# Modeling of Extreme Values via Exponential Normalization Compared with Linear and Power Normalization

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Results

- ${F}_{X}\in {D}_{l}\left({F}_{\xi}\right)\u27fa{F}_{exp\left(X\right)}\in {D}_{p}\left({F}_{exp\left(\xi \right)}\right)\u27fa{F}_{-exp(-X)}\in {D}_{p}\left({F}_{-exp(-\xi )}\right),$ where ${F}_{\xi}$ is an l-max stable DF, and ${F}_{exp\left(\xi \right)}$ and ${F}_{-exp(-\xi )}$ are p-max stable DFs.
- ${F}_{X}\in {D}_{p}\left({F}_{\xi}\right)\u27fa{F}_{exp\left(X\right)}\in {D}_{e}\left({F}_{exp\left(\xi \right)}\right)\u27fa{F}_{-exp(-X)}\in {D}_{e}\left({F}_{-exp(-\xi )}\right),$ where ${F}_{\xi}$ is an p-max stable DF, and ${F}_{exp\left(\xi \right)}$ and ${F}_{-exp(-\xi )}$ are e-max stable laws.

## 3. BM Approach and GPDEs

**Theorem**

**1.**

- a.
- ${Q}_{1}\left(x\right)\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}{Q}_{1;\gamma}(x;\overline{b})\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1+log{W}_{1;\gamma}(x;1,\overline{b}),$$\overline{b}=\frac{b}{c}$and$c=1+\gamma loga,$if$r\left(F\right)>1$;
- b.
- ${Q}_{2}\left(x\right)\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}{Q}_{2;\gamma}(x;\underline{b})\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1+log{W}_{2;\gamma}(x;1,\underline{b}),$$\underline{b}=\frac{b}{\underline{c}}$and$\underline{c}=1-\gamma loga,$if$0<r\left(F\right)\le 1.$

**Proof.**

**Theorem**

**2**

**the peak over threshold stability property**). The left truncated GPDE again yields a GPDE. This means that, for every $1<L<x,$ we have ${Q}_{1;\gamma}^{\left[L\right]}(x;\sigma )={Q}_{1;\gamma}(\frac{x}{L};\overline{\sigma}),$ where $\phantom{\rule{3.33333pt}{0ex}}\overline{\sigma}=\frac{\sigma}{c}$ and $c=1+\gamma \sigma logL.$ Moreover, for every $0<L<x<1,$ we have ${Q}_{2;\gamma}^{\left[L\right]}(x;\sigma )={Q}_{2;\gamma}(\frac{x}{L};\overline{\sigma}),$ where $\phantom{\rule{3.33333pt}{0ex}}\overline{\sigma}=\frac{\sigma}{\overline{c}}$ and $\overline{c}=1-\gamma \sigma log(-L).$

**Proof.**

**Theorem**

**3.**

- c.
- ${Q}_{3;\gamma}(x;\overline{b})=1+log{W}_{3;\gamma}(x;1,\overline{b}),$if$-1<r\left(F\right)<0$;
- d.
- ${Q}_{4;\gamma}(x;\underline{b})=1+log{W}_{4;\gamma}(x;1,\underline{b}),$if$r\left(F\right)<-1.$

**Proof.**

#### Estimation of the EVI via GPDE Model

## 4. Simulation Study

## 5. Comparison Study between the Linear, Power and Exponential Models

- we accept ${H}_{0},$ if $H=0,$$KSSTAT\phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}CV$ and $P>$ level of significant and
- we reject ${H}_{0},$ if $H=1,$$KSSTAT\phantom{\rule{3.33333pt}{0ex}}>\phantom{\rule{3.33333pt}{0ex}}CV$ and $P\le $ level of significant.

