New Approaches to Extremal Index Estimation

M. CRISTINA MIRANDA1,2,3, MANUELA SOUTO DE MIRANDA2,M. IVETTE GOMES3

1ISCA,2CIDMA

University of Aveiro

Campus Santiago, 3810-193 Aveiro

3CEAUL

University od Lisbon

PORTUGAL

Abstract: The extremal index is a parameter associated with the extreme value distributions of dependent

stationary sequences. Under certain local dependence conditions, exceedances above a specified threshold tend

to occur in isolated clusters. The reciprocal of the extremal index can be interpreted as the limiting size of these

clusters. Accurately estimating the size of such clusters is crucial for analyzing real data and can significantly

influence decision making processes that impact population well being. The paper presents a recent method for

the estimation of the extremal index which starts by the estimation of the parameter itself and, only then, to use

that estimate in the cluster mean size estimation. The procedure starts with the estimation of a specific proportion

by the corresponding relative frequency. Thus, it is very simple, intuitive, it has good statistical properties, and it

does not depend on the method used for the mean cluster estimation. The interpretation of the extremal index as

a proportion is known, but it has not been used directly as an estimation method. In recent years, various authors

have proposed different estimators for the extremal index. This paper applies some of the latest estimation methods

for the extremal index to real data and analyses their performance using training and test samples. The results

are compared with other well known estimators, for which Rpackages are available. The results show a better

performance of the Proportion estimator, followed by the Gaps estimator, when compared to the other considered

index estimators.

Key-Words: Extremal Value Theory, Extremal index, Robust estimation, Proportion estimator, Negative

Binomial, Stationary sequences, Clusters of exceedances

Received: March 15, 2024. Revised: August 6, 2024. Accepted: September 7, 2024. Published: October 22, 2024.

1 Introduction

Extreme value theory (EVT) addresses the study of

extreme value distributions. The occurrence of rare

events in real life, such as floods, strong winds,

extreme temperatures, epidemic peaks, or stock

market crashes, often has catastrophic consequences

for populations. This reality spurred the

development of EVT around the 1950´s, particularly

in the Netherlands, due to its geographic

characteristics and the need to survive in a territory

below the sea level. In the 1980’s, the classical

independent and identically distributed (IID)

approach was extended to stationary sequences, [1].

Under suitable local dependence conditions, it is also

possible to obtain the limit distribution for the

maximum of stationary sequences. In fact, this limit

distribution presents the same shape parameter but

the location and scale are affected by an extra

parameter, the Extremal Index (EI), hereby denoted

by θ.

In an IID scenario, extreme values typically

manifest as rare and isolated occurrences. However,

in dependent sequences, the dynamics change

significantly. The structure of dependence leads to

clusters of extreme values rather than isolated

events. As the strength of dependence increases, so

does the size of these clusters. The EI can be

interpreted as the reciprocal of the mean size of

clusters of exceedances above a specified threshold.

Therefore, EI estimation is crucial as it provides a

measure of the duration of periods with consistently

high or low values in the sequence under study.

A different approach is to consider the sequence as a

compound Poisson Process. Extreme events occur as

rare phenomena as in a Poisson Process, [2]. Under

suitable local dependence conditions, when the

sequence is stationary but not I.I.D., the process of

exceedances converges to a compound Poisson

Process with intensity related to the EI, [3]. When

analysed in the dynamical systems framework, the

EI may be seen as a measure about the dynamics of

the underlying systems, [4].

Since the first proposed estimators, in the 90’s,

estimating the EI is an issue that has been addressed

with increased attention. Some of the existing

proposals are based on the size of the clusters and

how these clusters are identified. This is the case

with the runs estimator and the blocks estimator

which require additional parameters in turn, [5], [6],

[7]. The intervals estimator, [8], is one of the most

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popular in the literature and is based on the

interexceedances times of overpassing a high

threshold. The intervals estimator motivated the

work of Süveges, who proposed a maximum

likelihood estimator, the so called Kgaps estimator,

[9], which also depends of a K parameter. A detailed

synthesis of the EI estimation options may be found

at the book by [10], or in the recent paper of [11].

Variations of the initial blocks and runs estimators

include the disjoint blocks estimator and the sliding

blocks estimator, [12]. Based on the maxima

method, [13], proposed a semiparametric approach

producing more efficient estimates. Relevant

asymptotic results and a moments EI estimator is

presented in the paper from [14]. To reduce bias,

[15], applies the Jackknife methodology. A key

problem with EI estimation is the choice of the

threshold to be considered, originating the process of

exceedances. Some of the existent estimators are too

sensitive to the choice of that parameter, [16].

