# Asymptotic Distribution of Certain Types of Entropy under the Multinomial Law

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Entropies and the Multinomial Distribution

- The Shannon entropy$${H}_{S}\left(\widehat{\mathit{p}}\right)=-\sum _{\ell =1}^{k}{\widehat{p}}_{\ell}log{\widehat{p}}_{\ell},$$
- The Tsallis entropy with index $q\in \mathbb{R}\backslash \left\{1\right\}$$${H}_{T}^{q}\left(\widehat{\mathit{p}}\right)=\sum _{\ell =1}^{k}\frac{{\widehat{p}}_{\ell}-{\widehat{p}}_{\ell}^{q}}{q-1},$$
- The Rényi entropy of order $q\in {\mathbb{R}}^{+}\backslash \left\{1\right\}$$${H}_{R}^{q}\left(\widehat{\mathit{p}}\right)=\frac{1}{1-q}log\sum _{\ell =1}^{k}{\widehat{p}}_{\ell}^{q},$$
- The Fisher information, also termed “Fisher Information Measure” in the literature, with renormalization coefficient ${F}_{0}=4$$${H}_{F}\left(\widehat{\mathit{p}}\right)={F}_{0}\sum _{\ell =1}^{k-1}{\left(\right)}^{\sqrt{{\widehat{p}}_{\ell +1}}}2$$Among other possibilities, we used Equation (2.7) from Ref. [12].

## 3. Asymptotic Distributions of Entropies

**Theorem**

**1.**

**Theorem**

**2.**

- For $\ell ,j=1,2,\dots ,k-2$ and $\ell \ne j-1,j,j+1$:$${\left({\mathsf{\Sigma}}_{\mathit{p}}^{\mathsf{\Delta}F}\right)}_{\ell j}=\left(\right)open="("\; close=")">\sqrt{{p}_{\ell +1}}-\sqrt{{p}_{\ell}}\left(\right)open="("\; close=")">\sqrt{{p}_{\ell +1}{p}_{j}}+\sqrt{{p}_{\ell}{p}_{j+1}}-\sqrt{{p}_{\ell}{p}_{j}}-\sqrt{{p}_{\ell +1}{p}_{j+1}}$$
- For $\ell =1,2,\dots ,k-2$:$$\begin{array}{c}{\left({\mathsf{\Sigma}}_{\mathit{p}}^{\mathsf{\Delta}F}\right)}_{\ell ,\ell -1}=\left(\right)open="("\; close=")">\sqrt{{p}_{\ell +1}}-\sqrt{{p}_{\ell}}\hfill \\ \left(\right)open="("\; close=")">\sqrt{{p}_{\ell}}-\sqrt{{p}_{\ell -1}}\hfill \end{array}\hfill & \left(\right)open="("\; close=")">\sqrt{{p}_{\ell +1}{p}_{\ell -1}}+{p}_{\ell}-1-\sqrt{{p}_{\ell}{p}_{\ell -1}}-\sqrt{{p}_{\ell +1}{p}_{\ell}}.\hfill $$
- For $\ell =1,2,\dots ,k-2$:$${\left({\mathsf{\Sigma}}_{\mathit{p}}^{\mathsf{\Delta}F}\right)}_{\ell \ell}={\left(\right)}^{\sqrt{{p}_{\ell +1}}}2.$$
- For $\ell =1,2,\dots ,k-2$:$${\left({\mathsf{\Sigma}}_{\mathit{p}}^{\mathsf{\Delta}F}\right)}_{\ell ,\ell +1}=\left(\right)open="("\; close=")">\sqrt{{p}_{\ell +1}}-\sqrt{{p}_{\ell}}\left(\right)open="("\; close=")">{p}_{\ell +1}-1+\sqrt{{p}_{\ell}{p}_{\ell +2}}-\sqrt{{p}_{\ell}{p}_{\ell +1}}-\sqrt{{p}_{\ell +1}{p}_{\ell +2}}$$
- For $j=1,2,\dots ,k-2$:$${\left({\mathsf{\Sigma}}_{\mathit{p}}^{\mathsf{\Delta}F}\right)}_{k-1,j}=(\sqrt{{p}_{k}}-\sqrt{{p}_{k-1}})(\sqrt{{p}_{j+1}}-\sqrt{{p}_{j}})\left(\right)open="("\; close=")">\frac{{p}_{k}{p}_{j+1}}{\sqrt{{p}_{j+1}}}-\frac{{p}_{k-1}{p}_{j}}{\sqrt{{p}_{j}}}$$
- Finally,$${\left({\mathsf{\Sigma}}_{\mathit{p}}^{\mathsf{\Delta}F}\right)}_{k-1,k-1}={(\sqrt{{p}_{k}}-\sqrt{{p}_{k-1}})}^{2}(1-{p}_{k-1}).$$

