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

: Several new asymmetric distributions have arisen naturally in the modeling extreme values are uncovered and elucidated. The present paper deals with the extreme value theorem (EVT) under exponential normalization. An estimate of the shape parameter of the asymmetric generalized value distributions that related to this new extension of the EVT is obtained. Moreover, we develop the mathematical modeling of the extreme values by using this new extension of the EVT. We analyze the extreme values by modeling the occurrence of the exceedances over high thresholds. The natural distributions of such exceedances, new four generalized Pareto families of asymmetric distributions under exponential normalization (GPDEs), are described and their properties revealed. There is an evident symmetry between the new obtained GPDEs and those generalized Pareto distributions arisen from EVT under linear and power normalization. Estimates for the extreme value index of the four GPDEs are obtained. In addition, simulation studies are conducted in order to illustrate and validate the theoretical results. Finally, a comparison study between the different extreme models is done throughout real data sets. under exponential normalization was proposed. New four generalized Pareto distributions related to the EVT under exponential normalization are obtained and their properties are elucidated. Estimates for the extreme value index of these distributions are suggested. The linear, power, and the suggested exponential models were applied, with a comparison, to several real data sets. The comparison between the three models revealed that the skewness and kurtosis of the data have an impact on the kind of the extreme model that describes the data.


Introduction
It has become necessary to study statistical models that have the ability to evaluate these rare phenomena to avoid its dangers due to the sudden rise of some natural harmful phenomena, such as earthquakes, Tsunami, air pollution, and other phenomena. In the last two decades, the EVT has emerged as one of the most significant statistical modeling disciplines for the applied sciences. The EVT can be applied to environmental studies, such as hydrology, pollution, rainfall, floods, wind gusts, and corrosion, in order to develop models for describing the distribution of extreme events. The distributional properties of the extreme and intermediate order statistics and exceedances over (below) high (low) thresholds are determined by the upper and lower tails of the underlying distribution. The most important challenges in any application of such extreme value models is the scarcity of extreme data, choosing the threshold, or beginning of the tail, and choosing the methods of estimating the unknown parameters. Much of the classical EVT is concerned substantially with distribution properties of the maximum X n:n = max{X 1 , X 2 , . . . , X n } of iid RVs X 1 , X 2 , . . . ., X n and all of the results obtained for maximum of course lead to anologous results for minimum through the obvious relation X 1:n = min{X 1 , X 2 , . . . , X n } = − max{−X 1 , −X 2 , . . . , −X n }. The core of the EVT is the extreme value distributions, which are well known in the literature (cf. [1]), and they are used as approximations to DFs of normalized partial maximum X n:n of iid RVs. A DF F is said to belong to the l-max domain of attraction of an extreme value distribution G under linear normalization, denoted by F ∈ D l (G), if there exist norming constants a n > 0 and b n ∈ R such that P X n:n − b n a n ≤ x = F n (a n x + b n ) where " w −→ n " stands for weak convergence, as n → ∞. It is well known that the asymptotic relation (1) yields only three possible types of non-degenerate limiting DFs, which are Frèchet, Weibull, and Gumbel DFs. Moreover, any non-degenerate DF G is an extreme value distrbution (i.e., it is a limit in (1)) if and only if it satisfies the stability relation G n (a n x + b n ) = G(x), x ∈ R, n ≥ 1, for every integer n, where a n > 0 and b n ∈ R are some suitable constants (cf. [1,2]). For this reason, these limits are called l-max-stable laws. On the other hand, these l-max stable laws may be written in the von Mises−Jenkinson format where µ and σ > 0 are the location and scale parameters, respectively, while γ ∈ R is a shape parameter that is known as the extreme value index (EVI), which is the central issue in empirical research dealing with extreme events. It is obviously found that the DF G(x; µ, σ, γ), which is known as the generalized extreme value distribution under linear normalization (GEVL), describes the Gumbel, Frèchet, and Weibull types with respect to the cases γ = 0 (interpreted as γ → 0), γ > 0 and γ < 0.