Recent research addresses to this problem, proposing

a new way to select the threshold in a context of non

parametric estimation of the extremal index of

stochastic processes, [17]. When academics are

called to apply their results to real data, it is frequent

not to encounter the conditions assumed to validate

the results. In that sense, there is the recognized need

for robust procedures which are not so vulnerable in

the absence of those conditions. The estimator

proposed in [18], is a robust proposal to estimate the

EI.

More recently, an interesting proposal uses artificial

censoring of interexceedances times with some

evidence of stability improvement and less

sensitivity to the choice of the parameter evolved, D,

in this case, and to the choice of the high threshold,

[19]. In the present paper, two different proposals

are considered to estimate the EI: a robust estimator,

based on the Negative Binomial (NB) model, and a

proportion estimator, also based in the

interexceedances times.

Following the introduction, Section 2 refers to

the basic principles of EVT in the case of IID

samples and for specific dependent sequences. In

Section 3 and 4, two recently proposed EI estimators

are presented. Section 5 presents other known

estimators to be considered in the data analysis.

Section 6 contains the results of applying both

robust and non robust estimators to a real data set.

The paper concludes in Section 7 with comments

and conclusions.

2 EVT classic theory

2.1 IID case

The theory of extreme values is highly developed

when dealing with IID samples. In this case, there

exists the well known theorem in EVT that allows us

to obtain the distribution of the limit of the

maximum when appropriately normalized, and

whenever such a limit exists. This theorem provides

a convenient way to understand the behaviour of

extreme values in large samples drawn

independently and identically distributed,

(Y1, Y2, ..., Yn), with some unknown common

cumulative distribution function (CDF), F(y). The

Fisher, Tippett, Gnedenko extremal types theorem

determines the limiting CDF of the maximum as a

member of the general extreme value (EV)

distribution family. If Yn:n=max{Y1, Y2, ..., Yn}

and if there are constants an>0and bn∈Rsuch

that the standardized maximum Z= (Yn:n−bn)/an

converges for some non degenerate distribution,

then, such a limit CDF is of the type of

EVξ(y) = 









exp(−(1 + ξy)−1/ξ),1 + ξy > 0,

if ξ= 0

exp(−exp(−y)), y ∈R,

if ξ= 0.

(1)

Several advancements have expanded the

applicability of EVT to dependent sequences, just as

sketched in Section 2.2. This is especially relevant

in the analysis of time series, where dependencies

among observations are typical. In such cases, recent

developments in EVT have adapted to account for

these dependencies, offering insights into the

distribution of extremes and enhancing our

understanding of their behaviour over.

2.2 EVT for dependent structures

When we have an IID sequence, extreme values tend

to be rare events, usually far apart from each other in

time. But when some dependence structure is

present, this affects their behaviour, originating

clusters of high values close to each other.

Dependent observations tend to exhibit similar

patterns. When an extreme value is observed, the

next observation is likely to be close to the previous

one. The presence of autocorrelation induces

proximity between observations, leading to clusters

of high values rather than isolated extremes.

In some sequences, those clusters appear

separated in time. So in limit, the clusters

ocuurrence times tend to be independent. This paper

addresses to the problem of estimating a parameter

that measures the interdependence among sequences

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of random variables. For sequences that verify

certain stationary and mixing conditions, the classic

EVT remains applicable with few changes, [1].

Definition 2.1. D(un)The dependent sequence

{Xn}n≥1,with marginal CDF F, verifies the

mixing condition D(un)of [1], if, for each

i1, ..., ip, j1, ..., jqsuch that

1≤i1< ... < ip< j1< ... < jq≤n, j1−ip≥l,

and the notation

Fk1...kr(un) := P(Xk1≤un, ..., Xkr≤un),



Fi1...ipj1...jq(un)−Fi1...ip(un)Fj1...jq(un)

< αn,l

where αn,ln→0as n→ ∞ for some sequence ln

with ln=o(n).

If the Dcondition in Definition (2.1) is assumed

for a stationary sequence {Xn}n≥1, then the limit

distribution of the maximum will be one of the three

types, Fréchet (ξ > 0), Gumbel (ξ= 0) or Weibull

(ξ < 0), depending on the ξvalue, in (1).

