# A Bimodal Model for Extremes Data

Cira E. G. Otiniano<EMAIL_ADDRESS>Departamento de Estatística,
Universidade de Brasília, 70910-900, Brasília, Brazil  Bianca Sousa
<EMAIL_ADDRESS>Departamento de Estatística, Universidade de Brasília,
70910-900, Brasília, Brazil  Roberto Vila<EMAIL_ADDRESS>Departamento de
Estatística, Universidade de Brasília, 70910-900, Brasília, Brazil  Marcelo
Bourguignon<EMAIL_ADDRESS>Departamento de Estatística,
Universidade Federal do Rio Grande do Norte, 59078-970, Natal/RN, Brazil

###### Abstract

In extreme values theory, for a sufficiently large block size, the maxima
distribution is approximated by the generalized extreme value (GEV)
distribution. The GEV distribution is a family of continuous probability
distributions, which has wide applicability in several areas including
hydrology, engineering, science, ecology and finance. However, the GEV
distribution is not suitable to model extreme bimodal data. In this paper, we
propose an extension of the GEV distribution that incorporate an additional
parameter. The additional parameter introduces bimodality and to vary tail
weight, i.e., this proposed extension is more flexible than the GEV
distribution. Inference for the proposed distribution were performed under the
likelihood paradigm. A Monte Carlo experiment is conducted to evaluate the
performances of these estimators in finite samples with a discussion of the
results. Finally, the proposed distribution is applied to environmental data
sets, illustrating their capabilities in challenging cases in extreme value
theory.

Keywords. Bimodality $\cdot$ Environmental data $\cdot$ Generalized extreme
value distribution $\cdot$ Maximum likelihood
Mathematics Subject Classification (2010). MSC 60E05 $\cdot$ MSC 62Exx $\cdot$
MSC 62Fxx.

##  1 Introduction

The generalized extreme value (GEV) distribution is widely used to model
extreme events in several areas such as finance, insurance, hydrology,
bioinformatics, among others. In journals of statistics and applied areas, a
huge amount of works with applications of the GEV distribution can be found.
Theory and applications of extreme value distributions can be found in the
books by Kotz and Nadarajah (2000), Coles (2001), Beirlant et al. (2004), Haan
and Ferreira (2010) , Embrechts et al. (2013), Longin (2016), Scheirer (2017)
, among others.

A random variable $Y$ has a Generalized Extreme Value (GEV) distribution with
shape parameter $\xi$, location parameter $\mu\in\mathbb{R}$, and scale
paramer $\sigma>0$ denoted by $X\sim F(\cdot;\xi,\mu,\sigma)$, if its
probability density function (PDF) is given by

$\displaystyle
f(y;\xi,\mu,\sigma)=\begin{cases}\displaystyle\frac{1}{\sigma}\Big{[}1+\xi\Big{(}\frac{y-\mu}{\sigma}\Big{)}\Big{]}^{(-1/\xi)-1}\,\exp\biggl{\\{}-\Big{[}1+\xi\Big{(}\frac{y-\mu}{\sigma}\Big{)}\Big{]}^{-1/\xi}\biggr{\\}},&\xi\neq
0,\\\\[11.38092pt]
\noindent\displaystyle\frac{1}{\sigma}\exp\biggl{\\{}-\Big{(}\frac{y-\mu}{\sigma}\Big{)}-\exp\Big{[}-\Big{(}\frac{y-\mu}{\sigma}\Big{)}\Big{]}\biggr{\\}},&\xi=0,\end{cases}\quad$

valid for $y>\mu-\sigma/\xi$ in the case $\xi>0$, and $y<\mu-\sigma/\xi$ in
the case $\xi<0$.

The cumulated distribution function (CDF) of the GEV distribution is given by

$\displaystyle
F(y;\xi,\mu,\sigma)=\begin{cases}\displaystyle\exp\biggl{\\{}-\Big{[}1+\xi\Big{(}\frac{y-\mu}{\sigma}\Big{)}\Big{]}^{-1/\xi}\biggr{\\}},&\xi\neq
0,\\\\[11.38092pt]
\displaystyle\exp\biggl{\\{}-\exp\Big{[}-\Big{(}\frac{y-\mu}{\sigma}\Big{)}\Big{]}\biggr{\\}},&\xi=0.\end{cases}$
(1.1)

When $\xi=0$ the pdf (1.1) is also known as the Gumbel distribution.

Here a GEV random variable with $\xi\neq 0$ and standar scale we denote by
$Y\sim F_{\rm G}(\cdot;\xi,\mu)=F(\cdot;\xi,\mu,1)$. In this case, the PDF and
CDF functios are given, respectively, by

$\displaystyle f_{\rm
G}(y;\xi,\mu)=\left[1+\xi(y-\mu)\right]^{(-1/\xi)-1}\,\exp\left\\{-\left[1+\xi(y-\mu)\right]^{-1/\xi}\right\\}$
(1.2)

and

$\displaystyle F_{\rm
G}(y;\xi,\mu)=\exp\left\\{-\left[1+\xi(y-\mu)\right]^{-1/\xi}\right\\}.$

In extreme value modeling, in special, environmental data (see Section 7) data
with bimodality has appeared more and more. The statistical modeling of this
type of data requires distributions that capture bimodality. In this context,
Nascimento et al. (2016) provided some new extended models to the GEV
distribution as a baseline function and the transmuted GEV distribution
introduced by Aryal and Tsokos (2009) , showing advantages compared with the
standard GEV distribution. Eljabri and Nadarajah (2017) studied the
Kumaraswamy GEV distribution. However, all the models cited above are not
suitable for capturing bimodality. In this sense, the mixture of GEV
distributions and the mixture of extreme value distributions has been an
alternative ( Otiniano et al. (2016) ). The difficulty with these models is in
the estimation process, as six sub-models must be considered.

This paper presents a new model for extremes, based on transformation of the
standard GEV, with the hope it yields a “better fit” in certain extremes
analysis (see Sect. 5). The inclusion of additional parameter in the GEV role
is to introduce bimodality, that is, the new distribution is more flexible
than the GEV distribution, and without computational complications for the
estimation of its parameters in the case $\xi\neq 0$.

In Sect. 2, we define the BGEV model. In Sect. 3, we derive the main
properties of the bimodal GEV distribution. In Sect. 4, we provide graphical
illustrations. In Sect. 5, inference procedure is carried out under the
likelihood paradigm. Already, in Sect. 6 discusses some simulation results for
the estimation method. In Sect. 7, two illustrative applications in
environmental data sets are investigated. Conclusions are addressed in Sect.
8.

##  2 The Bimodal GEV distribution

In this section, we introduce a bimodal distribution of extreme value (BGEV)
as Rathei and Swamme (2007) .

We define a random variable with BGEV distribution and parameters $\xi\neq
0,\mu,\sigma,\delta$, denoted $X\sim F_{\rm
BGEV}(\cdot;\xi,\mu,\sigma,\delta)$, if its PDF is given by

$\displaystyle f_{\rm BGEV}(x;\xi,\mu,\sigma,\delta)$ $\displaystyle=f_{\rm
G}\big{(}T_{\sigma,\delta}(x);\xi,\mu\big{)}\,T_{\sigma,\delta}^{\prime}(x)$
$\displaystyle=\sigma(\delta+1)|x|^{\delta}\big{[}1+\xi(\sigma
x|x|^{\delta}-\mu)\big{]}^{(-1/\xi)-1}\exp\left\\{-\big{[}1+\xi(\sigma
x|x|^{\delta}-\mu)\big{]}^{-1/\xi}\right\\},$ (2.1)

where

$\displaystyle T_{\sigma,\delta}(x)=\sigma x|x|^{\delta}\,,\quad
x\in\mathbb{R},\ \ \delta\geqslant-1,\ \ \sigma>0$ (2.2)

is an invertible transformation with derivatives

$\displaystyle
T_{\sigma,\delta}^{\prime}(x)=\sigma(\delta+1)|x|^{\delta}\,,\quad
T_{\sigma,\delta}^{\prime\prime}(x)={\rm
sign}(x)\,\sigma(\delta+1)\delta|x|^{\delta-1}\,,$ (2.3)

and

$T_{\sigma,\delta}^{(k)}(x)=\big{[}{\rm
sign}(x)\big{]}^{k-1}\,\sigma\,\Bigg{[}\prod_{i=-1}^{k-2}(\delta-i)\Bigg{]}|x|^{\delta-(k-1)}\,,\quad
k\geqslant 2\,,$

where ${\rm sign}(\cdot)$ denotes the sign function.

Since $T_{\sigma,\delta}$ is non-decreasing for $\delta>-1$, then, its inverse
function, denoted by $T_{\sigma,\delta}^{-1}$, exists and is given by

$\displaystyle T_{\sigma,\delta}^{-1}(x)={\rm
sign}(x)\biggl{(}\frac{|x|}{\sigma}\biggr{)}^{1/(\delta+1)}.$ (2.4)

The PDF (2) is valid for $x>T_{\sigma,\delta}^{-1}(\mu-1/\xi)$ in the case
$\xi>0$, and for $x<T_{\sigma,\delta}^{-1}(\mu-1/\xi)$ in the case $\xi<0$.

Note that, for $\xi>0$, the function $f_{\rm BGEV}$ is a PDF, because

$\displaystyle\int_{T_{\sigma,\delta}^{-1}(\mu-1/\xi)}^{\infty}f_{\rm
BGEV}(x;\xi,\mu,\sigma,\delta)\,{\rm d}x=\int_{\mu-1/\xi}^{\infty}f_{\rm
G}\big{(}T_{\sigma,\delta}(x);\xi,\mu\big{)}\,{\rm
d}T_{\sigma,\delta}(x)=1\,.$

Similarly, it is verified that, for $\xi<0$, $f_{\rm BGEV}$ is a PDF.

The expression (2) is equivalent to

$f_{\rm BGEV}(x;\xi,\mu,\sigma,\delta)=(F_{\rm G}\circ
T_{\sigma,\delta})^{\prime}(x)\,,$

where

$(F_{\rm G}\circ T_{\sigma,\delta})(x;\xi,\mu)=F_{\rm
BGEV}(x;\xi,\mu,\sigma,\delta)$ (2.5)

is the CDF of $X$.

When $\delta=0$ the function (2) is of type (1.1); $F_{\rm
BGEV}(x;\xi,\mu,\sigma,0)=F_{\rm G}(x;\xi,\mu/\sigma,1/\sigma)$.

##  3 Some properties of the BGEV distribution

In this section, some mathematical properties as monotonicity, bimodality
property, stochastic representation, moments, quantiles and tail behavior of
the BGEV distribution are discussed.

###  3.1 Monotonicity

Let $m_{\xi}$ be the unique mode for the GEV distribution (1.2). To state the
next result we define the following quantities:
$x^{\min}_{\xi}=\min\\{0,T_{\sigma,\delta}^{-1}(m_{\xi})\\}$ and
$x_{\xi}^{\max}=\max\\{0,T_{\sigma,\delta}^{-1}(m_{\xi})\\}$.

######  Proposition 3.1.

The following monotonicity properties it hold:

  * 1)

If $\xi>0$ and $\delta<0$, then the BGEV PDF is increasing for each
$x<x^{\min}_{\xi}$;

  * 2)

If $\xi>0$ and $\delta>0$, then the BGEV PDF is decreasing for each
$x<x^{\min}_{\xi}$;

  * 3)

If $\xi<0$ and $\delta<0$, then the BGEV PDF is decreasing for each
$x>x^{\max}_{\xi}$;

  * 4)

If $\xi<0$ and $\delta>0$, then the BGEV PDF is increasing for each
$x>x^{\max}_{\xi}$.