The GEVL provides a prevailing parametric approache for modeling extreme events, which is known as the block maxima (BM). Its application consists of partitioning a data set into blocks of equal length, and fitting the GEVL to the set of block maxima. An extension of the BM approach is the peak over threshold (POT) approach (see [1]), where we only consider the observations which lie above an appropriate threshold. The generalized Pareto distribution under linear normalization (GPDL) introduced by [3,4] is considered as a foremost pillar of the POT approach. The GPDL is the limit distribution of scaled excesses over high thresholds, which has the form 1 + log G(x; µ, σ, γ).
In order to widen the class of limit laws in EVT for solving more approximation problems, the authors of [5] extended the EVT under power normalization X n:n α n 1 βn S(X n:n ), where S(x) = sign(x) = −1, 0, 1 according to x < 0, x = 0, x > 0, respectively. Another reason for using the power normalization in EVT is concerning the possibility of getting a better rate of convergence in EVT (cf. [6]). Clearly, the power normalization is a strictly monotone continuous transformation. Therefore, this transformation does not give rise to any wastage of information that the data contains (e.g., the sufficiency property is preserved under one to one transformation). Nevertheless, we might lose some flexibility if we used such normalization. For example, under this normalization we can not change the sign of the data or get rid of zero. The DF F is said to belong to the p-max domain of attraction of a non-degenerate DF H under power normalization, denoted byF ∈ D p (H), if for some norming constants α n > 0 and β n > 0, The possible p-types of limiting DFs H in (3) are the p-max stable laws satisfying the stability relation H n (α n |x| β n S(x)) = H(x), x ∈ R, for every n ≥ 1, where α n > 0 and β n > 0 are some suitable sequences of constants. Here, two DFs, F and G, are of the same p-type if we can find α > 0 and β > 0, for which F(x) = G(α |x| β S(x)), for all x. Consequently, any non-degenerate DF H is a p-max stable, or equivalently H is a limit in (3), if and only if for every n ≥ 1 the two DFs H and H n are of the same p-type. In [7] the author has exemplified these types by the von Mises representation Each of these families is called generalized extreme value distribution under power normalization (GEVP). It is well known that the p-max-stable laws attract more distributions than the l-max-stable laws. This fact virtually means that the linear model may be unsuccessful for fitting an extreme data set; on the contrary, the power model succeed to fit it (see [1]). The authors of [8] applied the BM approach under power normalization using the GEVPs. Moreover, in a series of papers, refs. [9][10][11][12][13] developed the modeling of extreme values under power normalization by defining and using the generalized Pareto distributions under power normalization (GPDPs), 1 + log P i;γ (x; 1, 1), i = 1, 2, to a real extreme-value data (for more details regarding the power transformation, see [14][15][16][17][18]).
Once more, in order to widen the class of the limit laws in EVT, in [19] the authors extended the EVT under exponential normalization T n (x) = T u n ,v n (x) = exp{u n (| log |x||) v n S(log |x|)}S(x), u n , v n > 0. Under this transformation, we can say that the DFs F and G are of the same e-type if F(x) = G(exp{(u(| log |x||) v S(log |x|))}S(x)) = G(T u,v (x)), for some constants u > 0, v > 0. In this case, a non-degenerate DF Λ(.) is said to be an e-max-stable laws if there exists a DF F and norming constants u n > 0, v n > 0, such that If (4) is satisfied, then we can say that the DF F belongs to the e-max-domain of attraction of the non-degenerate DF Λ under e-normalization, denoted by F ∈ D e (Λ). The authors of [19], showed that the possible limiting DFs Λ in (4) are the e-max stable laws that satisfy the stability property that any non-degenerate DF Λ is an e-max stable, or equivalently Λ is a limit in (4), if and only if for every n ≥ 1 the two DFs Λ and Λ n are of the same e-type (for more details about the exponential transformation, see [20]).
In [19], the authors showed that the possible limit laws arisen from (4) attract more DFs than the p-max-stable laws. This fact virtually means that the linear and power models may fail to fit the given extreme data, while the exponential model succeeds. This fact gives us a sufficient motivation for developing the modeling of extreme values via the exponential model, denoted by the e-model. The aimed development is the first object of this paper and it will be achieved within two stages. The first stage is to infer the generalized extreme value distributions related to the EVT under exponential normalization. These asymmetric DFs enable us to apply the BM approach. The second stage is deriving the possible generalized Pareto families of asymmetric distributions relating to the EVT under exponential normalization. These families will pave the way to applying the POT approach. The second object of this paper is comparing between the EVT under linear, power, and exponential normalization via a real data sets of air pollution.