Other local dependence conditions are usually

necessary in the framework of the EI estimation so

that the asymptotic distribution of the sample

maximum can be calculated. These may be

formulated in terms of the Dkcondition, [20], [21],

which may be stated as follows:

Definition 2.2. Dk(un)Suppose that a sequence

{Xn}n≥1verifies the DLeadbetter condition, given

above in Definition (2.1). The Dk(un)condition is

said to hold if for some {kn}n≥1such that

kn→ ∞, knαn,ln→0, knln/n→0

as n→ ∞,we have

nP (X1> un, M1,k ≤un< Mk,rn)→

n→∞ 0

with {rn= [n/kn]}n≥1,[x]denoting the integer part

of xand Mi,j =max {Xi, ..., Xj},(j > i).

Understanding the joint distribution of k

consecutive terms enables us to derive the limiting

distribution of the maximum for these sequences.

2.3 The extremal index

Consider a strictly stationary process {Xn}n≥1with

CDF F, with finite or infinite right endpoint. The

process is said to have an extremal index θ∈[0,1]

if, for each τ > 0, there is a sequence {un}n≥1such

that, as n→ ∞:

a. n¯

F(un)→τand

b. P[Mn≤un]→exp(−τθ),

where ¯

F:= 1 −Fand Mn:= M0,n denotes the

maximum of nconsecutive observed variables,

defined by Mk,l := max{Xi:i=k+ 1, . . . , l}.

Let {Yn}n≥1be an IID sequence, and suppose

{Xn}n≥1is a stationary sequence with the same

marginal F. Then, if there exist real constants anand

bn, such that

FYn:n(any+bn)→H(y),

where H(y)needs to be of the type of the EV

distribution, in (1), then, for the stationary sequence

{Xn}n≥1, with an extreme value index, θ,

FXn:n(any+bn)→Hθ(y).

So, the EI value has an effect of clustering and

shrinking the extreme values when we compare the

two types of samples. The EI is a parameter that may

be interpreted as a measure of the dimension of the

clusters of exceedances over some high threshold. In

fact, it may be interpreted as the limit of the

reciprocal of the cluster mean size, [22]. For

independent sequences or asymptotically

independent sequences, θ= 1. This means that

exceedances do not form clusters as it happens with

dependent samples. The value of θrelates to the

dimension of the clusters: the closest to zero, the

greater the clusters size will be (in average). Figure

1 illustrates the difference between sequences for

which θ= 1 and an associated dependent sample

with a small value of θ.

Figure 1: Circles represent a sample with θ= 1;

triangles denote a dependent sample, with θ= 0.2.

3 Robust estimation based on the NB

A recent proposed method also based in the limit

mean size estimation of the clusters uses the

uniparametric NB model and robust estimation of its

parameter. The model was suggested by [23], upon

the NB2 reparametrization of the model, [24], which

turns easier robust estimation in the framework of

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the generalized linear regression. The process was

investigated aiming to improve robustness of the

runs estimator in the sense that we are interested in

modelling the number of exceedances that occur

before a non exceedance and to bypass difficulties

caused by the presence of too many zeros and

distributional overdispersion, [18].

Assume a GLM with independent response

variables Yi,i= 1, . . . , n, such that Yifollow a NB

distribution. With the NB2 reparametrization, the

probability function NB can be expressed in terms of

µiand σ, where σis a shape parameter and µiis the

conditional mean, being given by

f(yi;µi, σ) =

Γ(yi+σ−1)

Γ(yi+ 1)Γ(σ−1)1

1 + σµiσ−1σµi

1 + σµiyi

(2)

where yi= 0,1, ..., n,Γ(.)is the Gamma

function and σ > 0is the overdispersion parameter

of the Yidistribution. Under that parametrization

(and denoting by xian observation of a potential k

dimensional regressor), the conditional variance is:

µi=E[Yi|x

xi], V (µi) = V(Yi|x

xi) = µi+σµ2

i.(3)

Expression (3) shows that when σ→0,V(µi)

converges to the variance of a Poisson distribution.

Thus, we will work on the GLM setting

Yi∼NB(µi, σ), i = 1, . . . , n,

g(µi) = g(E[Yi|x

xi]) =

iβ

βxT

i+ϵi=β0+β1x1+· · · +βkxk+ϵi,(4)

with the link function g(.) = log(.)and where

β∈Rk+1 is the unknown parameter. Robust

estimators for β

βand σare based on maximum

likelihood estimators (MLE), substituting estimating

equations by bounded functions. They are already

disposable from Rpackage robNB for the model

parameter that we are interested in. Particularly, we

only need to estimate the intercept, thus without

regressors. So, from (4), the cluster mean size

estimate will be ˆµi=exp ˆ

β0and the EI estimate will

be its reciprocal, i.e.,

θNB = 1/ˆµi=exp{− ˆ

β0}.