###### Proof.

Since $T_{\sigma,\delta}^{\prime}(x)=\sigma(\delta+1)|x|^{\delta}$, for
$\delta<0$, $T_{\sigma,\delta}^{\prime}(x)$ is decreasing (resp. increasing)
for each $x>0$ (resp. $x<0$). For $\delta>0$, $T_{\sigma,\delta}^{\prime}(x)$
is increasing (resp. decreasing) for each $x>0$ (resp. $x<0$). On the other
hand, since $m_{\xi}$ is the mode of the GEV distribution (1.2), $f_{\rm
G}\big{(}T_{\sigma,\delta}(x);\xi,\mu\big{)}$ is increasing (resp. decreasing)
for each $x<x^{\min}_{\xi}$ (resp. $x>x^{\max}_{\xi}$) and $\xi>0$ (resp.
$\xi<0$). Since $T_{\sigma,\delta}^{\prime}(x)$ is increasing and nonnegative
for all $x<0$, and $f_{\rm G}\big{(}T_{\sigma,\delta}(x);\xi,\mu\big{)}$ is
increasing for each $x<x^{\min}_{\xi}$, $\xi>0$ and $\delta<0$, from
definition (2) of BGEV density, we have that $f_{\rm
BGEV}(x;\xi,\mu,\sigma,\delta)=f_{\rm
G}\big{(}T_{\sigma,\delta}(x);\xi,\mu\big{)}\,T_{\sigma,\delta}^{\prime}(x)$
is the product of two increasing and nonnegative functions. Thus, the BGEV PDF
is increasing for each $x<x^{\min}_{\xi}$. This completes the proof of first
item.

The proof of the other items follows the same reasoning as the one of Item 1).
∎

###  3.2 Bimodality property

######  Proposition 3.2.

The point $x$ is a critical point of BGEV density (2) if it is solution of the
following equation:

${T_{\sigma,\delta}^{\prime\prime}(x)}-{1+\xi-\big{[}1+\xi(T_{\sigma,\delta}(x)-\mu)\big{]}^{-1/\xi}\over
1+\xi(T_{\sigma,\delta}(x)-\mu)}\,\big{[}T_{\sigma,\delta}^{\prime}(x)\big{]}^{2}=0,$

where $T_{\sigma,\delta}(x)$, $T_{\sigma,\delta}^{\prime}(x)$ and
$T_{\sigma,\delta}^{\prime\prime}(x)$ are given in (2.2) and (2.3).

###### Proof.

The proof is trivial and omitted. ∎

######  Lemma 3.3.

Let $1/\xi$ be a natural number and $\delta\geqslant 2$. The PDF of the BGEV
distribution has at most three real critical points.

###### Proof.

Let $y=1+\xi(T_{\sigma,\delta}(x)-\mu)$. Since $\xi$ is a positive number,
$y>0$. By using definitions of $T_{\sigma,\delta}(x)$,
$T_{\sigma,\delta}^{\prime}(x)$ and $T_{\sigma,\delta}^{\prime\prime}(x)$ in
(2.2) and (2.3), a simple algebraic manipulation shows that the equation of
Proposition 3.2 is equivalent to

$\displaystyle|x|^{-2}T_{\sigma,\delta}(x)\big{[}\delta
y^{1+1/\xi}-(\xi+1)y^{1/\xi}+1\big{]}=0,\quad y>0.$

For $\delta\geqslant 2$, $x=0$ is a critical point of $f_{\rm
BGEV}(x;\xi,\mu,\sigma,\delta)$. So the number of positive roots of the
polynomial $p(y)=\delta y^{1+1/\xi}-(\xi+1)y^{1/\xi}+1$, when $1/\xi$ a
natural number, determines the number of remaining roots of the BGEV PDF.

By Descartes’ rule of signs (Griffiths 1947 ; Xue 2012 ), the polynomial
$p(y)$ has two sign changes (the sequence signs is $+$, $-$, $+$), meaning
that this polynomial has two or zero positive roots. This completes the proof.
∎

######  Theorem 3.4 (Uni- or bimodality).

If $1/\xi\geqslant 2$ is an even natural number and $\delta\geqslant 2$, then
the PDF of the BGEV distribution is uni- or bimodal.

###### Proof.

If $1/\xi$ is even, the polynomial $p(y)$ defined in Lemma 3.3 has odd degree.
Let $p(x)$ be the extension of polynomial $p(y)$ to $x$ in $\mathbb{R}$.
Again, by Descartes’ rule of signs the polynomial $p(-x)$ has one sign change
(the sequence signs is $-$, $-$, $+$), meaning that this one has one negative
root. But as a sub-product of the proof of Lemma 3.3 we get that $p(x)$ has
two or zero positive roots. Since $p(x)$ has odd degree and the complex roots
are given in pairs, we only have two possibilities: (a) $p(x)$ has one
negative root and zero positive roots or (b) $p(x)$ has one negative root and
two positive roots. From case (a) it follows that $p(y)$, $y>0$, has no
positive roots, and by Lemma 3.3, $x=0$ is the only critical point of BGEV
density. Already, from case (b) it follows that $p(y)$, $y>0$, has exactly two
positive roots, and therefore, the BGEV PDF has exactly three real critical
points: $x=0$, $x=T_{\sigma,\delta}^{-1}(\mu+(y_{1}-1)/\xi)$ and
$x=T_{\sigma,\delta}^{-1}(\mu+(y_{2}-1)/\xi)$.

Finally, since $\lim_{x\to\pm\infty}f_{\rm BGEV}(x;\xi,\mu,\sigma,\delta)=0$,
the uni- or bimodality of the BGEV density follows. ∎

###  3.3 Stochastic representation of the BGEV distribution

######  Proposition 3.5.

Let $X\sim F_{\rm BGEV}(\cdot;\xi,\mu,\sigma,\delta)$.

  * 1)

If $X\sim F_{\rm BGEV}(\cdot;\xi,\mu,\sigma,\delta)$, then
$Y=T_{\sigma,\delta}(X)\sim F_{\rm G}(\cdot;\xi,\mu)$.

  * 2)

If $Y\sim F_{\rm G}(\cdot;\xi,\mu)$, then $X=T_{\sigma,\delta}^{-1}(Y)\sim
F_{\rm BGEV}(\cdot;\xi,\mu,\sigma,\delta)$.

  * 3)

If $X\sim F_{\rm BGEV}(\cdot;\xi,\mu,\sigma,\delta)$ and $c\neq 0$ a constant,
then $X\sim F_{\rm BGEV}(\cdot;\xi,|c|\mu,c^{2}\sigma,\delta)$.

###### Proof.

If $X\sim F_{\rm BGEV}(\xi,\mu,\sigma,\delta)$, by (2.5), follows that

$\displaystyle\mathbb{P}(Y\leqslant y)=\mathbb{P}(X\leqslant
T_{\sigma,\delta}^{-1}(y))=F_{\rm
BGEV}(T_{\sigma,\delta}^{-1}(y);\xi,\mu,\sigma,\delta)=(F_{\rm G}\circ
T_{\sigma,\delta})(T_{\sigma,\delta}^{-1}(y);\xi,\mu)=F_{\rm G}(y;\xi,\mu).$

This prove the Item 1). Analogously, the proof of second item follows.

Already, the proof of Item 3) follows by combining the Item 1), the relation
$|c|T_{\sigma,\delta}(x)=T_{|c|\sigma,\delta}(x)$, the following well-known
fact: If $Y\sim F_{\rm G}(\cdot;\xi,\mu)=F_{\rm G}(\cdot;\xi,\mu,1)$, then
$cY\sim F_{\rm G}(\cdot;\xi,c\mu,c)$ for $c\neq 0$; with Item 2). ∎

###  3.4 Moments

Here we give a closed analytical formula for the $k$th moment of a random
variable with BGEV distribution.

######  Proposition 3.6.

If $X\sim F_{\rm BGEV}(\xi,\mu,\sigma,\delta)$ and $\xi<{1}/{k}$, then

$\displaystyle\mathbb{E}(X^{k(\delta+1)})=\begin{cases}\displaystyle\frac{1}{\xi^{k}\sigma^{k}}\sum_{i=0}^{k}\binom{k}{i}(\xi\mu-1)^{k-i}\Gamma(1-\xi{i}),&\mu-\frac{1}{\xi}>0,\\\\[19.91684pt]
\displaystyle\frac{(-1)^{k(\delta+1)}}{\xi^{k}\sigma^{k}}\sum_{i=0}^{k}\binom{k}{i}\left(\xi\mu-1\right)^{k-i}\gamma\big{(}1-\xi
i;(1-\xi\mu)^{-1/\xi}\big{)}\\\ \displaystyle\hskip
19.0633pt+\frac{1}{\xi^{k}\sigma^{k}}\sum_{i=0}^{k}\binom{k}{i}\left(\xi\mu-1\right)^{k-i}\Gamma\big{(}1-\xi
i;(1-\xi\mu)^{-1/\xi}\big{)},&\mu-\frac{1}{\xi}<0,\end{cases}$

where $\gamma(a;x)=\int_{x}^{\infty}t^{a-1}e^{-t}\,{\rm d}t$ and
$\Gamma(a;x)=\int_{0}^{x}t^{a-1}e^{-t}\,{\rm d}t$ are the incomplete gamma
functions.

###### Proof.

For $\xi>0$, by definition of expectation and the expression (1.2), we have

$\displaystyle\mathbb{E}(X^{k(\delta+1)})$ $\displaystyle=$
$\displaystyle\int_{T_{\sigma,\delta}^{-1}(\mu-1/\xi)}^{\infty}x^{k(\delta+1)}f_{\rm
BGEV}(x;\xi,\mu,\sigma,\delta)\,{\rm d}x$ (3.1) $\displaystyle=$
$\displaystyle\int_{T_{\sigma,\delta}^{-1}(\mu-1/\xi)}^{\infty}x^{k(\delta+1)}f_{\rm
G}\big{(}T_{\sigma,\delta}(x);\xi,\mu\big{)}\,{\rm d}T_{\sigma,\delta}(x)$
$\displaystyle=$
$\displaystyle\int_{\mu-1/\xi}^{\infty}\big{[}T^{-1}_{\sigma,\delta}(y)\big{]}^{k(\delta+1)}f_{\rm
G}\big{(}y;\xi,\mu\big{)}\,{\rm d}y.$

When replacing the expression (2.4) of $T_{\sigma,\delta}^{-1}$ in (3.1) we
obtain two cases:
(i) If $\mu-{1}/{\xi}>0$,

$\displaystyle\mathbb{E}(X^{k(\delta+1)})=\frac{1}{\sigma^{k}}\int_{\mu-1/\xi}^{\infty}y^{k}f_{\rm
G}\big{(}y;\xi,\mu\big{)}\,{\rm d}y=\frac{1}{\sigma^{k}}\,\mathbb{E}(Y^{k}),\
\ \ \ Y\sim F_{\rm G}(\cdot;\xi,\mu),$

where

$\displaystyle\mathbb{E}(Y^{k})=\sum_{i=1}^{k}\binom{k}{i}\Big{(}\mu-\frac{1}{\xi}\Big{)}^{k-i}\frac{1}{\xi^{i}}\,\Gamma(1-\xi
i),\quad\xi<1/k.$

(ii) If $\mu-{1}/{\xi}<0$,

$\displaystyle\mathbb{E}(X^{k(\delta+1)})$ $\displaystyle=$
$\displaystyle\frac{(-1)^{{k}{(\delta+1)}}}{\sigma^{k}}\int_{\mu-1/\xi}^{0}y^{k}f_{\rm
G}\big{(}y;\xi,\mu\big{)}\,{\rm
d}y+\frac{1}{\sigma^{k}}\int_{0}^{\infty}y^{k}f_{\rm
G}\big{(}y;\xi,\mu\big{)}\,{\rm d}y$ (3.2) $\displaystyle=$
$\displaystyle\frac{(-1)^{{k}{(\delta+1)}}}{\sigma^{k}}\,{I_{1}}+\frac{1}{\sigma^{k}}\,{I_{2}}.$

Using the PDF (1.2) and the substitution $w^{-1}=1+\xi(y-\mu)$, we obtain

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\xi^{k+1}}\int_{(1-\xi\mu)^{-1}}^{\infty}[w^{-1}+(\xi\mu-1)]^{k}\
w^{\frac{1}{\xi}-1}{\rm e}^{-w^{1/\xi}}\,{\rm d}w.$ (3.3)

To solve the integral of (3.3) we use the Newton’s binomial formula, thus

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\xi^{k+1}}\sum_{i=1}^{k}\binom{k}{i}\left(\xi\mu-1\right)^{k-i}\int_{(1-\xi\mu)^{-1}}^{\infty}w^{\frac{1}{\xi}-i-1}{\rm
e}^{-w^{1/\xi}}\,{\rm d}w$ (3.4) $\displaystyle=$
$\displaystyle\frac{1}{\xi^{k}}\sum_{i=1}^{k}\binom{k}{i}\left(\xi\mu-1\right)^{k-i}\int_{(1-\xi\mu)^{-1/\xi}}^{\infty}z^{-\xi
i}{\rm e}^{-z}\,{\rm d}z,$

where in the last line we used the new substitution $z=w^{1/\xi}$.