## 4. Analysis and Validation

- Linear: ${p}_{\ell}=2\ell /\left(k(k+1)\right)$, $1\le \ell \le k$.
- One-Almost-Zero: ${p}_{\ell}=1/k$ for $1\le \ell \le k-2$, ${p}_{k-1}={\u03f5}_{0}$, and ${p}_{k}=2/k-{\u03f5}_{0}$ with ${\u03f5}_{0}=2.220446\times {10}^{-16}$ (the smallest positive number for which, in our computer platform, $1+{\u03f5}_{0}>1$).
- Half-and-Half: ${p}_{\ell}=1/k+\u03f5/k$ for $1\le \ell \le k/2$, and ${p}_{\ell}=1/k-\u03f5/k$ for $k/2+1\le \ell \le k$, with $\u03f5\in \{0.1,0.3,0.5,0.8\}$.

## 5. Application

**1998 and 2008:**$3.43\times {10}^{-22}$,**1998 and 2018:**$2.01\times {10}^{-8}$,**2008 and 2018:**$1.06\times {10}^{-3}$.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

$\mathit{p}$ | vector of probabilities |

${\mathit{p}}^{\prime}$ | the transpose of $\mathit{p}$ |

$\widehat{\mathit{p}}$ | an estimator of $\mathit{p}$ |

$\mathit{N}$ | multivariate discrete random variable |

$\mathit{n}$ | a sample from $\mathit{N}$ |

${H}_{S}$ | Shannon entropy |

${H}_{T}^{q}$ | Tsallis entropy of order q |

${H}_{R}^{q}$ | Rényi entropy of order q |

${H}_{F}$ | Fisher information measure |

$\mathsf{\Sigma}$ | covariance matrix |

## Appendix A

#### Appendix A.1. Matrix Operations

#### Appendix A.2. Ordinal Patterns

#### Appendix A.3. Computational Information

## References

- Johnson, N.L.; Kotz, S.; Balakrishnan, N. Discrete Multivariate Distributions; Wiley-Interscience: Hoboken, NJ, USA, 1997. [Google Scholar]
- Modis, T. Links between entropy, complexity, and the technological singularity. Technol. Forecast. Soc. Chang.
**2022**, 176, 121457. [Google Scholar] [CrossRef] - Hutcheson, K. A test for comparing diversities based on the Shannon formula. J. Theor. Biol.
**1970**, 29, 151–154. [Google Scholar] [CrossRef] [PubMed] - Hutcheson, K.; Shenton, L.R. Some moments of an estimate of Shannon’s measure of information. Commun. Stat. Theory Methods
**1974**, 3, 89–94. [Google Scholar] [CrossRef] - Jacquet, P.; Szpankowski, W. Entropy computations via analytic depoissonization. IEEE Trans. Inf. Theory
**1999**, 45, 1072–1081. [Google Scholar] [CrossRef] - Cichoń, J.; Golębiewski, Z. On Bernoulli Sums and Bernstein Polynomials. In Proceedings of the 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms, Montreal, QC, Canada, 18–22 June 2012; pp. 179–190. [Google Scholar]
- Cook, G.W.; Kerridge, D.F.; Pryce, J.D. Estimations of Functions of a Binomial Parameter. Sankhyā Indian J. Stat. Ser. A
**1974**, 36, 443–448. [Google Scholar] - Chagas, E.T.C.; Frery, A.C.; Gambini, J.; Lucini, M.M.; Ramos, H.S.; Rey, A.A. Statistical Properties of the Entropy from Ordinal Patterns. Chaos Interdiscip. J. Nonlinear Sci.
**2022**, 32, 113118. [Google Scholar] [CrossRef] [PubMed] - Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys.