- Only the power and exponential models are favorable in describing the pollutant $NO$ that is monitored by LB6. The power model is the best one.
- The linear model is only the favorable model to describe the pollutant $N{O}_{2}$ that is monitored by LB6.
- All of the models are favorable to describe the pollutant $PM10$ that is monitored by LB6. The best model is the linear model followed by the power model.
- Only the e-model is favorable to describe the pollutant $NO$, which is monitored by GR4.
- None of the three models is favorable to describe the pollutant $N{O}_{2}$ that is monitored by GR4.
- All of the models are favorable to describe the pollutant $PM10$ that is monitored by GR4. The best model is the e-model followed by the linear model.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GPDE | Generalized Pareto distributions under exponential normalization |

EVT | Extreme value theory |

iid | Independent identically distributed |

RVs | Random variables |

DF | Distribution function |

EVI | Extreme value index |

GEVL | Generalized extreme value distribution under linear normalization |

BM | Block maxima |

POT | Peak over threshold |

GPDL | Generalized Pareto distribution under linear normalization |

GEVP | Generalized extreme value distribution under power normalization |

GPDP | Generalized Pareto distributions under power normalization |

GEVE | Generalized extreme value distribution under exponential normalization |

GPDE | Generalized Pareto distribution under exponential normalization |

ML | Maximum likelihood |

K-S | Kolmogorov-Smirnov |

CV | Critical value |

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**Table 1.**Estimating the extreme value index (EVI) $\gamma $ via ${W}_{1;\gamma}(x;2\times {10}^{-3},10)$ by using the maximum likelihood (ML) and (9) estimators.

$\mathit{\gamma}$ | ML Estimate | The Estimate (9) | |||||
---|---|---|---|---|---|---|---|

$\mathit{q}=\mathbf{0.6}$ | $\mathit{q}=\mathbf{0.68}$ | $\mathit{q}=\mathbf{0.7}$ | $\mathit{q}=\mathbf{0.75}$ | $\mathit{q}=\mathbf{0.8}$ | |||

0.08 | $\widehat{{\gamma}_{e}}$ | 0.0800 | −0.0267 | 0.0496 | 0.0520 | 0.0411 | 0.0819 |

MSE | $5.22\times {10}^{-4}$ | 1.1394 | 0.0926 | 0.0786 | 0.1513 | $3.54\times {10}^{-4}$ | |

0.09 | $\widehat{{\gamma}_{e}}$ | 0.0896 | −0.0718 | 0.0062 | 0.0256 | 0.0871 | 0.1349 |

MSE | $4.98\times {10}^{-4}$ | 2.6172 | 0.7018 | 0.4141 | $8.15\times {10}^{-4}$ | 0.2015 | |

0.1 | $\widehat{{\gamma}_{e}}$ | 0.1037 | 0.0341 | 0.1062 | 0.0899 | 0.1182 | 0.1700 |

MSE | $5.06\times {10}^{-4}$ | 0.4344 | 0.0039 | 0.0101 | 0.0330 | 0.4899 | |

0.11 | $\widehat{{\gamma}_{e}}$ | 0.1095 | 0.0327 | 0.1096 | 0.1094 | 0.1435 | 0.1992 |

MSE | $6.41\times {10}^{-4}$ | 0.5977 | $1.73\times {10}^{-5}$ | $3.44\times {10}^{-5}$ | 0.1125 | 0.7959 | |

0.12 | $\widehat{{\gamma}_{e}}$ | 0.1215 | 0.0775 | 0.1517 | 0.1476 | 0.1872 | 0.2714 |

MSE | $4.57\times {10}^{-4}$ | 0.1809 | 0.1003 | 0.0762 | 0.4511 | 2.2913 |

**Table 2.**Estimating the EVI $\gamma $ in the generalized Pareto distributions under exponential normalization (GPDE) $\phantom{\rule{3.33333pt}{0ex}}{Q}_{1;\gamma},$ by using the ML method.

k | 5000 | 4500 | 4000 | 3500 | 3000 | 2500 | 2000 | 1000 |
---|---|---|---|---|---|---|---|---|

The GPDE $\phantom{\rule{3.33333pt}{0ex}}{Q}_{1;\gamma},$ with $\gamma =0.08$ | ||||||||

$\widehat{\gamma}$ | 0.0486 | 0.0522 | 0.0618 | 0.0665 | 0.0679 | 0.0738 | 0.0756 | $0.0761{\phantom{\rule{3.33333pt}{0ex}}}^{\star}$ |

MSE | 0.0011 | 0.0009 | 0.0006 | 0.0003 | 0.0003 | 0.0003 | 0.0004 | 0.0004 |

The GPDE $\phantom{\rule{3.33333pt}{0ex}}{Q}_{1;\gamma},$ with $\gamma =0.09$ | ||||||||

$\widehat{\gamma}$ | 0.0488 | 0.0475 | 0.0576 | 0.0653 | 0.0724 | 0.0729 | 0.0727 | $0.0832{\phantom{\rule{3.33333pt}{0ex}}}^{\star}$ |