4 Proportion estimator

The EI proportion estimator appeared recently, [25],

on the basis of a completely different estimation

approach. While other methods are developed

aiming to find the best estimator for the limiting

clusters mean size, and then to take the reciprocal of

that estimate as the EI estimate, the proportion

estimator goes directly to the estimation of the EI as

a distributional parameter by its own. The procedure

emerged taking a particular attention to the

distribution of the interexceedances times, like it is

presented in [8]. Those authors proved that the

distribution of the interexceedances times is the

mixture:

(1 −θ)ϵ0+θfD(θ),(5)

where ϵ0is the degenerate probability distribution at

0and fD(θ)represents an exponential distribution

with mean θ−1. Thus, as referred in that paper, the

EI is the proportion of strictly positive

interexceedances times.

More formally, let {Xn}n≥1be a strictly stationary

sequence of random variables with marginal

cumulative function F(.)and consider a chosen high

threshold u. Suppose the exceedances occur at times

j1, ..., jN,N being the number of exceedances, i.e.,

i=1

I{Xn>u},(6)

Consider also Si(i= 1, . . . , N, N ≤n), the times

of isolated exceedances or of the first element of a

cluster, and

T∗

i=Si+1 −Siwith i= 1, . . . , N −1.. (7)

In the limit distribution, clusters occur at the

singularities of the compound Poisson process.

Sequential exceedances will belong to the same

cluster with null interexceedances times and they

represent the multiplicity of each singularity of the

compound Poisson process. So, in the limit

distribution, the T∗

ii= 1, . . . , N −1) correspond to

the realizations of a random variable Tθ, whose

distribution is characterized by (5).

The simplest way and also the most intuitive method

for estimating a proportion is to use the relative

frequency. Besides, the estimator has good

properties, as strong convergence results are known,

without requirements of differentiability or even

continuity, for the EI parameter space is the interval

[0,1] and the estimates will take values in (0,1).

Particularly, they will take values in (0,1) and they

are defined by

θP=#{T∗

i>0}

N−1,(8)

where N(N < n)is the number of exceedances to

the fixed threshold observed in the sample.

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In what concerns robustness, things are not so clear.

The most commonly used criteria for evaluating

robustness are B robustness and the breakdown point

(BP) value. B robustness ensures that a robust

estimator has a bounded influence function, which,

in simple terms, measures the sensitivity of the

estimator to changes in individual observed values.

The breakdown point, on the other hand, is related to

the estimator’s resistance, where a strictly positive

BP indicates that the estimator can handle a certain

proportion of altered observations without leading to

a breakdown in the estimate. Actually, the

proportion estimator is not B robust. Nevertheless, it

has a positive breakdown point, thus it is robust in

the last sense.

Perhaps even more interesting is the fact that the

type of dependence structure is ignored in the

estimation process. The proportion estimator seems

not to be very sensitive to the type of dependence

within clusters of exceedances, like it happens with

real data sequences. This is because the proportion

of strictly positive interexceedance times is

unaffected by the numerical values of the

exceedances. In fact, what truly matters are the times

between clusters of exceedances, rather than the

times within different clusters.

There are only a few technical details: the

convergence is based on the number of exceedances

in the sample and not directly in the sample size, as

usual. Different samples with the same ndimension

may have different numbers of exceedances. So,

with an unique sample there is only one observation

of the relative frequency. It is impossible to interpret

the EI as the binomial parameter (exception for the

Bernoulli model). Fortunately, the nature of extreme

values assure that exceedances are rare events. Most

real data sets have the size enough for splitting the

original sample into several subsamples with

constant number of exceedances N. In fact, some

big data collections contain too many zero values

and they are already cyclic. Thus, some subsamples

will turn possible to define a particular Nvalue. The

properties of the relative frequency (consistency and

positive BP) will assure good results or, at least,

reasonable results even in the worst cases. In

simulations studies the last question is overpassed

using only samples with the same number of

exceedances (which are random realizations of a

Poisson distribution), eventually discarding other

generated samples. Another technical suggestion

deals with the choice of N, which must cover the

dependence structure in such a large part as possible.

That is an issue in current investigation by the

authors.