Similarly we obtain

$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\xi^{k}}\sum_{i=1}^{k}\binom{k}{i}\left(\xi\mu-1\right)^{k-i}\int_{0}^{(1-\xi\mu)^{-1/\xi}}z^{-\xi
i}{\rm e}^{-z}\,{\rm d}z.$ (3.5)

The proof is completed by expressing as integrals of (3.4) and (3.5) in terms
of the incomplete gamma functions and then updating the equation (3.2).

For $\xi<0$ the proof is similar. ∎

######  Corollary 3.7.

Let $X\sim F_{\rm BGEV}(\xi,\mu,\sigma,\delta)$ and $\delta\leq 0$, then

$\displaystyle\mathbb{E}(X)=\begin{cases}\displaystyle\frac{1}{(\xi\sigma)^{\llbracket
1/(\delta+1)\rrbracket}}\sum_{i=0}^{\llbracket
1/(\delta+1)\rrbracket}\binom{\llbracket
1/(\delta+1)\rrbracket}{i}\left(\xi\mu-1\right)^{\llbracket
1/(\delta+1)\rrbracket-i}\Gamma(1-\xi{i}),&\\!\\!\mu-\frac{1}{\xi}>0,\\\\[19.91684pt]
\displaystyle\frac{(-1)^{k(\delta+1)}}{(\xi\sigma)^{\llbracket
1/(\delta+1)\rrbracket}}\sum_{i=0}^{\llbracket
1/(\delta+1)\rrbracket}\binom{\llbracket
1/(\delta+1)\rrbracket}{i}\left(\xi\mu-1\right)^{\llbracket
1/(\delta+1)\rrbracket-i}\gamma\big{(}1-\xi i;(1-\xi\mu)^{-1/\xi}\big{)}\\\
\displaystyle+\frac{1}{(\xi\sigma)^{\llbracket
1/(\delta+1)\rrbracket}}\sum_{i=0}^{\llbracket
1/(\delta+1)\rrbracket}\binom{\llbracket
1/(\delta+1)\rrbracket}{i}\left(\xi\mu-1\right)^{\llbracket
1/(\delta+1)\rrbracket-i}\Gamma\big{(}1-\xi
i;(1-\xi\mu)^{-1/\xi}\big{)},&\\!\\!\mu-\frac{1}{\xi}<0.\end{cases}$

###### Proof.

The result follows directly of Proposition 3.6 by considering
$\delta=({1}/{k})-1$ and $k\in\mathbb{Z}^{+}$. ∎

###  3.5 Quantiles

In the next section, through Monte Carlo simulation studies, we test the
behavior of the parameter estimates of the BGEV model. The random samples of
the population are simulated using the inverse transformation method based on
quantiles. The quantile function of a random variable $X\sim F_{\rm
BGEV}(\cdot;\xi,\mu,\sigma,\delta)$ is defined by

$x_{Q}=\inf\\{x\in\mathbb{R}:Q\leq F_{\rm BGEV}(x;\xi,\mu,\sigma,\delta)\\}$

or equivalent by

$\displaystyle x_{Q}$ $\displaystyle=$ $\displaystyle
T_{\sigma,\delta}^{-1}(F_{\rm G}^{-1}(Q)).$

From (2.5), we have $F_{\rm G}^{-1}(y)=\mu+{[(-\ln y)^{-\xi}-1]}/{\xi}$. Now,
by using the expression (2.4) of $T_{\sigma,\delta}^{-1}$, we get

$\displaystyle
x_{Q}=\begin{cases}\displaystyle\biggl{[}\frac{\mu}{\sigma}+\frac{(-\ln
Q)^{-\xi}-1}{\sigma\xi}\biggr{]}^{{1}/{(\delta+1)}},&\displaystyle\mu+\frac{(-\ln
Q)^{-\xi}-1}{\xi}>0,\\\\[11.38092pt]
(-1)^{\frac{2+\delta}{1+\delta}}\displaystyle\biggl{[}\frac{\mu}{\sigma}+\frac{(-\ln
Q)^{-\xi}-1}{\sigma\xi}\biggr{]}^{{1}/{(\delta+1)}},&\displaystyle\mu+\frac{(-\ln
Q)^{-\xi}-1}{\xi}<0.\end{cases}$ (3.6)

###  3.6 Tail behavior of BGEV

According to Embrechts et al. (2013) , asymptotic estimates of probability
models are omnipresent in statistics, insurance mathematics, mathematical
finance, and hydrology. In this sense, the theory of regular variation plays a
crucial role. Concepts and properties of regularly varying can be found in the
book of Bingham et al. (1987) In order to study the asymptotic behavior of the
tail of the BGEV distribution, here we show some of the results of regular
variation.

######  Definition 3.1.

A positive Lebesgue measurable function $h$ on $(0,\infty)$ is regularly
varying at $\infty$ of index $\alpha\in\mathbb{R}$
($h\in\mathcal{R}_{\alpha}$) if

$\lim\limits_{t\rightarrow\infty}\frac{h(tx)}{h(t)}=x^{\alpha}.$

Suppose $F$ is a distribution function such that $F(x)<1$ for all $x>0$. Let
$\overline{F}(x)=1-F(x)$ and $\overline{F}\in\mathcal{R}_{-\alpha}$ for some
$\alpha>0$ , then

$\displaystyle\overline{F}(x)=o(x^{-\alpha}L(x))$ (3.7)

for some $L\in\mathcal{R}_{0}$, $L_{0}$ a slowly varying function. The
expression (3.7) indicates that the tail of the distribution $F$ slowly
decays, as $x^{-\alpha}$, that is $F$ has heavy (right) tail with index
$\alpha$. Thus, $F$ is Pareto-like distribution.

######  Proposition 3.8.

Let $X\sim F_{\rm BGEV}(\cdot;\xi,\mu,\sigma,\delta)$ with $\xi>0$ and
$\delta>-1$ . Then

$\displaystyle\overline{F}_{\rm
BGEV}(x;\xi,\mu,\sigma,\delta)=o\left(x^{-\frac{\delta+1}{\xi}}\right),\ \
x>0.$ (3.8)

###### Proof.

The proof follows of the straightforward calculation of

$\lim\limits_{t\rightarrow\infty}\frac{\overline{F}_{\rm
BGEV}(tx;\xi,\mu,\sigma,\delta)}{\overline{F}_{\rm
BGEV}(t;\xi,\mu,\sigma,\delta)}=x^{-\frac{\delta+1}{\xi}},$

where ${F}_{\rm BGEV}(tx;\xi,\mu,\sigma,\delta)$ is given in (2.5). ∎

From (3.8) we have that ${F}_{\rm BGEV}$ is Pareto-like distribution with tail
index ${(\delta+1)}/{\xi}$. The new $\delta$ parameter affects the weight of
the tail.

##  4  Graphical study

The role that the parameters $\xi$, $\mu$, $\sigma$, $\delta$, and play in the
BGEV distribution were investigated graphically by generating different
densities with variations in each parameter. Fig. 1 shows that $\sigma$ is a
scale parameter. According to Fig. 2, the shape of the BGEV PDF changed by
varying the $\mu$ parameter. That is, $\mu$ is a shape parameter. Fig. 2
suggests that depending on the combinatation of the parameters values $\xi$
and $\mu$ the BGEV density can be unimodal or bimodal. For example, $f_{\rm
BGEV}(.;2,\mu,1,2)$ is unimodal and $f_{{\rm BGEV}}(.;0.5,\mu,1,2)$ is
bimodal. In the bimodal case, the first mode value is greater than the second
mode value for $\mu\in[0,0.25)$, while for $\mu>0.25$ the second mode value is
greater than the first mode. Figures 3 and 4 correspond to variations of the
new parameter $\delta$. When $\delta=0$ the BGEV coincides with the basic GEV
distribution. Two scenarios were considered. Scenario 1 for $\xi<0$ (see Fig.
3), and scenario 2 for $\xi>0$ (see Fig. 4 ). In scenario 1 , Fig. 3, when
$\delta\leq 0$ the PDF is unimodal and for $\delta>0$ the PDF is bimodal. In
bimodal case, the mode values increases as delta grows. In scenario 2, the
Fig. 4 shows that when delta takes positive or negative values close to zero
and $\xi>1$ the PDF is bimodal. In this case, the mode value also increases as
delta increases.

Figure 1:

Left: BGEV PDF, $f_{\rm BGEV}(x;0.5,0,\sigma,0.4)$, with $\sigma$ varying;
Right: BGEV PDF, $f_{\rm BGEV}(x;0.5,0,\sigma,2)$ with $\sigma$ varying.

Figure 2:

Left: BGEV PDF, $f_{\rm BGEV}(x;2,\mu,1,2)$, with $\mu$ varying; Right: BGEV
PDF, $f_{\rm BGEV}(x;0.5,\mu,1,2)$, with $\mu$ varying.

Figure 3:

Top left and right: BGEV PDF, $f_{\rm BGEV}(x;-0.25,0,1,\delta)$ with $\delta$
varying; Bottom left and right: BGEV PDF, $f_{\rm BGEV}(x;-1,0,1,\delta)$ with
$\delta$ varying.

Figure 4:

Top left and right: BGEV PDF, $f_{\rm BGEV}(x;0.5,0,1,\delta)$ with $\delta$
varying; Bottom left and right: BGEV PDF, $f_{\rm BGEV}(x;1.5,0,1,\delta)$
with $\delta$ varying.

##  5  Maximum likelihood estimation

In this section we present the expressions to be calculated to obtain the
maximum likelihood estimators for the parameters of a BGEV distribution.