MSE | 0.0018 | 0.0019 | 0.0012 | 0.0008 | 0.0005 | 0.0005 | 0.0006 | 0.0003 |

The GPDE $\phantom{\rule{3.33333pt}{0ex}}{Q}_{1;\gamma},$ with $\gamma =0.1$ | ||||||||

$\widehat{\gamma}$ | 0.0700 | 0.0702 | 0.0724 | 0.0720 | 0.0794 | 0.0835 | $0.0924{\phantom{\rule{3.33333pt}{0ex}}}^{\star}$ | 0.0924 |

MSE | 0.0011 | 0.0012 | 0.0010 | 0.0010 | 0.0006 | 0.0005 | 0.0002 | 0.0003 |

The GPDE $\phantom{\rule{3.33333pt}{0ex}}{Q}_{1;\gamma},$ with $\gamma =0.11$ | ||||||||

$\widehat{\gamma}$ | 0.0785 | 0.0854 | 0.0884 | 0.0920 | 0.0945 | 0.1006 | $0.1021{\phantom{\rule{3.33333pt}{0ex}}}^{\star}$ | 0.1189 |

MSE | 0.0011 | 0.0008 | 0.0007 | 0.0004 | 0.0004 | 0.0002 | 0.0002 | 0.0004 |

The GPDE $\phantom{\rule{3.33333pt}{0ex}}{Q}_{1;\gamma},$ with $\gamma =0.12$ | ||||||||

$\widehat{\gamma}$ | 0.0894 | 0.0949 | 0.0983 | 0.1038 | 0.1113 | 0.1133 | 0.1143 | $0.1173{\phantom{\rule{3.33333pt}{0ex}}}^{\star}$ |

MSE | 0.0011 | 0.0007 | 0.0006 | 0.0004 | 0.0004 | 0.0011 | 0.0007 | 0.0005 |

**Table 3.**Estimating the EVI $\gamma $ in the GPDE $\phantom{\rule{3.33333pt}{0ex}}{Q}_{1;\gamma},$ by using the estimator (16).

m | 125 | 250 | 375 | 500 | 625 | 750 | 1000 | 1250 |
---|---|---|---|---|---|---|---|---|

The GPDE $\phantom{\rule{3.33333pt}{0ex}}{Q}_{1;\gamma},$ with $\gamma =0.08$ | ||||||||

$\widehat{\gamma}$ | 0.0794 | 0.0858 | 0.0883 | 0.0818 | 0.0784 | 0.0819 | $0.0801{\phantom{\rule{3.33333pt}{0ex}}}^{\star}$ | 0.0808 |

MSE | 0.0047 | 0.0026 | 0.0021 | 0.0013 | 0.0012 | 0.0009 | 0.0007 | 0.0006 |

The GPDE $\phantom{\rule{3.33333pt}{0ex}}{Q}_{1;\gamma},$ with $\gamma =0.09$ | ||||||||

$\widehat{\gamma}$ | 0.0924 | 0.0865 | 0.0845 | 0.0895 | 0.0911 | $0.0905{\phantom{\rule{3.33333pt}{0ex}}}^{\star}$ | 0.0913 | 0.0917 |

MSE | 0.0048 | 0.0024 | 0.0019 | 0.0016 | 0.0012 | 0.0010 | 0.0008 | 0.0007 |

The GPDE $\phantom{\rule{3.33333pt}{0ex}}{Q}_{1;\gamma},$ with $\gamma =0.1$ | ||||||||

$\widehat{\gamma}$ | 0.1041 | 0.0967 | 0.0957 | 0.0992 | $0.1002{\phantom{\rule{3.33333pt}{0ex}}}^{\star}$ | 0.0995 | 0.0987 | 0.1011 |

MSE | 0.0061 | 0.0027 | 0.0018 | 0.0013 | 0.0015 | 0.0010 | 0.0009 | 0.0007 |

The GPDE $\phantom{\rule{3.33333pt}{0ex}}{Q}_{1;\gamma},$ with $\gamma =0.11$ | ||||||||

$\widehat{\gamma}$ | $0.1099{\phantom{\rule{3.33333pt}{0ex}}}^{\star}$ | 0.1166 | 0.1132 | 0.1103 | 0.1103 | 0.1137 | 0.1136 | 0.1136 |