**1988**, 52, 479–487. [Google Scholar] [CrossRef] - Rényi, A. On Measures of Entropy and Information. In Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, USA, 20 June–30 July 1961; Volume 1, pp. 547–561. [Google Scholar]
- Frieden, B.R. Science from Fisher Information: A Unification; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Sánchez-Moreno, P.; Yáñez, R.J.; Dehesa, J.S. Discrete densities and Fisher information. In Proceedings of the 14th International Conference on Difference Equations and Applications, Istanbul, Turkey, 19–23 October 2009; pp. 291–298. [Google Scholar]
- Lehmann, E.L.; Casella, G. Theory of Point Estimation; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2006. [Google Scholar]
- Lehman, E.L.; Romano, J.P. Testing Statistical Hypothesis, 3rd ed.; Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar]
- Freedman, D.; Diaconis, P. On the histogram as a density estimator: L2 theory. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
**1981**, 57, 453–476. [Google Scholar] [CrossRef] - Agresti, A. An Introduction to Categorical Data Analysis; Wiley-Interscience: Hoboken, NJ, USA, 2007. [Google Scholar]
- Borges, J.B.; Ramos, H.S.; Loureiro, A.A.F. A Classification Strategy for Internet of Things Data Based on the Class Separability Analysis of Time Series Dynamics. ACM Trans. Internet Things
**2022**, 3, 1–30. [Google Scholar] [CrossRef] - Beranger, B.; Lin, H.; Sisson, S. New models for symbolic data analysis. Adv. Data Anal. Classif.
**2022**, 1–41. [Google Scholar] [CrossRef] - Borges, J.B.; Medeiros, J.P.S.; Barbosa, L.P.A.; Ramos, H.S.; Loureiro, A.A. IoT Botnet Detection based on Anomalies of Multiscale Time Series Dynamics. IEEE Trans. Knowl. Data Eng. 2022; Early Access. [Google Scholar] [CrossRef]
- Chagas, E.T.C.; Frery, A.C.; Rosso, O.A.; Ramos, H.S. Analysis and Classification of SAR Textures using Information Theory. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2021**, 14, 663–675. [Google Scholar] [CrossRef] - Bandt, C.; Pompe, B. Permutation entropy: A natural complexity measure for time series. Phys. Rev. Lett.
**2002**, 88, 174102. [Google Scholar] [CrossRef] [PubMed] - Zanin, M.; Zunino, L.; Rosso, O.A.; Papo, D. Permutation Entropy and Its Main Biomedical and Econophysics Applications: A Review. Entropy
**2012**, 14, 1553–1577. [Google Scholar] [CrossRef] - Sigaki, H.Y.D.; Perc, M.; Ribeiro, H.V. History of art paintings through the lens of entropy and complexity. Proc. Natl. Acad. Sci. USA
**2018**, 115, E8585–E8594. [Google Scholar] [CrossRef] [PubMed] - Chagas, E.T.C.; Queiroz-Oliveira, M.; Rosso, O.A.; Ramos, H.S.; Freitas, C.G.S.; Frery, A.C. White Noise Test from Ordinal Patterns in the Entropy-Complexity Plane. Int. Stat. Rev.
**2022**, 90, 374–396. [Google Scholar] [CrossRef]

**Figure 1.**Linear, One-Almost-Zero, and Half-and-Half probability functions for $k=6$ and $\u03f5=0.3$.

**Figure 2.**Empirical densities and normal QQ-plots of the Fisher information in situations that fail to pass the normality test at 1%.