5 Other EI estimators under

consideration

In this paper the authors choose to give particular

attention to the EI estimators based on the

interexceedance times initially proposed by [8]. In

previous simulation studies they included the Blocks

and the Runs estimators, [26], [7], also. In this paper

the option was to analyse the EI estimators that are

based on the interexceedance times only. To

compare the performance of the author’s proposals

referred above, namely, ˆ

θNB and ˆ

θP rop, the

following were also computed at the same high

thresholds:

• the Intervals estimator, [8], hereby denoted by

θInt;

• the K generalized Gap estimator, ˆ

θKGaps, [9];

• the DGaps estimator, ˆ

θDGaps, [19].

5.1 Intervals estimator

The authors in [8], pointed out the nature of the

asymptotic distribution of the interexceedance times

Ti=ji+1 −jias part of a parametric family of

distributions indexed by the extremal index. Under

the same notation used in Section 4, with a

preliminary estimator of θ, on the basis of

θn(u) =

2PN−1

i=1 Ti2

(N−1) PN−1

i=1 T2

(9)

and

θ∗

n(u) = 2{PN−1

i=1 (Ti−1)}2

(N−1) PN−1

i=1 (Ti−1)(Ti−2),(10)

the Intervals estimator is ˆ

θInt(u)is defined by

θInt(u) = 









min{1,ˆ

θn(u)}if max Ti:

1≤i≤N−1≤2

min{1,ˆ

θ∗

n(u)}if max Ti:

1≤i≤N−1>2

(11)

5.2 K Gaps estimator

The ˆ

θKGaps estimator eliminates small

interexceedance times by setting them to zero. This

approach appears reasonable at a first glance: if a

period of high sales is briefly interrupted by an

external event, the total count of high sales days

shouldn’t be markedly impacted. This interruption

can be viewed as an outlier within the larger cluster

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of high sales days. Therefore, a more appropriate

strategy might involve disregarding such values and

treating the entire set of observations as a single

cluster. The K Gaps estimator is a maximum

likelihood estimator, turned possible due to the

inclusion of a new variable in the likelihood

function, namely, the K Gap,

SuK(un) = max{T(un)−K; 0},[9], [19], with

T(un) = min{j≥1 : Xj+1 > u|X1> u}.

5.3 D Gaps estimator

The ˆ

θDGaps estimator is presented by [19], as a

generalization of the ˆ

θKGaps estimator. An artificial

scheme of censoring is introduced where the small

interexceedance times are censored. This procedure

requires a new time parameter Dto be conveniently

chosen. Consider the order statistics of the

interexceedance times as defined in Section(5.1).

The authors in [19], apply a type I left censoring

after choosing a time censor D≥0and consider the

loglikelihood function of the censored sample. Their

estimator is the value which minimizes that function.

6 Application to real data

Recent years brought new challenges to international

markets in several fields like electronics and

pharmaceutical products, either due to COVID or

justified by the constraints of war. In times of

COVID, pharmaceutical laboratory suffered extra

stress and this is still presently pointed occasionally

at news. High values of drug sales frequently happen

in clusters. This can be explained by some

epidemiology or by some other phenomena, not so

easy to identify. Being able to estimate the size of

the clusters allows for better planning the production

and, on the other side, may help to study eventual

hidden causes for the peaks of sales. High levels of

air pollution tend to be associated with high levels of

antihistaminic sales. Available since the 40’s, this

type of drugs represent an option of treatment that

can improve life quality to a vast number of people,

[27]. The sales numbers of this type of drugs show

clusters of high values which may be relevant, either

to pharma business or to the medical research. The

data set used in the present work was obtained from

Kaggle [28], and consists of weekly pharma sales

data from antihistamines for systemic use (R06,

accordingly to Anatomical Therapeutic Chemical

(ATC) Classification System), since 2014 to 2019.

As shown in Figure 2 , this data set presents clusters

of high values separated from each other. We’ll

assume the validity of the conditions of local

dependence.

Following the holdout procedure, which is typical

in time series analysis, [29], the sample was divided

Figure 2: Sales data from antihistamines for systemic

use (R06), since 2014 to 2019

into a training sub sample containing the first 60% of

observations and a test sample, consisting of the

latest 40% observations (Initially, 70% of the first

observations were considered for the training

sample. However, the decision to use 60% instead

was made, to ensure that the test sample visually

preserved the behaviour of the entire series more

accurately). The resulting samples are illustrated in

Figure 3. To obtain the results presented in this

paper the R packages extRemes,fExtremes and exdex

were used.