MSE | 0.0047 | 0.0028 | 0.0013 | 0.0014 | 0.0009 | 0.0010 | 0.0009 | 0.0008 |

The GPDE $\phantom{\rule{3.33333pt}{0ex}}{Q}_{1;\gamma},$ with $\gamma =0.12$ | ||||||||

$\widehat{\gamma}$ | 0.1232 | $0.1198{\phantom{\rule{3.33333pt}{0ex}}}^{\star}$ | 0.1209 | 0.1143 | 0.1168 | 0.1170 | 0.1167 | 0.1206 |

MSE | 0.0058 | 0.0031 | 0.0017 | 0.0017 | 0.0011 | 0.0010 | 0.0009 | 0.0008 |

n | Minimum | Maximum | Median | Mean | SD | Skewness | Kurtosis | |
---|---|---|---|---|---|---|---|---|

$NO$ | 1601 | 3 | 466.10 | 29.05 | 45.94 | 50.84904 | 3.209211 | 14.44866 |

$N{O}_{2}$ | 838 | 5.5 | 171.1 | 53.75 | 55.21205 | 24.54528 | 0.768683 | 1.277585 |

$PM10$ | 369 | 11 | 353 | 33 | 41.19241 | 33.964229 | 5.128280 | 34.681366 |

Estimate parameters of the GEVL $G(x;\mu ,\sigma ,\gamma )$ via BM approach for LB6 | |||

Pollutant | $\widehat{\gamma}$ | $\widehat{\sigma}$ | $\widehat{\mu}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$ |

$NO$ | 0.5794 | 16.6815 | 21.5258 |

$N{O}_{2}$ | −0.0719 | 20.9257 | 44.5482 |

$PM10$ | 0.3048 | 11.7537 | 28.6365 |

Parameter estimations of the GEVP ${P}_{1;\gamma}(x;a,b)$ via BM approach for LB6 | |||

Pollutant | $\widehat{\gamma}$ | $\widehat{b}$ | $\widehat{a}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$ |

$NO$ | −0.2034 | 1.1971 | 0.0251 |

$N{O}_{2}$ | −0.3729 | 1.8877 | $8.49\times {10}^{-4}$ |

$PM10$ | −0.0585 | 2.3911 | $3.35\times {10}^{-4}$ |

Estimate parameters of the GEVE ${W}_{1;\gamma}(x;a,b)$ via BM approach for LB6 | |||

Pollutant | $\widehat{\gamma}$ | $\widehat{b}$ | $\widehat{a}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$ |

$NO$ | −0.3939 | 3.5125 | 0.0200 |

$N{O}_{2}$ | −0.4589 | 6.7161 | $1.46\times {10}^{-4}$ |

$PM10$ | −0.1567 | 7.9032 | $7.24\times {10}^{-5}$ |

Fitting data of LB6 by the GEVL $G(x;\widehat{\mu},\widehat{\sigma},\widehat{\gamma})$ | |||

Pollutant | P | $KSSTAT$ | Decision |

$NO$ | 0.0414 | 0.0347 | reject ${H}_{0}$ |

$N{O}_{2}$ | 0.4084 | 0.0305 | accept ${H}_{0}$ |

$PM10$ | 0.9506 | 0.0266 | accept ${H}_{0}$ |

Fitting data of LB6 by the GEVP ${P}_{1;\widehat{\gamma}}(x;\widehat{a},\widehat{b})$ | |||

Pollutant | P | $KSSTAT$ | Decision |

$NO$ | 0.3141 | 0.0189 | accept ${H}_{0}$ |

$N{O}_{2}$ | 0.0199 | 0.0481 | reject ${H}_{0}$ |

$PM10$ | 0.5204 | 0.0293 | accept ${H}_{0}$ |

Fitting data of LB6 by the GEVE ${W}_{1;\widehat{\gamma}}(x;\widehat{a},\widehat{b})$ | |||

Pollutant | P | $KSSTAT$ | Decision |

$NO$ | 0.1271 | 0.0253 | accept ${H}_{0}$ |

$N{O}_{2}$ | 0.0011 | 0.0635 | reject ${H}_{0}$ |

$PM10$ | 0.4131 | 0.0342 | accept ${H}_{0}$ |

n | Minimum | Maximum | Median | Mean | SD | Skewness | Kurtosis | |
---|---|---|---|---|---|---|---|---|