Let $X\sim F_{\rm BGEV}$ be a random variable with PDF $f_{\rm
BGEV}(x;\Theta)$ defined in (2), where $\Theta=(\xi,\mu,\sigma,\delta)$. Let
$(X_{1},\dots,X_{n})$ be a random sample of X and $x=(x_{1},\dots,x_{n})$ the
corresponding observed values of the random sample $(X_{1},\dots,X_{n})$. The
log-likelihood function for $\Theta=(\xi,\mu,\sigma,\delta)$ is given by

$\displaystyle\ell_{n}(\Theta;x)=n\ln\sigma+n\ln(\delta+1)+\sum_{i=1}^{n}\Big{[}\delta\ln|x_{i}|-\Big{(}1+{1\over\xi}\Big{)}\ln\Psi_{i}(\Theta)-\Psi_{i}^{-1/\xi}(\Theta)\Big{]},$
(5.1)

where $\Psi_{i}(\Theta)=1+\xi\big{(}\sigma x_{i}|x_{i}|^{\delta}-\mu\big{)},\
i=1,\ldots,n.$ Note that, for all $i=1,\ldots,n$,

$\displaystyle\begin{array}[]{lllll}&\frac{\partial\Psi_{i}(\Theta)}{\partial\mu}=-\xi,&\frac{\partial\Psi_{i}(\Theta)}{\partial\sigma}=\xi
x_{i}|x_{i}|^{\delta},\\\\[5.69046pt]
&\frac{\partial\Psi_{i}(\Theta)}{\partial\delta}=\xi\sigma
x_{i}|x_{i}|^{\delta}\ln|x_{i}|,&\frac{\partial\Psi_{i}(\Theta)}{\partial\xi}=\sigma
x_{i}|x_{i}|^{\delta}-\mu.\end{array}$

Then, the maximum likelihood estimates of $\mu,\sigma,\delta,\xi$ are the
solutions of the following system of equations

$\displaystyle\begin{array}[]{llllll}\frac{\partial\ell_{n}(\Theta;x)}{\partial\mu}&=&\mu\sum_{i=1}^{n}\Omega_{i}(\Theta)=0,\\\\[5.69046pt]
\frac{\partial\ell_{n}(\Theta;x)}{\partial\sigma}&=&\frac{n}{\sigma}-\sum_{i=1}^{n}x_{i}|x_{i}|^{\delta}\Omega_{i}(\Theta)=0,\\\\[5.69046pt]
\frac{\partial\ell_{n}(\Theta;x)}{\partial\delta}&=&\frac{n}{\delta+1}+\sum_{i=1}^{n}[\ln|x_{i}|+\sigma
x_{i}|x_{i}|^{\delta}\ln|x_{i}|\,\Omega_{i}(\Theta)]=0,\\\\[5.69046pt]
\frac{\partial\ell_{n}(\Theta;x)}{\partial\xi}&=&\xi^{-2}\sum_{i=1}^{n}(\sigma
x_{i}|x_{i}|^{\delta}-\mu)\left\\{\ln\Psi_{i}(\Theta)-\xi\Omega_{i}(\Theta)\right\\}=0,\end{array}$
(5.6)

where
$\Omega_{i}(\Theta)=\Psi_{i}^{-1}(\Theta)[1+\xi-\Psi_{i}^{-{1}/{\xi}}(\Theta)],\
i=1,\ldots,n.$ Since

$\displaystyle\begin{array}[]{lllll}{\partial\Omega_{i}(\Theta)\over\partial\theta}=-\xi^{-1}\Psi_{i}^{-1}(\Theta)\big{[}\xi\Omega_{i}^{-1}(\Theta)-\Psi_{i}^{-1/\xi}(\Theta)\big{]}\frac{\partial\Psi_{i}(\Theta)}{\partial\theta},\quad\theta\in\\{\xi,\mu,\sigma,\delta\\},\end{array}$

the second-order partial derivatives of the log-likelihood function
$\ell_{n}(\Theta;x)$ are given by

$\displaystyle\begin{array}[]{lllll}\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\mu^{2}}=\sum_{i=1}^{n}\\{\Omega_{i}(\Theta)+\mu\Psi^{-1}_{i}(\Theta)[\xi\Omega_{i}(\Theta)-\Psi_{i}^{-{1}/{\xi}}(\Theta)]\\},\\\\[5.69046pt]
\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\sigma^{2}}=-\frac{n}{\sigma^{2}}+\sum_{i=1}^{n}x_{i}^{2}|x_{i}|^{2\delta}\Psi^{-1}_{i}(\Theta)[\xi\Omega_{i}(\Theta)-\Psi_{i}^{-{1}/{\xi}}(\Theta)],\\\\[5.69046pt]
\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\delta^{2}}=-\frac{n}{(\delta+1)^{2}}+\sigma\sum_{i=1}^{n}x_{i}|x_{i}|^{2\delta}\left(\ln|x_{i}|\right)^{2}\big{\\{}\Omega_{i}(\Theta)-\sigma
x_{i}\Psi^{-1}_{i}(\Theta)[\xi\Omega_{i}(\Theta)-\Psi_{i}^{-{1}/{\xi}}(\Theta)]\big{\\}},\\\\[5.69046pt]
\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\xi^{2}}=-2\xi^{-3}\sum_{i=1}^{n}(\sigma
x_{i}|x_{i}|^{\delta}-\mu)[\ln\Psi_{i}(\Theta)-\xi\Omega_{i}(\Theta)]\\\\[5.69046pt]
\qquad\qquad\quad\ -\xi^{-2}\sum_{i=1}^{n}(\sigma
x_{i}|x_{i}|^{\delta}-\mu)^{2}\big{\\{}\Omega_{i}(\Theta)-\Psi^{-1}_{i}(\Theta)[\xi\Omega_{i}(\Theta)-\Psi_{i}^{-{1}/{\xi}}(\Theta)+1]\big{\\}},\end{array}$
(5.12)

and, by Schwarz’s theorem or Clairaut’s theorem on equality of mixed partials
(James 1966), the second-order mixed derivatives of $\ell_{n}(\Theta;x)$ can
be written as

$\displaystyle\begin{array}[]{lllll}\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\mu\partial\sigma}=\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\sigma\partial\mu}=-\mu\sum_{i=1}^{n}x_{i}|x_{i}|^{\delta}\Psi^{-1}_{i}(\Theta)[\xi\Omega_{i}(\Theta)-\Psi_{i}^{-{1}/{\xi}}(\Theta)],\\\\[5.69046pt]
\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\mu\partial\delta}=\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\delta\partial\mu}=-\mu\sigma\sum_{i=1}^{n}x_{i}|x_{i}|^{\delta}\ln|x_{i}|\Psi^{-1}_{i}(\Theta)[\xi\Omega_{i}(\Theta)-\Psi_{i}^{-{1}/{\xi}}(\Theta)],\\\\[5.69046pt]
\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\mu\partial\xi}=\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\xi\partial\mu}=-\mu\xi^{-1}\sum_{i=1}^{n}(\sigma
x_{i}|x_{i}|^{\delta}-\mu)\Psi^{-1}_{i}(\Theta)[\xi\Omega_{i}(\Theta)-\Psi_{i}^{-{1}/{\xi}}(\Theta)],\\\\[5.69046pt]
\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\sigma\partial\delta}=\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\delta\partial\sigma}=-\sum_{i=1}^{n}x_{i}|x_{i}|^{2\delta}\ln|x_{i}|\big{\\{}\Omega_{i}(\Theta)-\sigma
x_{i}\Psi^{-1}_{i}(\Theta)[\xi\Omega_{i}(\Theta)-\Psi_{i}^{-{1}/{\xi}}(\Theta)]\big{\\}},\\\\[5.69046pt]
\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\sigma\partial\xi}=\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\xi\partial\sigma}=-\xi^{-1}\sum_{i=1}^{n}x_{i}|x_{i}|^{\delta}\big{(}\sigma
x_{i}|x_{i}|^{\delta}-\mu\big{)}\Psi^{-1}_{i}(\Theta)[\xi\Omega_{i}(\Theta)-\Psi_{i}^{-{1}/{\xi}}(\Theta)],\\\\[5.69046pt]
\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\delta\partial\xi}=\frac{\partial^{2}\ell_{n}(\Theta;x)}{\partial\xi\partial\delta}=-\sigma\xi^{-1}\sum_{i=1}^{n}x_{i}|x_{i}|^{\delta}\ln|x_{i}|\big{(}\sigma
x_{i}|x_{i}|^{\delta}-\mu\big{)}\Psi^{-1}_{i}(\Theta)[\xi\Omega_{i}(\Theta)-\Psi_{i}^{-{1}/{\xi}}(\Theta)].\end{array}$
(5.19)

If $X\sim F_{\rm BGEV}$, under certain regularity conditions, the Fisher
information matrix (FIM) may also be written as

$\displaystyle[\mathcal{I}(\Theta)]_{i,j}=-\mathbb{E}\biggl{[}\frac{\partial^{2}\ell_{1}(\Theta;X)}{\partial\theta_{i}\partial\theta_{j}}\,\bigg{|}\,\Theta\biggr{]},\quad\theta_{i},\theta_{j}\in\\{\xi,\mu,\sigma,\delta\\},$

where $\partial^{2}\ell_{1}(\Theta;x)/\partial\theta_{i}\partial\theta_{j}$
are given in (5.12) and (5.19) by taking $n=1$. It is clear that the
analytical calculations of the FIM above are difficult. But by using the Monte
Carlo estimates of the Hessian of the negative log-likelihood function we can
form an average of these ones as an estimate of the FIM (Spall 2005).

##  6 Numerical Illustrations

The performance of maximum likelihood estimators calculated was tested by
Monte Carlo simulation with 45 combinations of $\xi,\mu$ and $\delta$. We
consider the scale parameter always equal to 1; $\sigma=1$.

We use the following procedure:
(i) Generate random samples of $X\sim F_{\rm
BGEV}(\cdot;\xi,\mu,\sigma,\delta)$ by using the quantile function (3.6). Were
generated M=100 random samples (replications) of size N=50, 100, 250, 1000,
for each choice of $\theta$.
(ii) Calculate the maximum likelihood estimates (MLEs) of
$\theta=(\xi,\mu,\delta)$ using the equations (5.1)- (5.6). Here, use the
Nelder and Mead’ optimization method implemented in the R statistical program
(Nelder and Mead 1965).
(iii) For the initial value, consider the real parameter value added by a
value of a auniform variable in $(0,1)$.

The estimation results are found in Tables 1 \- 5. We have five different
configurations. In configuration 1, the shape parameters $\xi=1$,
$\mu\in\\{1,0,-1\\}$ and $\delta\in\\{0,2,4\\}$. Its mean estimates, the bias
and the mean square error (MSE) of the estimates are present in Table 1. In
configuration 2, $\xi=0.5$, $\mu\in\\{1,0,-1\\}$ and $\delta\in\\{0,2,4\\}$.
For this configuration, the mean estimates, the bias and the MSE of the
estimates are present in Table 2. In cofigurations 3, 4, and 5,
$\mu\in\\{1,0,-1\\}$, $\delta\in\\{0,2,4\\}$ and $\xi=0.25,\ -0.25$, and
$-0.5$, respectively. The mean estimates, the bias and the MSE of the
estimates for these three configurations are shown in the Tables 3, 4, and 5.
In the five configurations, the bias and MSE of the mean estimates are small.
That is, the algorithm has obtained good estimates. The bias and the MSE
decreased as the sample size was increased from 50 to 1000.

