# f-Gintropy: An Entropic Distance Ranking Based on the Gini Index

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## Abstract

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## 1. Introduction

## 2. Gintropy, f-Gintropy and Gintropic Divergence

- It is never negative, ${\sigma}_{f}\ge 0$.
- It is zero for $x=0$ and $x=\infty $ only, or at $\overline{C}=0$ and $\overline{C}=1$.
- It is maximal at the $x={x}_{m}$ value, fulfilling $f\left({x}_{m}\right)=\u2329f\u232a$.
- It has a single maximum only, either as a function of x or $\overline{C}$ or ${\overline{F}}_{f}$.
- The f-gintropy, ${\sigma}_{f}$, like the entropy, is everywhere concave on $\overline{C}$ and ${\overline{F}}_{f}$.

## 3. Example: Gintropic Distance Ranking of Income Data

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Probability density function for the distributions of the normalized income for some countries and geographical regions. The income for each region is normalized to the respective average value. Please note that we use log-log scales.

**Figure 2.**Normalized gintropy $\widehat{\sigma}\left(\overline{C}\right)$ calculated from the income distribution data in comparison with the one expected for the natural distribution (27).

**Figure 3.**Gintropy for different regions fitted with the one derived for the Tsallis–Pareto distribution ${\widehat{\sigma}}_{q}\left(\overline{C}\right)$ (30). In the figures we illustrate the best best fit and also give the best-fit parameter, q. There is no separate panel for Hungary since the experimental gintropy for Hungary and Cluj are very close, as already seen in Figure 2.

**Figure 4.**Normalized f-gintropy ${\widehat{\sigma}}_{f}\left(\overline{C}\right)$ with $f\left(x\right)={x}^{2}$ calculated from the income distribution data. Note the more evident separation of the studied geographical regions.

**Table 1.**Gintropic distances for income distributions calculated by using the (28) gintropic Kullback–Leibler divergences. We also indicate this distance relative to the natural distribution.

$\times {10}^{-3}$ | Natural | Australia | USA | Cluj | Hungary | Japan |

Natural | 0 | 0 | $0.4$ | $2.9$ | $2.9$ | 17 |

Australia | 0 | 0 | $0.4$ | $2.9$ | $2.9$ | 17 |

USA | $0.4$ | $0.4$ | 0 | $1.1$ | $1.1$ | 13 |

Cluj | 3 | 3 | $1.1$ | 0 | 0 | $6.4$ |

Hungary | 3 | 3 | $1.1$ | 0 | 0 | $6.4$ |

Japan | 19 | 19 | 14 | $6.8$ | $6.8$ | 0 |

**Table 2.**f-gintropic distances for income distributions calculated using the (28) generalized Kullback–Leibler divergences for $f\left(x\right)={x}^{2}$.

$\times {10}^{-3}$ | Australia | USA | Cluj | Hungary | Japan |

Australia | 0 | $1.8$ | 36 | 19 | 62 |

USA | $1.7$ | 0 | 27 | 13 | 50 |

Cluj | 48 | 39 | 0 | $3.8$ | 4 |

Hungary | 22 | 16 | $3.5$ | 0 | 14 |

Japan | 86 | 74 | $4.3$ | 17 | 0 |

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**MDPI and ACS Style**

Biró, T.S.; Telcs, A.; Józsa, M.; Néda, Z.
f-Gintropy: An Entropic Distance Ranking Based on the Gini Index. *Entropy* **2022**, *24*, 407.
https://doi.org/10.3390/e24030407

**AMA Style**

Biró TS, Telcs A, Józsa M, Néda Z.
f-Gintropy: An Entropic Distance Ranking Based on the Gini Index. *Entropy*. 2022; 24(3):407.
https://doi.org/10.3390/e24030407

**Chicago/Turabian Style**

Biró, Tamás Sándor, András Telcs, Máté Józsa, and Zoltán Néda.
2022. "f-Gintropy: An Entropic Distance Ranking Based on the Gini Index" *Entropy* 24, no. 3: 407.
https://doi.org/10.3390/e24030407