The rest of this paper is structured, as follows: In Section 2, we deduce the generalized extreme value distributions relating to the EVT under exponential normalization (GEGEs). In Section 3, which is devoted to the theoretical details, we first suggest an estimate for the EVI in each of the GEGEs. This estimate corresponds to a Dubey estimate in the GEVL model (3) and the GEVP models P iγ (x; a, b), i = 1, 2 (cf. [8]). Secondly, we derive the generalized Pareto distributions under exponential normalization (GPDEs). Finally, we propose estimators for the EVI in these GPDEs. Section 4 is devoted to a simulation study, which illustrates and corroborates the theoretical results. In Section 5, the EVT under linear, power, and exponential normalization is applied, with comparisons to several real data sets.

Preliminary Results
In [19], the authors derived the following a chain of equivalences between l-max-domains, p-max-domains and e-max-domains of attraction: stable DF, and F exp(ξ) and F − exp(−ξ) are p-max stable DFs.
In the above implications, F X denotes the DF of the RV X and "⇐⇒" stands for "if and only if". Moreover, ref. [19] used these implications in order to determine e-max-stable laws, wherein the first six e-max-stable DFs have right endpoint r(Λ) = sup{x : Λ(x) < 1} > 0 and the subsequent six e-max-stable DFs have r(Λ) ≤ 0.
We now totalize the limit laws (5) using the von Mises type representations. For any a, b > 0, the types (5) are totalized by the following general von Mises type forms: where γ is a given real number. When γ = 0, (6) is called generalized extreme value distribution under exponential normalization (GEVE), denoted by GEVE(γ, a, b). Clearly, the parametric models in (6) enable us to apply the BM approach under exponential normalization, where, in this case, we have to assume that the data in hand form a random sample drwan from an exact GEVE(γ, a, b).
In Section 4, we will compare the ML method and estimateγ e for estimating γ via the W −1 1;γ . Moreover, we will detect the value of q, which gives the best estimate for γ. It will be revealed that the estimate (9) is very poor for large values of γ (γ > 0.1). Regardless of the fact that this estimate is based on the BM approach, this approach also suffers some other problems, among them is only considering several maxima within several blocks and ignoring most the other data. In a spirit of the result of [3,4], we propose applying the POT approach based on the EVT under exponential normalization, where we deal with the right tail F(x) = 1 − F(x), for large x, i.e., we deal with top-order observations. In order to adapt this approach for the e-model we derive the GPDE. Our focus will be mainly on the case r(F) > 0 via Theorem 1. Clearly, the case r(F) > 0 covers most of the important practical applications of the EVT. However, the case r(F) < 0 will be briefly discussed in Theorem 3. In the next theorems and throughout the paper, we adopt the notations F [A] (x) = P(X ≤ x) | X > A) and " −→ n " to mean convergence as n → ∞.
where " w −→ u " means weak convergence, as e (−1) t+1 u ↑ r(F) and Proof. The proof of Part [a]: In view of the EVT, we obtain which, in view of the assumption r(F) > 1, implies that On the other hand, (11) cannot be true unless F(exp (u n (log x) v n )) −→ n 1, for all x for which W 1;γ (x; a, b) > 0. Thus, we can write By using the modified Khinchin's Theorem (cf. [1]), the relations (11) and (12) yield Now, let n be chosen, such that u n ≤ u ≤ u n+1 , where u is any real number such that e u < r(F) (note that by putting x = e in (11), we get e u n ↑ r(F)). Subsequently, (13) vn → 1. Thus, put α(u) ≡ v n and apply again the modified Khinchin's Theorem, (11) may be written in the form n(1 − F(exp (u(log x) α(u) ))) −→ n − log W 1;γ (x; a, b).
Therefore, by putting x = e in (14), we get By combining (14) and (15), we get, as n → ∞, or equivalently as e u ↑ r(F), which was to be proved. The proof of Part [b] is very similar to the proof of Part [a], with the exception of only of obvious changes. This completes the proof of Theorem 1.