Table 1: Empirical means, bias and mean squared errors (MSE) of the estimates
of the parameters for $(\xi,\sigma)=(1,1)$ and some values of $\mu,\delta$ and
$n$.

| Empirical means |  | Bias |  | MSE
---|---|---|---|---|---
$n$ | $\xi$ | $\widehat{\xi}$ | $\mu$ | $\widehat{\mu}$ | $\delta$ | $\widehat{\delta}$ |  | $\widehat{\xi}$ | $\widehat{\mu}$ | $\delta$ |  | $\widehat{\xi}$ | $\widehat{\mu}$ | $\widehat{\delta}$
| 1 | 1.018 |  $-$1 |  $-$1.017 | 0 | 0.036 |  |  $-$0.018 | 0.017 |  $-$0.036 |  | 0.049 | 0.0214 | 0.0171
| 1 | 1.038 |  |  $-$1.013 | 2 | 2.082 |  |  $-$0.038 | 0.013 |  $-$0.082 |  | 0.039 | 0.0206 | 0.1167
| 1 | 1.034 |  $-$1 |  $-$1.053 | 4 | 4.292 |  |  $-$0.034 | 0.053 |  $-$0.292 |  | 0.0449 | 0.0201 | 0.3219
50 | 1 | 1.085 | 0 | 0.007 | 0 | 0.048 |  |  $-$0.085 |  $-$0.007 |  $-$0.048 |  | 0.0527 | 0.0081 | 0.0229
| 1 | 1.085 | 0 | 0.009 | 2 | 2.126 |  |  $-$0.085 |  $-$0.009 |  $-$0.126 |  | 0.0433 | 0.007 | 0.1972
| 1 | 1.049 | 0 | 0.028 | 4 | 4.152 |  |  $-$0.049 |  $-$0.028 |  $-$0.152 |  | 0.0546 | 0.0078 | 0.4478
| 1 | 1.148 | 1 | 0.97 | 0 | 0.096 |  |  $-$0.148 | 0.03 |  $-$0.096 |  | 0.0797 | 0.029 | 0.0668
| 1 | 1.083 | 1 | 0.995 | 2 | 2.14 |  |  $-$0.083 | 0.005 |  $-$0.14 |  | 0.0867 | 0.0279 | 0.4675
| 1 | 1.125 | 1 | 0.973 | 4 | 4.452 |  |  $-$0.125 | 0.027 |  $-$0.452 |  | 0.0954 | 0.0295 | 2.3334
| 1 | 1.018 |  $-$1 |  $-$1.002 | 0 | 0.009 |  |  $-$0.018 | 0.002 |  $-$0.009 |  | 0.0195 | 0.0073 | 0.0069
| 1 | 1.037 |  $-$1 |  $-$1.007 | 2 | 2.025 |  |  $-$0.037 | 0.007 |  $-$0.025 |  | 0.0151 | 0.0103 | 0.0635
| 1 | 1.022 |  $-$1 |  $-$1.007 | 4 | 4.114 |  |  $-$0.022 | 0.007 |  $-$0.114 |  | 0.019 | 0.0092 | 0.1591
100 | 1 | 1.038 | 0 | 0.005 | 0 | 0.024 |  |  $-$0.038 |  $-$0.005 |  $-$0.024 |  | 0.0189 | 0.0027 | 0.0086
| 1 | 1.07 | 0 |  $-$0.005 | 2 | 2.079 |  |  $-$0.07 | 0.005 |  $-$0.079 |  | 0.021 | 0.0034 | 0.089
| 1 | 1.029 | 0 | 0.005 | 4 | 4.079 |  |  $-$0.029 |  $-$0.005 |  $-$0.079 |  | 0.0177 | 0.0039 | 0.2877
| 1 | 1.08 | 1 | 0.97 | 0 | 0.071 |  |  $-$0.08 | 0.03 |  $-$0.071 |  | 0.034 | 0.0142 | 0.0295
| 1 | 1.076 | 1 | 0.978 | 2 | 2.199 |  |  $-$0.076 | 0.022 |  $-$0.199 |  | 0.037 | 0.0124 | 0.2379
| 1 | 1.036 | 1 | 0.995 | 4 | 4.199 |  | 0.036 | 0.052 |  $-$0.159 |  | 0.037 | 0.0124 | 0.0237
| 1 | 0.999 |  $-$1 |  $-$1.003 | 0 | 0.003 |  | 0.013 | 0.001 | 0.003 |  |  $-$0.013 | 0.0048 | 0.0038
| 1 | 1.006 |  $-$1 |  $-$1.002 | 2 | 2.026 |  |  $-$0.006 | 0.002 |  $-$0.026 |  | 0.0046 | 0.0029 | 0.0243
| 1 | 1.018 |  $-$1 |  $-$1.01 | 4 | 4.051 |  |  $-$0.018 | 0.01 |  $-$0.051 |  | 0.0069 | 0.0036 | 0.0744
250 | 1 | 1.028 | 0 |  $-$0.001 | 0 | 0.019 |  |  $-$0.028 | 0.001 |  $-$0.019 |  | 0.0065 | 0.0011 | 0.0042
| 1 | 1.031 | 0 | 0.001 | 2 | 2.043 |  |  $-$0.031 |  $-$0.001 |  $-$0.043 |  | 0.0078 | 0.0012 | 0.0318
| 1 | 1.016 | 0 | 0.001 | 4 | 4.065 |  |  $-$0.016 |  $-$0.001 |  $-$0.065 |  | 0.0068 | 0.0014 | 0.1125
| 1 | 1.044 | 1 | 0.983 | 4 | 0.039 |  |  $-$0.044 | 0.017 |  $-$0.039 |  | 0.0181 | 0.0079 | 0.0165
| 1 | 1.009 | 1 | 1.005 | 2 | 2.045 |  |  $-$0.009 |  $-$0.005 |  $-$0.045 |  | 0.0138 | 0.0067 | 0.1171
| 1 | 1.031 | 1 | 0.991 | 4 | 4.059 |  |  $-$0.031 | 0.009 |  $-$0.059 |  | 0.0125 | 0.0057 | 0.2744
| 1 | 1.005 | 1 | 0.996 | 0 | 0.012 |  |  $-$0.005 | 0.004 |  $-$0.012 |  | 0.0028 | 0.0014 | 0.0025
| 1 | 1.011 | 1 | 0.995 | 2 | 2.024 |  |  $-$0.011 | 0.005 |  $-$0.024 |  | 0.0025 | 0.0014 | 0.0213
| 1 | 1.002 | 1 | 1.001 | 4 | 3.998 |  |  $-$0.002 |  $-$0.001 | 0.002 |  | 0.0029 | 0.0014 | 0.0578
1000 | 1 | 1.01 | 0 | 0 | 0 | 0.005 |  |  $-$0.010 | 0.000 |  $-$0.005 |  | 0.002 | 4e$-$04 | 0.0010
| 1 | 1.011 | 0 |  $-$0.003 | 2 | 2.00 |  |  $-$0.011 | 0.003 |  $-$0.001 |  | 0.0019 | 4e$-$04 | 0.0064
| 1 | 1.008 | 0 | 0.001 | 4 | 4.039 |  |  $-$0.008 |  $-$0.001 |  $-$0.039 |  | 0.0015 | 4e$-$04 | 0.0216
| 1 | 1.001 |  $-$1 |  $-$1.006 | 0 | 0.006 |  |  $-$0.001 | 0.006 |  $-$0.006 |  | 0.0019 | 9e$-$04 | 6e$-$040
| 1 | 0.998 |  $-$1 |  $-$1.002 | 2 | 2.019 |  | 0.002 | 0.002 |  $-$0.019 |  | 0.0011 | 9e$-$04 | 0.0059
| 1 | 1.001 |  $-$1 |  $-$1.001 | 4 | 4.012 |  |  $-$0.001 | 0.001 |  $-$0.012 |  | 0.0014 | 6e$-$04 | 0.0140

Table 2: Empirical means, bias and mean squared errors (MSE) of the estimates
of the parameters for $(\xi,\sigma)=(0.5,1)$ and some values of $\mu,\delta$
and $n$.

| Empirical means |  | Bias |  | MSE
---|---|---|---|---|---
$n$ | $\xi$ | $\widehat{\xi}$ | $\mu$ | $\widehat{\mu}$ | $\delta$ | $\widehat{\delta}$ |  | $\widehat{\xi}$ | $\widehat{\mu}$ | $\delta$ |  | $\widehat{\xi}$ | $\widehat{\mu}$ | $\widehat{\delta}$
| 0.5 | 0.501 |  $-$1 |  $-$0.999 | 0 | 0.032 |  |  $-$0.001 |  $-$0.001 |  $-$0.032 |  | 0.0233 | 0.024 | 0.0129
| 0.5 | 0.524 |  $-$1 |  $-$1.032 | 2 | 2.15 |  |  $-$0.024 | 0.032 |  $-$0.15 |  | 0.0268 | 0.025 | 0.1033
| 0.5 | 0.486 |  $-$1 |  $-$1.019 | 4 | 4.215 |  | 0.014 | 0.019 |  $-$0.215 |  | 0.0254 | 0.023 | 0.2411
50 | 0.5 | 0.549 | 0 | 0.064 | 0 |  $-$0.049 |  | 0.028 |  $-$0.064 | 0.0375 |  | 0.0154 | 0.0282 | 0321
| 0.5 | 0.562 | 0 | 0.015 | 2 | 2.118 |  |  $-$0.062 |  $-$0.015 |  $-$0.118 |  | 0.045 | 0.0175 | 0.1552
| 0.5 | 0.551 | 0 | 0.019 | 4 | 4.282 |  |  $-$0.051 |  $-$0.019 |  $-$0.282 |  | 0.0303 | 0.0124 | 0.4835
| 0.5 | 0.541 | 1 | 1.029 | 0 | 0.057 |  |  $-$0.041 |  $-$0.029 |  $-$0.057 |  | 0.0725 | 0\. 0255 | 0.0515
| 0.5 | 0.559 | 1 | 1.006 | 2 | 2.094 |  |  $-$0.059 |  $-$0.006 |  $-$0.094 |  | 0.0786 | 0.0254 | 0.3663
| 0.5 | 0.563 | 1 | 0.998 | 4 | 4.379 |  |  $-$0.063 | 0.002 |  $-$0.379 |  | 0.083 | 0.0238 | 1.7225
| 0.5 | 0.503 |  $-$1 |  $-$1.011 | 0 | 0.024 |  |  $-$0.003 | 0.011 |  $-$0.024 |  | 0.0113 | 0.0142 | 0.0068
| 0.5 | 0.503 |  $-$1 |  $-$1.02 | 2 | 2.09 |  |  $-$0.003 | 0.02 |  $-$0.09 |  | 0.0088 | 0.0128 | 0.0469
| 0.5 | 0.518 |  $-$1 |  $-$1.017 | 4 | 4.145 |  |  $-$0.018 | 0.017 |  $-$0.145 |  | 0.0094 | 0.011 | 0.1269
100 | 0.5 | 0.525 | 0 | 0.005 | 0 | 0.025 |  |  $-$0.025 |  $-$0.005 |  $-$0.025 |  | 0.0115 | 0.0052 | 0.0084
| 0.5 | 0.511 | 0 | 0.021 | 2 | 2.067 |  |  $-$0.011 |  $-$0.021 |  $-$0.067 |  | 0.0105 | 0.0065 | 0.0663
| 0.5 | 0.52 | 0 | 0.012 | 4 | 4.136 |  |  $-$0.02 |  $-$0.012 |  $-$0.136 |  | 0.0131 | 0.0057 | 0.2212
| 0.5 | 0.53 | 1 | 1.012 | 0 | 0.011 |  |  $-$0.03 |  $-$0.012 |  $-$0.011 |  | 0.0281 | 0.01 | 0.0195
| 0.5 | 0.53 | 1 | 0.997 | 2 | 2.102 |  |  $-$0.03 | 0.003 |  $-$0.102 |  | 0.0519 | 0.0125 | 0.3862
| 0.5 | 0.516 | 1 | 1.002 | 4 | 4.093 |  |  $-$0.016 |  $-$0.002 |  $-$0.093 |  | 0.024 | 0.0084 | 0.3418
| 0.5 | 0.513 |  $-$1 |  $-$1.002 | 0 | 0.011 |  | 0.002 |  $-$0.011 | 0.0042 |  | 0.0033 | 0.002 | 0.232
| 0.5 | 0.505 |  $-$1 |  $-$1.009 | 2 | 2.027 |  |  $-$0.005 | 0.009 |  $-$0.027 |  | 0.0051 | 0.004 | 0.0164
| 0.5 | 0.501 |  $-$1 |  $-$1.009 | 4 | 4.044 |  |  $-$0.001 | 0.009 |  $-$0.044 |  | 0.0042 | 0.0054 | 0.0571
250 | 0.5 | 0.511 | 0 | 0.009 | 0 | 0.01 |  |  $-$0.011 |  $-$0.009 |  $-$0.01 |  | 0.0033 | 0.0024 | 0.0035
| 0.5 | 0.505 | 0 | 0.008 | 2 | 2.035 |  |  $-$0.005 |  $-$0.008 |  $-$0.035 |  | 0.0053 | 0.0026 | 0.0315
| 0.5 | 0.514 | 0 |  $-$0.005 | 4 | 4.051 |  |  $-$0.014 | 0.005 |  $-$0.051 |  | 0.006 | 0.003 | 0.0654
| 0.5 | 0.537 | 1 | 0.988 | 0 | 0.023 |  |  $-$0.037 | 0.012 |  $-$0.023 |  | 0.0108 | 0.0045 | 0.0066
| 0.5 | 0.511 | 1 | 1.005 | 2 | 2.008 |  |  $-$0.011 |  $-$0.005 |  $-$0.008 |  | 0.0109 | 0.0047 | 0.0639
| 0.5 | 0.528 | 1 | 0.998 | 4 | 4.062 |  |  $-$0.028 | 0.002 |  $-$0.062 |  | 0.0108 | 0.0046 | 0.1906
| 0.5 | 0.493 | 1 | 1.003 | 0 |  $-$0.003 |  | 0.007 |  $-$0.003 | 0.003 |  | 0.0026 | 0.0015 | 0.0020
| 0.5 | 0.512 | 1 | 0.992 | 2 | 2.041 |  |  $-$0.012 | 0.008 |  $-$0.041 |  | 0.0023 | 0.0013 | 0.0167
| 0.5 | 0.506 | 1 | 0.999 | 4 | 4.023 |  |  $-$0.006 | 0.001 |  $-$0.023 |  | 0.0025 | 0.0011 | 0.0497
1000 | 0.5 | 0.507 | 0 |  $-$0.005 | 0 |  $-$0.001 |  |  $-$0.007 | 0.005 | 0.001 |  | 8e$-$04 | 6e$-$04 | 9e$-$04
| 0.5 | 0.501 | 0 | 0.003 | 2 | 2.006 |  |  $-$0.001 |  $-$0.003 |  $-$0.006 |  | 0.0013 | 7e$-$04 | 0.0061
| 0.5 | 0.505 | 0 |  $-$0.001 | 4 | 3.998 |  |  $-$0.005 | 0.001 | 0.002 |  | 9e$-$04 | 7e$-$04 | 0.0135
| 0.5 | 0.501 |  $-$1 |  $-$0.998 | 0 | 0 |  |  $-$0.001 |  $-$0.002 | 0.000 |  | 0.0011 | 0.0012 | 6e$-$04
| 0.5 | 0.505 |  $-$1 |  $-$1.001 | 2 | 2.004 |  |  $-$0.005 | 0.001 |  $-$0.004 |  | 9e$-$04 | 9e$-$04 | 0.0045
| 0.5 | 0.499 |  $-$1 |  $-$0.997 | 4 | 4.007 |  | 0.001 |  $-$0.003 |  $-$0.007 |  | 0.001 | 0.0013 | 0.0149