$NO$ | 1706 | 1.2 | 380.60 | 8.6 | 23.265 | 39.721 | 4.043 | 21.225 |

$N{O}_{2}$ | 1706 | 4.3 | 120.6 | 34.6 | 36.82 | 17.8640 | 0.6906 | 0.5265 |

$PM10$ | 1471 | 7.4 | 325.29 | 25.2 | 30.529 | 18.527 | 4.7380 | 53.16945 |

Estimate parameters of the GEVL $G(x;\mu ,\sigma ,\gamma )$ via BM approach for GR4 | |||

Pollutant | $\widehat{\gamma}$ | $\widehat{\sigma}$ | $\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\widehat{\mu}$ |

$NO$ | 1.0065 | 5.6233 | 6.1006 |

$N{O}_{2}$ | −0.0537 | 14.9302 | 28.8916 |

$PM10$ | 0.3007 | 8.5229 | 22.2351 |

Parameter estimations of the GEVP ${P}_{1;\gamma}(x;a,b)$ via BM approach for GR4 | |||

Pollutant | $\widehat{\gamma}$ | $\widehat{b}$ | $\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\widehat{a}$ |

$NO$ | −0.0013 | 1.0922 | 0.1334 |

$N{O}_{2}$ | −0.3830 | 1.7425 | 0.0031 |

$PM10$ | −0.0739 | 2.5652 | $3.51\times {10}^{-4}$ |

Estimate parameters of the GEVE ${W}_{1;\gamma}(x;a,b)$ via BM approach for GR4 | |||

Pollutant | $\widehat{\gamma}$ | $\widehat{b}$ | $\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\widehat{a}$ |

$NO$ | −0.4550 | 1.7980 | 0.3384 |

$N{O}_{2}$ | −0.4795 | 5.4804 | 0.0015 |

$PM10$ | −0.1741 | 7.8801 | $1.352\times {10}^{-4}$ |

Fitting data of GR4 by the GEVL $G(x;\widehat{\mu},\widehat{\sigma},\widehat{\gamma})$ | |||

Pollutant | P | $KSSTAT$ | Decision |

$NO$ | 0.0086 | 0.0398 | reject ${H}_{0}$ |

$N{O}_{2}$ | 0.0184 | 0.0370 | reject ${H}_{0}$ |

$PM10$ | 0.2332 | 0.0269 | accept ${H}_{0}$ |

Fitting data of GR4 by the GEVP ${P}_{1;\widehat{\gamma}}(x;\widehat{a},\widehat{b})$ | |||

Pollutant | P | $KSSTAT$ | Decision |

$NO$ | 0.0061 | 0.0386 | reject ${H}_{0}$ |

$N{O}_{2}$ | 0.0098 | 0.0367 | reject ${H}_{0}$ |

$PM10$ | 0.1073 | 0.0274 | accept ${H}_{0}$ |

Fitting data of GR4 by the GEVE ${W}_{1;\widehat{\gamma}}(x;\widehat{a},\widehat{b})$ | |||

Pollutant | P | $KSSTAT$ | Decision |

$NO$ | 0.0817 | 0.0270 | accept ${H}_{0}$ |

$N{O}_{2}$ | $4.496\times {10}^{-4}$ | 0.0474 | reject ${H}_{0}$ |

$PM10$ | 0.1752 | 0.0242 | accept ${H}_{0}$ |

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**MDPI and ACS Style**

Barakat, H.M.; Khaled, O.M.; Rakha, N.K.
Modeling of Extreme Values via Exponential Normalization Compared with Linear and Power Normalization. *Symmetry* **2020**, *12*, 1876.
https://doi.org/10.3390/sym12111876

**AMA Style**

Barakat HM, Khaled OM, Rakha NK.
Modeling of Extreme Values via Exponential Normalization Compared with Linear and Power Normalization. *Symmetry*. 2020; 12(11):1876.
https://doi.org/10.3390/sym12111876

**Chicago/Turabian Style**

Barakat, Haroon Mohamed, Osama Mohareb Khaled, and Nourhan Khalil Rakha.
2020. "Modeling of Extreme Values via Exponential Normalization Compared with Linear and Power Normalization" *Symmetry* 12, no. 11: 1876.
https://doi.org/10.3390/sym12111876