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;γ (x; σ) = Q 1;γ ( x L ;σ), whereσ = σ c and c = 1 + γσ log L. Moreover, for every 0 < L < x < 1, we have Q On the other hand, we have . This completes the proof of Theorem 2.
Moreover, the limits Q 3;γ (x;b) and Q 4;γ (x; b) satisfy the peak over threshold stability property.
Proof. The proof is very similar to the proof of Theorems 1 and 2, with the exception of only of obvious changes.

Estimation of the EVI via GPDE Model
In this subsection, we derive estimates for the parametrs γ andσ in the GPDE Q 1;γ (x; σ). These estimates consort with the Pickand's estimates in the GEVL model (2) (cf. [4]). Let n be the sample size and m = m(n) be an integer much smaller than n. Let X i = X n−i+1:n be the ith largest observation in the sample, i = 1, 2, . . . , n. The values X i X 4m , i = 1, 2, . . . , 4m − 1, will be treated as though they were the descending order statistics from a sample of size 4m − 1 from the DF Q 1;γ (x; σ) for some 0 < σ < ∞ and −∞ < γ < ∞. Because, for any 0 ≤ y ≤ 1, we have Q −1 1,γ (y; σ) = exp exp 1 γσ ((1 − y) −γ − 1) , we get Q −1 . To estimate γ and σ, we replace the population quantiles Q −1 In the next section, we will consider the determination problem of m via a simulation study. Theoretically, the value m = m(n) should satisfy the two conditions lim n→∞ m = ∞ and lim n→∞ m n = 0 (cf. [4]).

Simulation Study
In Table 1, we compare the ML method and the Formula (9) for estimating the EVI γ via the first GEVE defined in (6). Additionally, from Table 1, we determine the value of q, which gives the best estimate for γ. In Table 1, we present estimates for each value of γ = 0.08, 0.09, 0.1, 0.11, 0.12 by applying the ML method and computing the estimateγ e that resulted from (9) for different quantiles q = 0.6, 0.68, 0.7, 0.75, 0.8. This procedure is repeated 1000 times to obtain the average estimates (for the given different values of q) for γ and their mean square errors (MSE's). Table 1 shows that the estimates (9) are poor when compared with the Ml estimates. Moreover, the precision of the estimatê γ e closely depends on the value of γ. It was revealed that when γ > 0.12, the estimates computed by (9) became very poor, for this reason in Table 1 we only considered the values γ ≤ 0.12. Table 1. Estimating the extreme value index (EVI) γ via W 1;γ (x; 2 × 10 −3 , 10) by using the maximum likelihood (ML) and (9)  In Table 2, for each value of γ = 0.08, 0.09, 0.1, 0.11, 0.12 we generate a random sample of size n = 20, 000 from Q 1;γ (x; 1). Moreover, we choose the threshold values k = 5000, 4500, . . . , 1000 (in the interval k ≤ n 4 ). In view of Theorem 2, the DF of the simulated data, which come after any threshold value k, has the same type of the DF Q 1;γ . Therefore, we can estimate the parameter γ by using the ML method for each of these threshold values. This procedure is repeated 1000 times to obtain the average estimates and their MSE's. Finally, we determine the value k, which gives the best estimate for the parameter γ by using the ML method. Table 3 is devoted to display the computed estimates of γ by using (16). In Table 3, the same procedure is applied with the exception that we choose m instead of k as m = 125, 250, . . . , 1250 (note that k = 4 m).
In both Tables 2 and 3, the asterisk in the superscript of a value means that this value is the best. Here, the "best" is according to the closeness to the actual value of γ and then according to the value of MSE in the case of equal closeness to true value of γ of two or more estimates. Moreover, Tables 2 and 3 show that the ML and (16) estimators for estimating the EVI γ via the GPDE Q 1;γ have high accuracy when comparing with the estimates of γ via W 1;γ .