**Figure 3.**Examples of cases where the null hypothesis of the Kolmogorov–Smirnov test is rejected. The histograms are computed with samples of size 300 using the Freedman–Diaconis rule [15], and the green lines are the asymptotic probability density functions. (

**a**) Type: ${H}_{S}$, Model: OAZ, $k=120$, $n={10}^{4}k$, $p-\mathrm{val}=0.00202$; (

**b**) Type: ${H}_{F}$, Model: OAZ, $k=120$, $n={10}^{4}k$, $p-\mathrm{val}=0.00013$; (

**c**) Type: ${H}_{R}^{1/3}$, Model: Linear, $k=24$, $n={10}^{4}k$, $p-\mathrm{val}\approx 0$; (

**d**) Type: ${H}_{T}^{1/2}$, Model: HaH, $\u03f5=0.8$, $k=6$, $n={10}^{4}k$, $p-\mathrm{val}=0.04297$.

**Table 1.**Situations for which the p-values of the Anderson–Darling test for the normality of samples of size 300 are less than $0.01$ (“HF” stands for the Fisher information; “HaH” and “OAZ” are the Half-And-Half and One-Almost-Zero models).

Type | Model | $\mathit{\u03f5}$ | k | n | p-Value |
---|---|---|---|---|---|

${H}_{F}$ | HaH | 0.8 | 6 | 600 | 0.0030 |

${H}_{F}$ | HaH | 0.1 | 6 | 600 | 0.0000 |

${H}_{F}$ | HaH | 0.1 | 24 | 2400 | 0.0000 |

${H}_{F}$ | HaH | 0.8 | 24 | 2400 | 0.0064 |

${H}_{F}$ | HaH | 0.1 | 6 | 6000 | 0.0000 |

${H}_{F}$ | HaH | 0.3 | 6 | 600 | 0.0000 |

${H}_{F}$ | HaH | 0.1 | 120 | 120,000 | 0.0089 |

${H}_{F}$ | HaH | 0.1 | 24 | 24000 | 0.0004 |

${H}_{F}$ | Linear | 0 | 6 | 600 | 0.0000 |

${H}_{F}$ | Linear | 0 | 24 | 2400 | 0.0000 |

${H}_{R}^{1/3}$ | HaH | 0.1 | 6 | 600 | 0.0000 |

${H}_{R}^{1/3}$ | HaH | 0.1 | 24 | 2400 | 0.0001 |

${H}_{R}^{1/3}$ | OAZ | 0 | 24 | 2400 | 0.0000 |

${H}_{R}^{2/3}$ | HaH | 0.1 | 6 | 600 | 0.0000 |

${H}_{R}^{2/3}$ | HaH | 0.1 | 24 | 2400 | 0.0001 |

${H}_{R}^{2/3}$ | OAZ | 0 | 24 | 2400 | 0.0000 |

${H}_{S}$ | HaH | 0.1 | 6 | 600 | 0.0001 |

${H}_{S}$ | HaH | 0.1 | 24 | 2400 | 0.0002 |

${H}_{S}$ | OAZ | 0 | 24 | 2400 | 0.0000 |

${H}_{T}^{1/2}$ | HaH | 0.1 | 6 | 600 | 0.0000 |

${H}_{T}^{1/2}$ | HaH | 0.1 | 24 | 2400 | 0.0001 |

${H}_{T}^{1/2}$ | OAZ | 0 | 24 | 2400 | 0.0000 |

${H}_{T}^{3/2}$ | HaH | 0.1 | 6 | 600 | 0.0001 |

${H}_{T}^{3/2}$ | HaH | 0.1 | 24 | 2400 | 0.0003 |

${H}_{T}^{3/2}$ | OAZ | 0 | 24 | 2400 | 0.0000 |

**Table 2.**Situations for which the p-values of the Kolmogorov–Smirnov test of samples of size 50 are larger than or equal to $0.05$ (“HaH” and “OAZ” are the Half-And-Half and One-Almost-Zero models).

