# Two Features of the GINAR(1) Process and Their Impact on the Run-Length Performance of Geometric Control Charts

## Abstract

**:**

_{2}transition probability matrix; Kalmykov order; statistical process control; run length

## 1. Introduction

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1.**

**Theorem**

**2.**

## 2. Proving the Two Features of the GINAR(1) Process

**Proof**

**of**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**2.**

## 3. Practical Implications in Statistical Process Control

**Definition**

**3.**

#### 3.1. Significance of $\mathbf{P}\in {TP}_{2}$

**Corollary**

**1.**

**Corollary**

**2.**

#### 3.2. Other Comparisons of Run Lengths

**Corollary**

**3.**

**Lemma**

**1.**

**Corollary**

**4.**

#### 3.3. An Illustration

## 4. Concluding Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

c.d.f. | cumulative distribution function |

DHR | decreasing hazard rate |

DTMC | discrete-time Markov chain |

GGINAR(1) | generalized geometric first-order integer-valued autoregressive process |

GINAR(1) | geometric first-order integer-valued autoregressive process |

i.i.d. | independent and identically distributed |

IHR | increasing hazard rate |

INAR(1) | first-order integer-valued autoregressive process |

NGINAR(1) | new geometric first-order integer-valued autoregressive process |

p.f. | probability function |

${\mathrm{P}\mathrm{F}}_{2}$ | Pólya frequency of order 2 |

RL | run length |

r.v. | random variable |

${\mathrm{T}\mathrm{P}}_{2}$ | totally positive of order 2 |

TPM | transition probability matrix |

UCL | upper control limit |

ZMGINAR(1) | zero-modified geometric first-order integer-valued autoregressive process |

## Appendix A

#### Appendix A.1. Auxiliary Definitions and Lemmas

**Definition**

**A1.**

**Lemma**

**A1.**

**Lemma**

**A2.**

**Definition**

**A2.**

**Definition**

**A3.**

**Lemma**

**A3.**

**Definition**

**A4.**

**Lemma**

**A4.**

**Definition**

**A5.**

**Definition**

**A6.**

#### Appendix A.2. Run Length Related Performance Measures

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**Figure 1.**Hazard rate functions of $R{L}^{0}({p}_{0},{\rho}_{0})$ and $R{L}^{U}({p}_{0},{\rho}_{0})$.

**Figure 2.**Survival function of $R{L}^{0}(p,{\rho}_{0})$, for $p=0.9{p}_{0}$ and $p={p}_{0}$ (black and gray solid lines); overall ARL function, $E\left[R{L}^{{X}_{1}}(p,{\rho}_{0})\right]$, for $\sqrt{\rho}/(\sqrt{\rho}+1)<p\le {p}_{0}$.

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**MDPI and ACS Style**

Morais, M.C.
Two Features of the GINAR(1) Process and Their Impact on the Run-Length Performance of Geometric Control Charts. *Entropy* **2023**, *25*, 444.
https://doi.org/10.3390/e25030444

**AMA Style**

Morais MC.
Two Features of the GINAR(1) Process and Their Impact on the Run-Length Performance of Geometric Control Charts. *Entropy*. 2023; 25(3):444.
https://doi.org/10.3390/e25030444

**Chicago/Turabian Style**

Morais, Manuel Cabral.
2023. "Two Features of the GINAR(1) Process and Their Impact on the Run-Length Performance of Geometric Control Charts" *Entropy* 25, no. 3: 444.
https://doi.org/10.3390/e25030444