Comparison Study between the Linear, Power and Exponential Models
Air pollution is a global problem, from which most countries across the world suffer (cf. [21][22][23]). In this section, we consider this problem via two data sets of pollutants, each of them consists of the maximum data of the three pollutants, nitric oxide (NO), nitrogen dioxide (NO 2 ), and particulate matter diameter less than 10 mm (PM10) (for some properties of these pollutants, see [1,22]). The first data set is taken from the site Lambeth-Streatham Green-Urban Background (denoted by LB6). The daily maximum of these pollutants was monitored and recorded every hour. Therefore, around 21,169 records are presented from 1 January 2014 to 31 July 2016. These data sets are publicly available from the following site: http://www.londonair.org.uk/london/asp/datadownload.asp. Table 4 shows the summary statistics for these maximum data sets. Table 5 is devoted to the estimate parameters of the generalized extreme value distributions for LB6.  We checked the fitting of any family by the Kolmogorov-Smirnov (K-S) test, where, in this test, we have four functions [H, P, KSSTAT, CV]. H is equal to 0 or 1, P is the p-value, KSSTAT is the maximum difference between the data and the fitting curve and CV is a critical value. Therefore,

•
we accept H 0 , if H = 0, KSSTAT ≤ CV and P > level of significant and • we reject H 0 , if H = 1, KSSTAT > CV and P ≤ level of significant. Table 6 gives the result of the Kolmogorov-Smirnov (K-S) test for fitting the three models G(x; µ, σ, γ), P 1;γ (x; a, b), and W 1;γ (x; a, b) to the maximum data sets from LB6. Table 7 illustrates the summary statistics for these maximum data set. Finally, the graphical representations of the data sets and the fitted distributions are given in Figures 1-9.          The second data set is taken from the site Greenwich-Eltham (denoted by GR4). The daily maxima of these pollutants are recorded every hour, so around 43,825 records are presented from 1 January 2014 to 31 December 2018. These data are publicly available from the following site: http: //www.londonair.org.uk/london/asp/datadownload.asp. Table 8 is devoted to the estimate parameters of the generalized extreme value distributions for GR4. Table 9 gives the result of the Kolmogorov-Smirnov (K-S) test for fitting the three models G(x; µ, σ, γ), P 1;γ (x; a, b) and W 1;γ (x; a, b) to the maximum data sets from GR4. Finally, the graphical representations of the data set and the fitted distributions are given in Figures 10-18.    Figure 11. The depiction of the data set of NO 2 and the fitted GEVL for GR4.        The result summary of this study is given below, where the more favorable model is chosen among accepted models and has a minimum KSSTAT value.
1. 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. 2. The linear model is only the favorable model to describe the pollutant NO 2 that is monitored by LB6. 3. 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. 4. Only the e-model is favorable to describe the pollutant NO, which is monitored by GR4. 5. None of the three models is favorable to describe the pollutant NO 2 that is monitored by GR4. 6. 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.
It is worth remarking that the study shows an interesting fact that the kurtosis of the data has an impact, to some extent, on the kind of the extreme model that describes the data, e.g., as the kurtosis increases, the e-model becomes more favorable. Moreover, the linear, power and exponential models become less favorable to fit the symmetric-platykurtic data set (for details about the description of data according to the skewness and kurtosis, see [24,25]), e.g., the case of pollutant NO 2 . Finally, a quick look at the Figures 1-18 reveals that the curves of the empirical DF and the tested family nearly coincide when we accept H 0 (e.g., 6,7,9,12,15,16,and 18), while, in the case of the rejection, the two curves diverge in some regions. This result endorses the results that are given in Tables 6 and 9.

Conclusions
In this paper, we developed the EVT under exponential normalization to model extreme values, which are arisen in different natural phenomena. An estimate of the shape parameter of the generalized value distributions that related to the EVT under exponential normalization was proposed. New four generalized Pareto distributions related to the EVT under exponential normalization are obtained and their properties are elucidated. Estimates for the extreme value index of these distributions are suggested. The linear, power, and the suggested exponential models were applied, with a comparison, to several real data sets. The comparison between the three models revealed that the skewness and kurtosis of the data have an impact on the kind of the extreme model that describes the data. Acknowledgments: This work was funded through the Plan Science UP Faculty of Science from the National Academy of Scientific Research and Technology of Egypt, ASRT is the 2nd affiliation of this research. The authors thank the anonymous referees for the valuable comments and suggestions, which significantly improved the quality of the paper.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: GPDE