# Prior Sensitivity Analysis in a Semi-Parametric Integer-Valued Time Series Model

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## Abstract

**:**

## 1. Introduction

## 2. A Generalization of the INAR(p) Model

## 3. Pitman–Yor Process

## 4. PY-INAR(p) Model

## 5. Prior Sensitivity

## 6. Simulated Data

## 7. Earthquake Data

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Formation of the elbows for $\sigma =0.5$ (left) and $\sigma =0.75$ (right). The red dotted lines indicate the chosen values of ${\lambda}_{\mathrm{max}}$.

**Figure 2.**Posterior distributions of the number of clusters K for the simulated time series with $\sigma =0$ and ${k}_{0}=4,10,16,30$. The red dotted lines indicate the value of ${k}_{0}$.

**Figure 3.**Posterior distributions of the number of clusters K for the simulated time series with $\sigma =0.25$ and ${k}_{0}=4,10,16,30$. The red dotted lines indicate the value of ${k}_{0}$.

**Figure 4.**Posterior distributions of the number of clusters K for the simulated time series with $\sigma =0.5$ and ${k}_{0}=4,10,16,30$. The red dotted lines indicate the value of ${k}_{0}$.

**Figure 5.**Posterior distributions of the number of clusters K for the simulated time series with $\sigma =0.75$ and ${k}_{0}=4,10,16,30$. The red dotted lines indicate the value of ${k}_{0}$.

True | |||
---|---|---|---|

Predicted | 1 | 2 | 3 |

1 | 297 | 32 | 0 |

2 | 11 | 217 | 42 |

3 | 0 | 84 | 316 |

**Table 2.**Out-of-sample MAE’s for the INAR(p) and the PY-INAR(p) models, with orders $p=1,2,\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}3$. The last column shows the relative variations of the MAE’s for the PY-INAR(p) models with respect to the corresponding MAE’s for the INAR(p) models.

INAR | PY-INAR | ${\mathbf{\Delta}}_{\mathbf{PY}\text{-}\mathbf{INAR}}$ | |
---|---|---|---|

$p=1$ | 3.861 | 3.583 | −0.072 |

$p=2$ | 3.583 | 3.417 | −0.046 |

$p=3$ | 3.972 | 3.305 | −0.202 |

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**MDPI and ACS Style**

Graziadei, H.; Lijoi, A.; Lopes, H.F.; Marques F., P.C.; Prünster, I. Prior Sensitivity Analysis in a Semi-Parametric Integer-Valued Time Series Model. *Entropy* **2020**, *22*, 69.
https://doi.org/10.3390/e22010069

**AMA Style**

Graziadei H, Lijoi A, Lopes HF, Marques F. PC, Prünster I. Prior Sensitivity Analysis in a Semi-Parametric Integer-Valued Time Series Model. *Entropy*. 2020; 22(1):69.
https://doi.org/10.3390/e22010069

**Chicago/Turabian Style**

Graziadei, Helton, Antonio Lijoi, Hedibert F. Lopes, Paulo C. Marques F., and Igor Prünster. 2020. "Prior Sensitivity Analysis in a Semi-Parametric Integer-Valued Time Series Model" *Entropy* 22, no. 1: 69.
https://doi.org/10.3390/e22010069