Table 3: Empirical means, bias and mean squared errors (MSE) of the estimates
of the parameters for $(\xi,\sigma)=(0.25,1)$ and some values of $\mu,\delta$
and $n$.

| Empirical means |  | Bias |  | MSE
---|---|---|---|---|---
$n$ | $\xi$ | $\widehat{\xi}$ | $\mu$ | $\widehat{\mu}$ | $\delta$ | $\widehat{\delta}$ |  | $\widehat{\xi}$ | $\widehat{\mu}$ | $\delta$ |  | $\widehat{\xi}$ | $\widehat{\mu}$ | $\widehat{\delta}$
| 0.25 | 0.256 |  $-$1 |  $-$1.008 | 0 | 0.04 |  |  $-$0.045 | 0.128 |  $-$0.04 |  | 0.0216 | 0.0322 | 0.0116
| 0.25 | 0.244 |  $-$1 |  $-$1.018 | 2 | 2.116 |  | 0.006 | 0.018 |  $-$0.116 |  | 0.0162 | 0.0323 | 0.0838
| 0.25 | 0.239 |  $-$1 |  $-$1.013 | 4 | 4.185 |  | 0.011 | 0.013 |  $-$0.185 |  | 0.0186 | 0.0262 | 0.2052
50 | 0.25 | 0.273 | 0 | 0.016 | 0 | 0.056 |  |  $-$0.023 |  $-$0.016 |  $-$0.056 |  | 0.0302 | 0.0155 | 0.0251
| 0.25 | 0.285 | 0 | 0.026 | 2 | 2.131 |  |  $-$0.035 |  $-$0.026 |  $-$0.131 |  | 0.035 | 0.0168 | 0.2199
| 0.25 | 0.275 | 0 | 0.02 | 4 | 4.213 |  |  $-$0.025 |  $-$0.02 |  $-$0.213 |  | 0.0247 | 0.0143 | 0.4635
| 0.25 | 0.267 | 1 | 1.001 | 0 | 0.037 |  |  $-$0.017 |  $-$0.001 |  $-$0.037 |  | 0.0803 | 0.029 | 0.0414
| 0.25 | 0.289 | 1 | 1.017 | 2 | 2.197 |  |  $-$0.039 |  $-$0.017 |  $-$0.197 |  | 0.0827 | 0.0283 | 0.5167
| 0.25 | 0.292 | 1 | 1.009 | 4 | 4.309 |  |  $-$0.042 |  $-$0.009 |  $-$0.309 |  | 0.0812 | 0.0226 | 1.3124
| 0.25 | 0.245 |  $-$1 |  $-$1.021 | 0 | 0.037 |  | 0.005 | 0.021 |  $-$0.037 |  | 0.0096 | 0.0199 | 0.0054
| 0.25 | 0.247 |  $-$1 |  $-$1.007 | 2 | 2.071 |  | 0.003 | 0.007 |  $-$0.071 |  | 0.0075 | 0.0157 | 0.048
| 0.25 | 0.241 |  $-$1 |  $-$1.004 | 4 | 4.045 |  | 0.009 | 0.004 |  $-$0.045 |  | 0.007 | 0.0114 | 0.1047
100 | 0.25 | 0.255 | 0 | 0.027 | 0 | 0.028 |  |  $-$0.005 |  $-$0.027 |  $-$0.028 |  | 0.0104 | 0.0074 | 0.007
| 0.25 | 0.262 | 0 | 0.01 | 2 | 2.067 |  |  $-$0.012 |  $-$0.01 |  $-$0.067 |  | 0.0107 | 0.0087 | 0.0579
| 0.25 | 0.263 | 0 | 0.007 | 4 | 4.115 |  |  $-$0.013 |  $-$0.007 |  $-$0.115 |  | 0.0101 | 0.0085 | 0.1793
| 0.25 | 0.249 | 1 | 1.009 | 0 | 0.011 |  | 0.001 |  $-$0.009 |  $-$0.011 |  | 0.025 | 0.0108 | 0.0124
| 0.25 | 0.272 | 1 | 1.098 | 2 | 2.058 |  |  $-$0.022 |  $-$0.058 | 0.0325 |  | 0.0117 | 0.1798 | 0.234
| 0.25 | 0.241 | 1 | 1.021 | 4 | 4.052 |  | 0.009 |  $-$0.021 |  $-$0.052 |  | 0.0232 | 0.0104 | 0.404
| 0.25 | 0.2255 |  $-$1 |  $-$0.994 | 0 | 0.007 |  |  $-$0.005 |  $-$0.006 |  $-$0.007 |  | 0.0022 | 0.0043 | 0.0017
| 0.25 | 0.243 |  $-$1 |  $-$1.003 | 2 | 2.053 |  | 0.007 | 0.003 |  $-$0.053 |  | 0.0026 | 0.0046 | 0.0186
| 0.25 | 0.254 |  $-$1 |  $-$1.014 | 4 | 4.072 |  |  $-$0.004 | 0.014 |  $-$0.072 |  | 0.003 | 0.0052 | 0.0436
250 | 0.25 | 0.267 | 0 | 0.002 | 0 | 0.02 |  |  $-$0.017 |  $-$0.002 |  $-$0.02 |  | 0.0028 | 0.0033 | 0.0032
| 0.25 | 0.255 | 0 | 0.005 | 2 | 2.01 |  |  $-$0.005 |  $-$0.005 |  $-$0.01 |  | 0.0034 | 0.0026 | 0.0229
| 0.25 | 0.249 | 0 | 0.001 | 4 | 4.038 |  | 0.001 |  $-$0.001 |  $-$0.038 |  | 0.0046 | 0.0031 | 0.0772
| 0.25 | 0.236 | 1 | 1.01 | 0 |  $-$0.003 |  | 0.014 |  $-$0.01 | 0.003 |  | 0.0079 | 0.005 | 0.0038
| 0.25 | 0.241 | 1 | 1.001 | 2 | 1.997 |  | 0.009 |  $-$0.001 | 0.003 |  | 0.0068 | 0.0037 | 0.0469
| 0.25 | 0.269 | 1 | 0.998 | 4 | 4.121 |  |  $-$0.019 | 0.002 |  $-$0.121 |  | 0.0087 | 0.0049 | 0.195
| 0.25 | 0.253 | 1 | 1.002 | 0 | 0.002 |  |  $-$0.003 |  $-$0.002 |  $-$0.002 |  | 0.0015 | 0.001 | 0.0010
| 0.25 | 0.25 | 1 | 0.996 | 2 | 2 |  | 0.000 | 0.004 | 0.000 |  | 0.002 | 9e$-$04 | 0.0132
| 0.25 | 0.256 | 1 | 0.996 | 4 | 4.018 |  |  $-$0.006 | 0.004 |  $-$0.018 |  | 0.0018 | 0.0013 | 0.0331
1000 | 0.25 | 0.254 | 0 |  $-$0.001 | 0 | 0.002 |  |  $-$0.004 | 0.001 |  $-$0.002 |  | 9e$-$04 | 8e$-$04 | 6e$-$04
| 0.25 | 0.249 | 0 | 0.004 | 2 | 2.013 |  | 0.001 |  $-$0.004 |  $-$0.013 |  | 8e$-$04 | 8e$-$04 | 0.0066
| 0.25 | 0.252 | 0 |  $-$0.002 | 4 | 4.012 |  |  $-$0.002 | 0.002 |  $-$0.012 |  | 8e$-$04 | 6e$-$04 | 0.0199
| 0.25 | 0.248 |  $-$1 |  $-$1.001 | 0 | 0.002 |  | 0.002 | 0.001 |  $-$0.002 |  | 6e$-$04 | 0.0014 | 5e$-$04
| 0.25 | 0.25 |  $-$1 |  $-$0.998 | 2 | 2.004 |  | 0.000 |  $-$0.002 |  $-$0.004 |  | 7e$-$04 | 0.0011 | 0.0037
| 0.25 | 0.253 |  $-$1 |  $-$1.002 | 4 | 4.015 |  |  $-$0.003 | 0.002 |  $-$0.015 |  | 7e$-$04 | 0.001 | 0.0106

Table 4: Empirical means, bias and mean squared errors (MSE) of the estimates
of the parameters for $(\xi,\sigma)=(-0.25,1)$ and some values of $\mu,\delta$
and $n$.