Figure 3: Sales data from antihistamines for systemic

use (R06): training sample (left) an test sample

(right).

Different estimates were computed using the

training sample and then tested by comparing these

estimates with the reciprocal of the average size of

the observed clusters of exceedances in the test

sample. These clusters were defined as groups of

exceedances above the specified high thresholds

after the first one observed. A set of thresholds

corresponding to the highest quantiles, from 0.6to

.98 was considered. For the KGaps and DGaps

estimators, the suitable combination of the pair

(u, K)(of the threshold and the tuning parameter)

was selected, accordingly to the graphical

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diagnostics based on the information matrix test, [9].

Figure 4 shows the behaviour of the obtained

estimates for the data. Apart from the intervals

estimate, it is possible to identify two main

tendencies: one group with higher values includes

the KGaps and the Proportion estimators, and

another group with lower values, with the NB and

the DGaps estimators. In the present case, it seems

that the former would perform better if used to

preview the future θvalue as the corresponding lines

are closer to the observed value of θ.

Figure 4: EI estimates for the antihistamines (R06)

data: estimates computed with the training sample

and the reciprocal of the mean size of the observed

clusters for the test sample (θT est).(K= 1 and D=

4) for the KGaps and DGaps, respectively.)

Considering the former group corresponding to

better performances the best results are provided by

the Proportion estimator herein suggested. Thus,

while ignoring completely the type of dependence

and being very simple to interpret, the well known

properties of the relative frequency assure a good

performance. The approach by the NB estimator

does not show advantages, though the trajectory is

similar to the θT est but with an almost constant

considerable bias. It is also relevant to notice that the

Intervals estimator seems to produce poor estimates.

7 Conclusions

This paper gives a contribution to the estimation of

the EI. Among different existing proposals in the

literature, two recent new approaches were applied

to a real data set: a robust estimator based on the

uniparametric NB model (3), and an estimator based

on the interexceedance times (4). The resulting

estimates were then compared for different high

thresholds showing a better performance from the

Proportion estimator. Assuming the ˆ

θT est is the most

close to the real value, the ˆ

θP rop estimator seems to

have better performance since their paths are the

closest to the ˆ

θtest path.

The simplicity, the properties, and the performance

of the Proportion estimator make it well suited for

application in more practical fields, particularly

when the dependence structure of the clusters is

unknown, which is often the case. However, some

disadvantages arise in simulation studies, potentially

due to the fact that the choice of Ndoes not depend

on the sample size, as well as limitations in the

application of resampling methods.

Local dependence conditions might be difficult to

prove. So, other alternative robust forms must be

found. The ˆ

θNB seems to be the one that shows

more stability concerning the threshold level.

The main conclusion after this paper is that there is

still much work to do for the academic community to

provide more accurate tools for the decision makers.

For a parameter that takes values between 0 and 1,

the results obtained by different approaches show a

range af values that is considerably high, like it

happens with the present data.

Future research will focus on identifying the

optimal number of exceedances for analysis (related

to the threshold choice), and conducting a deeper

study on robustness, particularly by considering

simulated contaminated samples. Additionally,

further investigations will explore the application of

the method in outlier detection and comparisons of

robust estimators. The variation in estimates

produced by different methods suggests the need for

developing confidence intervals for the EI.

Acknowledgment:

Research partially supported through the Portuguese

Foundation for Science and Technology (FCT -

Fundação para a Ciência e a Tecnologia), by the

Center for Research and Development in

Mathematics and Applications

(CIDMA),UIDB/04106/2020,

doi.org/10.54499/UIDB/04106/2020,

doi.org/10.54499/UIDP/04106/2020,and Centre

of Statistics and its Applications (CEAUL), within

project

UIDB/00006/2020

(https:

//doi.org/10.54499/UIDB/00006/2020).

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

All the authors contributed to the conceptualization

as well as to the the final writing.

M. Cristina Miranda was responsible for chosing the

data and for the code to produce and visualise the

results.

Manuela Souto de Miranda made a strong

contribution in the robustness area.

M. Cristina Miranda and M. Ivette Gomes gave

contributions in the EVT framework.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

Research partially supported through the Portuguese

Foundation for Science and Technology (FCT -

Fundação para a Ciência e a Tecnologia)

The authors have no conflicts of interest to

declare that are relevant to the content of this

article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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Conflict of Interest

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2024.23.25

M. Cristina Miranda,

Manuela Souto De Miranda, M. Ivette Gomes

E-ISSN: 2224-2678

231

Volume 23, 2024