Type | Model | $\mathit{\u03f5}$ | k | n | Type | Model | $\mathit{\u03f5}$ | k | n | |
---|---|---|---|---|---|---|---|---|---|---|

${H}_{S}$ | HaH | $0.1$ | $6,24$ | ${10}^{3}k,{10}^{4}k$ | ${H}_{T}^{3/2}$ | HaH | 0.1 | $6,24$ | ${10}^{3}k,{10}^{4}k$ | |

$120,720$ | ${10}^{4}k$ | $120,720$ | ${10}^{4}k$ | |||||||

$0.3$ | $6,120$ | for all | $0.3$ | $6,120$ | for all | |||||

24 | ${10}^{3}k$ | 24 | ${10}^{3}k$ | |||||||

720 | ${10}^{3}k,{10}^{4}k$ | 720 | ${10}^{3}k,{10}^{4}k$ | |||||||

$0.5$ | $6,24$ | for all | $0.5$ | $6,24$ | for all | |||||

$120,720$ | ${10}^{3}k,{10}^{4}k$ | $120,720$ | ${10}^{3}k,{10}^{4}k$ | |||||||

$0.8$ | 6 | ${10}^{2}k,{10}^{3}k$ | $0.8$ | 6 | ${10}^{2}k,{10}^{3}k$ | |||||

$24,720$ | for all | $24,720$ | for all | |||||||

120 | ${10}^{3}k,{10}^{4}k$ | 120 | ${10}^{3}k,{10}^{4}k$ | |||||||

Linear | 0 | $6,24,120$ | for all | Linear | 0 | $6,24,120$ | for all | |||

720 | ${10}^{3}k,{10}^{4}k$ | 720 | ${10}^{2}k,{10}^{4}k$ | |||||||

OAZ | 0 | $6,24$ | for all | OAZ | 0 | $6,24$ | for all | |||

${H}_{F}$ | HaH | $0.1$ | 6 | ${10}^{3}k$ | ${H}_{R}^{1/3}$ | HaH | $0.3$ | 6 | ${10}^{2}k,{10}^{3}k$ | |

$0.3$ | $6,24$ | for all | 0.5 | 6 | for all | |||||

$0.5$ | 6 | ${10}^{2}k,{10}^{3}k$ | $0.8$ | 6 | ${10}^{2}k,{10}^{3}k$ | |||||

24 | ${10}^{3}k,{10}^{4}k$ | Linear | 0 | 6 | for all | |||||

120 | ${10}^{4}k$ | OAZ | 0 | 6 | for all | |||||

$0.8$ | 6 | ${10}^{2}k,{10}^{3}k$ | 24 | ${10}^{2}k$ | ||||||

24 | for all | ${H}_{R}^{2/3}$ | HaH | $0.3$ | 6 | for all | ||||

Linear | 0 | 6 | ${10}^{3}k,{10}^{4}k$ | $0.5$ | 6 | for all | ||||

24 | ${10}^{4}k$ | $0.8$ | 6 | ${10}^{2}k,{10}^{3}k$ | ||||||

OAZ | 0 | $6,24$ | for all | Linear | 0 | 6 | for all | |||

${H}_{T}^{1/2}$ | HaH | $0.1$ | $6,24$ | ${10}^{3}k,{10}^{4}k$ | OAZ | 0 | 6 | ${10}^{2}k,{10}^{3}k$ | ||