| Empirical means |  | Bias |  | MSE
---|---|---|---|---|---
$n$ | $\xi$ | $\widehat{\xi}$ | $\mu$ | $\widehat{\mu}$ | $\delta$ | $\widehat{\delta}$ |  | $\widehat{\xi}$ | $\widehat{\mu}$ | $\delta$ |  | $\widehat{\xi}$ | $\widehat{\mu}$ | $\widehat{\delta}$
|  $-$0.25 |  $-$0.266 |  $-$1 |  $-$0.981 | 0 | 0.055 |  | 0.016 |  $-$0.019 |  $-$0.055 |  | 0.0071 | 0.0269 | 0.0197
|  $-$0.25 |  $-$0.288 |  $-$1 |  $-$1.012 | 2 | 2.151 |  | 0.038 | 0.012 |  $-$0.151 |  | 0.0082 | 0.0279 | 0.0849
|  $-$0.25 |  $-$0.296 |  $-$1 |  $-$0.997 | 4 | 4.288 |  | 0.046 |  $-$0.003 |  $-$0.288 |  | 0.0093 | 0.0259 | 0.2736
50 |  $-$0.25 |  $-$0.286 | 0 | 0.002 | 0 | 0.024 |  | 0.036 |  $-$0.002 |  $-$0.024 |  | 0.0159 | 0.0298 | 0.018
|  $-$0.25 |  $-$0.266 | 0 | 0.003 | 2 | 2.094 |  | 0.016 |  $-$0.003 |  $-$0.094 |  | 0.0125 | 0.0243 | 0.1833
|  $-$0.25 |  $-$0.283 | 0 | 0.031 | 4 | 4.079 |  | 0.033 |  $-$0.031 |  $-$0.079 |  | 0.0163 | 0.0252 | 0.439
|  $-$0.25 |  $-$0.256 | 1 | 1.01 | 0 | 0.028 |  | 0.006 |  $-$0.01 |  $-$0.028 |  | 0.0326 | 0.0312 | 0.0211
|  $-$0.25 |  $-$0.278 | 1 | 1.03 | 2 | 2.119 |  | 0.028 |  $-$0.03 |  $-$0.119 |  | 0.0332 | 0.0224 | 0.2411
|  $-$0.25 |  $-$0.257 | 1 | 1.037 | 4 | 4.165 |  | 0.007 |  $-$0.037 |  $-$0.165 |  | 0.0374 | 0.0359 | 0.6435
|  $-$0.25 |  $-$0.264 |  $-$1 |  $-$1.006 | 0 | 0.027 |  | 0.014 | 0.006 |  $-$0.027 |  | 0.0033 | 0.0149 | 0.0051
|  $-$0.25 |  $-$0.264 |  $-$1 |  $-$0.992 | 2 | 2.056 |  | 0.014 |  $-$0.008 |  $-$0.056 |  | 0.0032 | 0.0169 | 0.0435
|  $-$0.25 |  $-$0.272 |  $-$1 |  $-$1.002 | 4 | 4.156 |  | 0.022 | 0.002 |  $-$0.156 |  | 0.0041 | 0.0116 | 0.134
100 |  $-$0.25 |  $-$0.258 | 0 | 0.002 | 0 | 0.028 |  | 0.008 |  $-$0.002 |  $-$0.028 |  | 0.0053 | 0.0138 | 0.0095
|  $-$0.25 |  $-$0.271 | 0 | 0.015 | 2 | 2.02 |  | 0.021 |  $-$0.015 |  $-$0.02 |  | 0.0056 | 0.014 | 0.0709
|  $-$0.25 |  $-$0.255 | 0 |  $-$0.001 | 4 | 4.095 |  | 0.005 | 0.001 |  $-$0.095 |  | 0.005 | 0.0118 | 0.2409
|  $-$0.25 |  $-$0.256 | 1 | 1.018 | 0 | 0.004 |  | 0.006 |  $-$0.018 |  $-$0.004 |  | 0.0115 | 0.0118 | 0.0093
|  $-$0.25 |  $-$0.275 | 1 | 1.026 | 2 | 2.007 |  | 0.025 |  $-$0.026 |  $-$0.007 |  | 0.0082 | 0.013 | 0.0599
|  $-$0.25 |  $-$0.274 | 1 | 1.015 | 4 | 4.02 |  | 0.024 |  $-$0.015 |  $-$0.02 |  | 0.0103 | 0.0131 | 0.234
|  $-$0.25 |  $-$0.261 |  $-$1 |  $-$0.994 | 0 | 0.012 |  | 0.011 |  $-$0.006 |  $-$0.012 |  | 9e$-$04 | 0.0041 | 0.0014
|  $-$0.25 |  $-$0.255 |  $-$1 |  $-$0.985 | 2 | 2.03 |  | 0.005 |  $-$0.015 |  $-$0.03 |  | 0.0011 | 0.0043 | 0.0166
|  $-$0.25 |  $-$0.259 |  $-$1 |  $-$1.007 | 4 | 4.037 |  | 0.009 | 0.007 |  $-$0.037 |  | 0.0012 | 0.0058 | 0.0458
250 |  $-$0.25 |  $-$0.257 | 0 | 0.012 | 0 | 0.007 |  | 0.007 |  $-$0.012 | 0.0018 |  | 0.0045 | 0.0108 | 0.0769
|  $-$0.25 |  $-$0.257 | 0 | 0.014 | 2 | 2.03 |  | 0.007 |  $-$0.014 |  $-$0.03 |  | 0.0021 | 0.0052 | 0.027
|  $-$0.25 |  $-$0.258 | 0 | 0.014 | 4 | 4.023 |  | 0.008 |  $-$0.014 |  $-$0.023 |  | 0.0022 | 0.0067 | 0.1011
|  $-$0.25 |  $-$0.258 | 1 | 1.012 | 0 | 0.006 |  | 0.008 |  $-$0.012 |  $-$0.006 |  | 0.0044 | 0.0046 | 0.0035
|  $-$0.25 |  $-$0.268 | 1 | 0.997 | 2 | 1.974 |  | 0.018 | 0.003 | 0.026 |  | 0.0031 | 0.0038 | 0.026
|  $-$0.25 |  $-$0.256 | 1 | 1.007 | 4 | 4.003 |  | 0.006 |  $-$0.007 |  $-$0.003 |  | 0.0035 | 0.0345 | 0.0042
|  $-$0.25 |  $-$0.257 | 1 | 1.003 | 0 |  $-$0.002 |  | 0.007 |  $-$0.003 | 0.002 |  | 9e$-$04 | 0.0013 | 9e$-$04
|  $-$0.25 |  $-$0.248 | 1 | 1.003 | 2 | 2.01 |  |  $-$0.002 |  $-$0.003 |  $-$0.010 |  | 8e$-$04 | 0.0013 | 0.0061
|  $-$0.25 |  $-$0.253 | 1 | 1 | 4 | 4.003 |  | 0.003 | 0.000 |  $-$0.003 |  | 0.001 | 0.0012 | 0.0228
1000 |  $-$0.25 |  $-$0.251 | 0 | 0 | 0 | 0.002 |  | 0.001 | 0.000 |  $-$0.002 |  | 4e$-$04 | 0.0013 | 7e$-$04
|  $-$0.25 |  $-$0.252 | 0 | 0 | 2 | 2.004 |  | 0.002 | 0.000 |  $-$0.004 |  | 5e$-$04 | 0.0012 | 0.0087
|  $-$0.25 |  $-$0.251 | 0 | 0.001 | 4 | 4.003 |  | 0.001 |  $-$0.001 |  $-$0.003 |  | 4e$-$04 | 0.0011 | 0.0176
|  $-$0.25 |  $-$0.25 |  $-$1 |  $-$1.003 | 0 | 0.003 |  | 0.000 | 0.003 |  $-$0.003 |  | 2e$-$04 | 0.003 | 0.0013
|  $-$0.25 |  $-$0.253 |  $-$1 |  $-$1 | 2 | 1.998 |  | 0.003 | 0.000 | 0.002 |  | 3e$-$04 | 0.0016 | 0.0033
|  $-$0.25 |  $-$0.254 |  $-$1 |  $-$0.997 | 4 | 4.021 |  | 0.004 |  $-$0.003 |  $-$0.021 |  | 2e$-$04 | 0.0012 | 0.0088

Table 5: Empirical means, bias and mean squared errors (MSE) of the estimates
of the parameters for $(\xi,\sigma)=(-0.5,1)$ and some values of $\mu,\delta$
and $n$.

| Empirical means |  | Bias |  | MSE
---|---|---|---|---|---
$n$ | $\xi$ | $\widehat{\xi}$ | $\mu$ | $\widehat{\mu}$ | $\delta$ | $\widehat{\delta}$ |  | $\widehat{\xi}$ | $\widehat{\mu}$ | $\widehat{\delta}$ |  | $\widehat{\xi}$ | $\widehat{\mu}$ | $\widehat{\delta}$
|  $-$0.5 |  $-$0.553 |  $-$1 |  $-$0.955 | 0 | 0.044 |  | 0.053 |  $-$0.045 |  $-$0.044 |  | 0.0102 | 0.0305 | 0.0123
|  $-$0.5 |  $-$0.563 |  $-$1 |  $-$0.948 | 2 | 2.229 |  | 0.063 |  $-$0.052 |  $-$0.229 |  | 0.0129 | 0.0319 | 0.1927
|  $-$0.5 |  $-$0.579 |  $-$1 |  $-$0.952 | 4 | 4.25 |  | 0.079 |  $-$0.048 |  $-$0.25 |  | 0.0077 | 0.0209 | 0.3505
$50$ |  $-$0.5 |  $-$0.545 | 0 | 0.04 | 0 | 0.017 |  | 0.045 |  $-$0.04 |  $-$0.017 |  | 0.012 | 0.0259 | 0.0197
|  $-$0.5 |  $-$0.565 | 0 | 0.032 | 2 | 2.005 |  | 0.065 |  $-$0.032 |  $-$0.005 |  | 0.0112 | 0.0224 | 0.1354
|  $-$0.5 |  $-$0.553 |  $-$1 |  $-$0.553 | 0 | 0.0434 |  | 4.069 | 0.053 |  $-$0.043 |  |  $-$0.069 | 0.0127 | 0.0275
|  $-$0.5 |  $-$0.562 | 1 | 1.021 | 0 |  $-$0.012 |  | 0.062 |  $-$0.021 | 0.012 |  | 0.0258 | 0.0248 | 0.0211
|  $-$0.5 |  $-$0.532 | 1 | 1.025 | 2 | 2.067 |  | 0.032 |  $-$0.025 |  $-$0.067 |  | 0.0247 | 0.0286 | 0.1658
|  $-$0.5 |  $-$0.536 |  $-$1 |  $-$0.964 | 0 | 0.025 |  | 0.036 |  $-$0.036 |  $-$0.025 |  | 0.0036 | 0.0131 | 0.0052
|  $-$0.5 |  $-$0.528 |  $-$1 |  $-$0.982 | 2 | 2.076 |  | 0.028 |  $-$0.018 |  $-$0.076 |  | 0.003 | 0.0114 | 0.0457
|  $-$0.5 |  $-$0.531 |  $-$1 |  $-$0.984 | 4 | 4.182 |  | 0.031 |  $-$0.016 |  $-$0.182 |  | 0.0027 | 0.0093 | 0.1574
$100$ |  $-$0.5 |  $-$0.523 | 0 | 0.015 | 0 | 0.015 |  | 0.023 |  $-$0.015 |  $-$0.015 |  | 0.004 | 0.0113 | 0.008
|  $-$0.5 |  $-$0.532 | 0 | 0.026 | 2 | 2.053 |  | 0.032 |  $-$0.026 |  $-$0.053 |  | 0.0042 | 0.011 | 0.0643
|  $-$0.5 |  $-$0.513 | 0 | 0.008 | 4 | 4.153 |  | 0.013 |  $-$0.008 |  $-$0.153 |  | 0.0042 | 0.0101 | 0.2634
|  $-$0.5 |  $-$0.516 | 1 | 1.01 | 0 | 0.016 |  | 0.016 |  $-$0.01 |  $-$0.016 |  | 0.011 | 0.0113 | 0.0102
|  $-$0.5 |  $-$0.532 | 1 | 1.008 | 2 | 1.988 |  | 0.032 |  $-$0.008 | 0.012 |  | 0.0095 | 0.0116 | 0.0731
|  $-$0.5 |  $-$0.527 | 1 | 0.983 | 4 | 3.955 |  | 0.027 | 0.017 | 0.045 |  | 0.0089 | 0.0103 | 0.1963
|  $-$0.5 |  $-$0.516 |  $-$1 |  $-$0.985 | 0 | 0.018 |  | 0.016 |  $-$0.015 |  $-$0.018 |  | 7e$-$04 | 0.0033 | 0.002
|  $-$0.5 |  $-$0.516 |  $-$1 |  $-$0.982 | 2 | 2.049 |  | 0.016 |  $-$0.018 |  $-$0.049 |  | 0.0012 | 0.0096 | 0.0346
|  $-$0.5 |  $-$0.51 |  $-$1 |  $-$1.004 | 4 | 4.096 |  | 0.01 | 0.004 |  $-$0.096 |  | 8e$-$04 | 0.0035 | 0.0767
$250$ |  $-$0.5 |  $-$0.514 | 0 | 0.011 | 0 |  $-$0.003 |  | 0.014 |  $-$0.011 | 0.003 |  | 0.0012 | 0.0033 | 0.003
|  $-$0.5 |  $-$0.51 | 0 | 0.007 | 2 | 2.025 |  | 0.01 |  $-$0.007 |  $-$0.025 |  | 0.0015 | 0.0042 | 0.0352
|  $-$0.5 |  $-$0.509 | 0 |  $-$0.001 | 4 | 4.009 |  | 0.009 | 0.001 |  $-$0.009 |  | 0.0012 | 0.0038 | 0.0823
|  $-$0.5 |  $-$0.51 | 1 | 1.011 | 0 | 0.006 |  | 0.01 |  $-$0.011 |  $-$0.006 |  | 0.0031 | 0.0068 | 0.0031
|  $-$0.5 |  $-$0.513 | 1 | 0.999 | 2 | 1.99 |  | 0.013 | 0.001 | 0.01 |  | 0.0028 | 0.004 | 0.0283
|  $-$0.5 |  $-$0.511 | 1 | 1.007 | 4 | 3.995 |  | 0.011 |  $-$0.007 | 0.005 |  | 0.0033 | 0.0061 | 0.0712
|  $-$0.5 |  $-$0.505 | 1 | 1.006 | 0 | 0 |  | 0.005 |  $-$0.006 | 0.000 |  | 6e$-$04 | 0.0012 | 8e$-$04
|  $-$0.5 |  $-$0.502 | 1 | 1.003 | 2 | 2.002 |  | 0.002 |  $-$0.003 |  $-$0.002 |  | 5e$-$04 | 0.0014 | 0.0063
|  $-$0.5 |  $-$0.505 | 1 | 1 | 4 | 4 |  | 0.005 | 0.000 | 0.000 |  | 7e$-$04 | 0.0012 | 0.0226
$1000$ |  $-$0.5 |  $-$0.501 | 0 | 0.009 | 0 | 0.017 |  | 0.001 |  $-$0.009 |  $-$0.017 |  | 0.001 | 0.0048 | 0.0114
|  $-$0.5 |  $-$0.505 | 0 | 0.005 | 2 | 2.01 |  | 0.005 |  $-$0.005 |  $-$0.010 |  | 3e$-$04 | 0.0013 | 0.0065
|  $-$0.5 |  $-$0.505 | 0 | 0.004 | 4 | 4.005 |  | 0.005 |  $-$0.004 |  $-$0.005 |  | 4e$-$04 | 0.0011 | 0.0223
|  $-$0.5 |  $-$0.505 |  $-$1 |  $-$0.985 | 0 | 0.023 |  | 0.005 |  $-$0.015 |  $-$0.023 |  | 4e$-$04 | 0.0049 | 0.0152
|  $-$0.5 |  $-$0.503 |  $-$1 |  $-$1 | 2 | 2.02 |  | 0.003 | 0.000 |  $-$0.020 |  | 2e$-$04 | 0.0026 | 0.0204
|  $-$0.5 |  $-$0.505 |  $-$1 |  $-$0.996 | 4 | 4.006 |  | 0.005 |  $-$0.004 |  $-$0.006 |  | 1e$-$04 | 7e$-$04 | 0.0114

