}$: | 0 | 0.25 | 0.5 | 0.75 | 1 | |
---|---|---|---|---|---|---|---|---|---|

${\widehat{T}}_{n}^{min}$ | 0 | 500 | 250 | 0.074 | 0.118 | 0.069 | 0.127 | 0.885 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 500 | 250 | 0.062 | 0.064 | 0.06 | 0.068 | 0.777 | |

${\widehat{T}}_{n}^{min}$ | 0 | 1000 | 500 | 0.062 | 0.213 | 0.058 | 0.257 | 0.935 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 1000 | 500 | 0.049 | 0.05 | 0.049 | 0.057 | 0.992 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 500 | 250 | 0.036 | 0.033 | 0.041 | 0.516 | 0.914 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 500 | 250 | 0.037 | 0.037 | 0.04 | 0.268 | 0.961 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 1000 | 500 | 0.029 | 0.026 | 0.03 | 0.824 | 0.963 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 1000 | 500 | 0.023 | 0.023 | 0.025 | 0.859 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 500 | 250 | 0.038 | 0.034 | 0.038 | 0.487 | 0.865 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 500 | 250 | 0.04 | 0.042 | 0.046 | 0.321 | 0.612 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 1000 | 500 | 0.019 | 0.017 | 0.018 | 0.725 | 0.922 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 1000 | 500 | 0.015 | 0.015 | 0.015 | 0.244 | 0.616 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 500 | 250 | 0.035 | 0.032 | 0.036 | 0.351 | 0.661 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 500 | 250 | 0.039 | 0.039 | 0.042 | 0.13 | 0.211 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 1000 | 500 | 0.02 | 0.016 | 0.017 | 0.684 | 0.893 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 1000 | 500 | 0.012 | 0.012 | 0.012 | 0.085 | 0.161 |

**Table 12.**Empirical sizes and powers in Case 4 for the Poisson INGARCH(1,1) model when ${\theta}_{0}=(1,0.4,0.4)$, $p=0.1$ and $\lambda =30$.

$\mathit{\alpha}$ | n | $\mathit{\tau}$ | $\mathit{\delta}$: | 0 | 0.25 | 0.5 | 0.75 | 1 | |
---|---|---|---|---|---|---|---|---|---|

${\widehat{T}}_{n}^{min}$ | 0 | 500 | 250 | 0.05 | 0.796 | 0.958 | 0.989 | 0.996 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 500 | 250 | 0.048 | 0.078 | 0.118 | 0.173 | 0.219 | |

${\widehat{T}}_{n}^{min}$ | 0 | 1000 | 500 | 0.032 | 1 | 1 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0 | 1000 | 500 | 0.043 | 0.613 | 0.874 | 0.931 | 0.957 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 500 | 250 | 0.085 | 0.712 | 0.97 | 0.997 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 500 | 250 | 0.04 | 0.065 | 0.243 | 0.466 | 0.647 | |

${\widehat{T}}_{n}^{min}$ | 0.1 | 1000 | 500 | 0.242 | 0.978 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.1 | 1000 | 500 | 0.069 | 0.916 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 500 | 250 | 0.078 | 0.677 | 0.96 | 0.995 | 0.999 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 500 | 250 | 0.032 | 0.069 | 0.284 | 0.535 | 0.735 | |

${\widehat{T}}_{n}^{min}$ | 0.2 | 1000 | 500 | 0.229 | 0.965 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.2 | 1000 | 500 | 0.047 | 0.836 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 500 | 250 | 0.06 | 0.642 | 0.947 | 0.993 | 0.999 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 500 | 250 | 0.027 | 0.08 | 0.332 | 0.621 | 0.807 | |

${\widehat{T}}_{n}^{min}$ | 0.3 | 1000 | 500 | 0.201 | 0.962 | 0.999 | 1 | 1 | |

${\widehat{T}}_{n}^{cusum}$ | 0.3 | 1000 | 500 | 0.027 | 0.749 | 0.999 | 1 | 1 |

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**MDPI and ACS Style**

Lee, S.; Kim, D.
Monitoring Parameter Change for Time Series Models of Counts Based on Minimum Density Power Divergence Estimator. *Entropy* **2020**, *22*, 1304.
https://doi.org/10.3390/e22111304

**AMA Style**

Lee S, Kim D.
Monitoring Parameter Change for Time Series Models of Counts Based on Minimum Density Power Divergence Estimator. *Entropy*. 2020; 22(11):1304.
https://doi.org/10.3390/e22111304

**Chicago/Turabian Style**

Lee, Sangyeol, and Dongwon Kim.
2020. "Monitoring Parameter Change for Time Series Models of Counts Based on Minimum Density Power Divergence Estimator" *Entropy* 22, no. 11: 1304.
https://doi.org/10.3390/e22111304