$120,720$ | ${10}^{4}k$ | |||||||||

$0.3$ | $6,120$ | for all | ||||||||

24 | ${10}^{3}k$ | |||||||||

720 | ${10}^{3}k,{10}^{4}k$ | |||||||||

$0.5$ | $6,24$ | for all | ||||||||

$120,720$ | ${10}^{3}k,{10}^{4}k$ | |||||||||

$0.8$ | 6 | ${10}^{2}k,{10}^{3}k$ | ||||||||

24 | for all | |||||||||

$120,720$ | ${10}^{3}k,{10}^{4}k$ | |||||||||

Linear | 0 | for all | for all | |||||||

OAZ | 0 | $6,24$ | for all | |||||||

120 | ${10}^{3}k,{10}^{4}k$ | |||||||||

720 | ${10}^{4}k$ |

Year | 1998 | 2008 | 2018 |
---|---|---|---|

STRONGLY AGREE | 148 | 285 | 186 |

AGREE | 429 | 602 | 496 |

NOT AGREE/DISAGREE | 278 | 210 | 229 |

DISAGREE | 275 | 196 | 181 |

STRONG DISAGREE | 72 | 30 | 38 |

Total | 1202 | 1323 | 1130 |

Mean | Variance | |||||
---|---|---|---|---|---|---|

1998 | 2008 | 2018 | 1998 | 2008 | 2018 | |

${H}_{S}$ | 0.914 | 0.839 | 0.863 | 0.0000724 | 0.0001081 | 0.0001175 |

${H}_{R}^{1/3}$ | 0.967 | 0.933 | 0.947 | 0.0001172 | 0.0002375 | 0.0002122 |

${H}_{R}^{2/3}$ | 0.939 | 0.881 | 0.902 | 0.0000317 | 0.0000500 | 0.0000512 |

${H}_{T}^{1/2}$ | 0.932 | 0.868 | 0.892 | 0.0000500 | 0.0000871 | 0.0000842 |

${H}_{T}^{3/2}$ | 0.919 | 0.849 | 0.870 | 0.0000622 | 0.0001089 | 0.0001183 |

${H}_{F}$ | 0.516 | 0.702 | 0.642 | 0.0008028 | 0.0011355 | 0.0011917 |

1998–2008 | 1998–2018 | 2008–2018 | |
---|---|---|---|

${H}_{S}$ | 0.0000000 | 0.0002606 | 0.1029 |

${H}_{R}^{1/3}$ | 0.0705681 | 0.2520262 | 0.5316 |

${H}_{R}^{2/3}$ | 0.0000000 | 0.0000478 | 0.0404 |

${H}_{T}^{1/2}$ | 0.0000000 | 0.0004471 | 0.0690 |

${H}_{T}^{3/2}$ | 0.0000001 | 0.0002316 | 0.1672 |

${H}_{F}$ | 0.0000219 | 0.0045777 | 0.2118 |

Mean | Variance | |
---|---|---|

${H}_{S}$ | 0.955 | 0.0000678 |

${H}_{R}^{1/3}$ | 0.985 | 0.0000171 |

${H}_{R}^{2/3}$ | 0.970 | 0.0000083 |

${H}_{T}^{1/2}$ | 0.971 | 0.0000281 |

${H}_{T}^{3/2}$ | 0.949 | 0.0000896 |

${H}_{F}$ | 0.192 | 0.0000678 |

1998–2008 | 1998–2018 | |
---|---|---|

${H}_{S}$ | 0.00000 | 0.0000 |

${H}_{R}^{1/3}$ | 0.00115 | 0.0107 |

${H}_{R}^{2/3}$ | 0.00000 | 0.0000 |

${H}_{T}^{1/2}$ | 0.00000 | 0.0000 |

${H}_{T}^{3/2}$ | 0.00000 | 0.0000 |

${H}_{F}$ | 0.00000 | 0.0000 |

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**MDPI and ACS Style**

Rey, A.A.; Frery, A.C.; Lucini, M.; Gambini, J.; Chagas, E.T.C.; Ramos, H.S.
Asymptotic Distribution of Certain Types of Entropy under the Multinomial Law. *Entropy* **2023**, *25*, 734.
https://doi.org/10.3390/e25050734

**AMA Style**

Rey AA, Frery AC, Lucini M, Gambini J, Chagas ETC, Ramos HS.
Asymptotic Distribution of Certain Types of Entropy under the Multinomial Law. *Entropy*. 2023; 25(5):734.
https://doi.org/10.3390/e25050734

**Chicago/Turabian Style**

Rey, Andrea A., Alejandro C. Frery, Magdalena Lucini, Juliana Gambini, Eduarda T. C. Chagas, and Heitor S. Ramos.
2023. "Asymptotic Distribution of Certain Types of Entropy under the Multinomial Law" *Entropy* 25, no. 5: 734.
https://doi.org/10.3390/e25050734