##  7 Applications to environmental data

In this section, to illustrate the applicability of the model proposed and its
advantages, the BGEV distribution was fitted to two data sets of the climate
of the Federal District in Brazil. Furthermore, we compared the BGEV
distribution with the GEV distribution. We adopt the MLE method (as discussed
in Sect. 5) and all the computations were done using the R software ( R-Team
2020). For the parameters estimates of the GEV distribution we use the
fExtremes package (Wuertz et al. 2017).

The data series employed were taken from the of INMET (National Institute of
Meteorology) at www.inmet.gov.br.DataStations-AutomaticStations. The data here
utilized correspond to the wind speed during the period 10/12/2018 to
10/12/2019 and the temperature at the oval point during the period 24/10/2018
to 20/10/2019. We used the maximum block technique with blocks of size 24
based in daily maximum value. Furthermore, the transformation
$(x-\rm{Mean})/SD$, where SD is the standard deviation of the data, is to used
to adjust it in the BGEV distribution. We applied Ljung–Box’s test (Trapletti
2016) for verifying the null hypothesis of serial independence of the data
sets. The test statistics did not reject the null hypothesis at the
significance level of $1\%$. The descriptive statistics are given in Table 6.
Finally, from Fig. 6(b), we can see that the data set has bimodality. More
information about the applications can be found in Paiva (2020)

Figure 5:

(a) Temperature data set (b) Wind speed data set

Histogram of the two data sets.

Table 6: Descriptive statistics for the standardized data sets.

| Minimum | 1º Quartile | Median | Mean | 3º Quartile | Maximum
---|---|---|---|---|---|---
Wind speed |  $-$2.26 |  $-$1.40 |  $-$1.08 |  $-$0.75 |  $-$0.38 | 1.54
Temperature at the oval point |  $-$3.44 |  $-$0.43 | 0.71 | 0.32 | 1.12 | 1.88

The MLEs of the parameters, the value of $-2\ell(\Theta)$, the Kolmogorov-
Smirnov (KS) and Anderson-Darling (AD) statistics for the BGEV and GEV models
are listed in Table 7. In general, the smaller the values of these statistics,
the better the fit. The graphical study performed in Sect. 4 helped in
choosing of the initial values of the estimates. For the wind speed the
initial values were $\mu=0,\sigma=1,\xi=-0.5$ and $\delta=0.5$ and for the
temperature at the oval point were $\mu=0,\sigma=1,\xi=-0.25$ and
$\delta=0.5$. Since the values are smaller for the BGEV distribution compared
with those values of the other models, the new distribution seems to be a very
competitive model for these data sets. These results illustrate the
potentiality of the BGEV model and the importance of the additional parameter.

Table 7: MLEs of the model parameters and the statistics KS, AD and $-2\ell(\Theta)$ for the two data sets. Estimates |  | Statistic
---|---|---
Wind speed | $\widehat{\mu}$ | $\widehat{\sigma}$ | $\widehat{\xi}$ | $\widehat{\delta}$ |  | KS | AD | $-2\ell(\Theta)$
BGEV |  $-$1.0430 | 0.6516 |  $-$0.3042 | 0.9434 |  | 0.05396 | 47.802 | 22204.9
GEV |  $-$0.4957 | 0.6853 | 0.1330 | 0 |  | 0.24109 | 45.179 | 22214.6
Temperature | $\widehat{\mu}$ | $\widehat{\sigma}$ | $\widehat{\xi}$ | $\widehat{\delta}$ |  | KS | AD | $-2\ell(\Theta)$
BGEV | 0.1804 | 0.8197 |  $-$0.4839 | 0.4201 |  | 0.0738 | 66.297 |
GEV |  $-$0.1955 | 1.0599 |  $-$0.6045 | 0 |  | 0.3261 | 70.706 | 22111.0

The fitted density (for the BGEV distribution) for the first and second data
sets are displayed in Figures 6 and 7 (together with the QQ plots),
respectively. These results illustrate the potentiality of the BGEV
distribution and the importance of the additional parameter.

Figure 6:

Histogram with estimated PDF (first panel) and QQ plots (second panel) for the
BGEV distribution based on the Wind speed data set.

Figure 7:

Histogram with estimated PDF (first panel) and QQ plots (second panel) for the
BGEV distribution based on the Temperature data set.

##  8 Conclusions

When extreme value data show bimodality, despite its broad sense applicability
in many fields, the GEV distribution is not suitable. In this article, we
propose a generalization of the GEV distribution, called BGEV distribution,
with an additional parameter which modifies the behavior of the distribution,
composing as a alternative models for single maxima events. The GEV
distribution appears as a particular case. We present important properties of
the model and through graphical studies we have shown its wide flexibility.
The good performance of the MLEs of the parameters was tested via Monte Carlo
simulation. Applications of the BGEV distribution for two extreme data sets
show that the new distribution can be used to provide better adjustments than
the GEV to model when the data show bimodality. We hope this new distribution
may attract wider applications for extreme values analysis.

## References

  * [1] Aryal RG, Tsokos PC (2011). Transmuted Weibull Distribution: A Generalization of the Weibull Probability Distribution. European Journal of Pure and Applied Mathematics 4(2): 89-102.
  * [2] Beirlant J, Goegebeur Y, Segers J, Teugels J (2004) Statistics of Extreme Theory and Applications. John Wiley $\&$ Sons, Hoboken.
  * [3] Bingham NH, Goldie CM, Teugels JL (1987). Regular Variation. Cambridge University Press, Cambridge.
  * [4] Coles S (2001) An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics.
  * [5] Embrechts P, Klüppelberg C, Mikosch T (2013) Modelling Extremal Events: for Insurance and Finance. pringer Science $\&$ Business Media, Berlin.
  * [6] Griffiths L (1947) Introduction to the Theory of Equations. J. Wiley, New York.
  * [7] Haan L, Ferreira A (2010) Extreme Value Theory: An Introduction. Springer Science $\&$ Business Media.
  * [8] Nascimento F, Bourguignon, M, Leão J. (2016) Extended generalized extreme value distribution with applications in environmental data. Hacettepe Journal of Mathematics and Statistics 45: 1847-1864.
  * [9] James RC (1966) Advanced Calculus. Wadsworth Pub. Co. in Belmont, California.
  * [10] Kotz SL, Nadarajah S (2000) Extreme Value Distributions: Theory and Applications. Imperial College Press, London.
  * [11] Longin F (2016) Extreme Events in Finance: A Handbook of Extreme Value Theory and its Applications. ohn Wiley $\&$ Sons.
  * [12] Nelder JA, Mead R (1965) A simplex algorithm for function minimization. Computer Journal 7:308- 31.
  * [13] Otiniano CEG, Gonçalves CR, Dorea CCY (2016). Mixture of extreme-value distributions: identifiability and estimation. Communications in Statistics - Theory and Methods 46:6528-6542.
  * [14] Paiva BS (2020) Distribuição generalizada de valor extremo bimodal. Dissertation, Universidade de Brasília https://repositorio.unb.br/bitstream/10482/40484/1/2020BiancaSouzadePaiva.pdf.
  * [15] Rathie PN, Swamee PK (2007). Invertible Alternatives to Normal and Lognormal Distributions. Journal of Hifrologic engineering, 12(2): 218-221.
  * [16] Eljabri S, Nadarajah S (2017) The Kumaraswamy GEV distribution, Communications in Statistics - Theory and Methods 46: 10203-10235.
  * [17] Scheirer WJ (2017) Extreme Value Theory-Based Methods for Visual Recognition. Morgan $\&$ Claypool Publishers.
  * [18] Spall JC (2005) Monte Carlo Computation of the Fisher Information Matrix in Nonstandard Settings. Journal of Computational and Graphical Statistics, 14: 889-909.
  * [19] R-Team (2020) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
  * [20] Trapletti A (2016) R: Box-Pierce and Ljung-Box Tests. stat.ethz.ch. Retrieved 2016-06-05. URL https://stat.ethz.ch/R-manual/R-devel/library/stats/html/box.test.html
  * [21] Wuertz D, Setz T, Chalabi Y (2017). Package fExtremes Package: Rmetrics - Modelling Extreme Events in Finance. Retrieved 2017-11-12. URL https://cran.r-project.org/web/packages/fExtremes/fExtremes.pdf.
  * [22] Xue J (2012) Loop Tiling for Parallelism. The Springer International Series in Engineering and Computer Science. Springer, US.