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The transmuted GEV distribution: properties and application
Communications for Statistical Applications and Methods 2019;26:239-259
Published online May 31, 2019
© 2019 Korean Statistical Society.

Cira E. G. Otiniano1,a, Bianca S. de Paivaa, and Daniele S. B. Martins Netob

aDepartment of Statistics, University of Brasília, Brazil;
bDepartment of Mathematics, University of Brasília, Brazil
Correspondence to: 1Department of Statistics, University of Brasília, Brasília, DF, Brazil.
E-mail: cira@unb.br
Received September 29, 2018; Revised February 19, 2019; Accepted April 23, 2019.
 Abstract

The transmuted generalized extreme value (TGEV) distribution was first introduced by Aryal and Tsokos (Nonlinear Analysis: Theory, Methods & Applications, 71, 401–407, 2009) and applied by Nascimento et al. (Hacettepe Journal of Mathematics and Statistics, 45, 1847–1864, 2016). However, they did not give explicit expressions for all the moments, tail behaviour, quantiles, survival and risk functions and order statistics. The TGEV distribution is a more flexible model than the simple GEV distribution to model extreme or rare events because the right tail of the TGEV is heavier than the GEV. In addition the TGEV distribution can adjusted various forms of asymmetry. In this article, explicit expressions for these measures of the TGEV are obtained. The tail behavior and the survival and risk functions were determined for positive gamma, the moments for nonzero gamma and the moment generating function for zero gamma. The performance of the maximum likelihood estimators (MLEs) of the TGEV parameters were tested through a series of Monte Carlo simulation experiments. In addition, the model was used to fit three real data sets related to financial returns.

Keywords : TGEV, moments, extreme events, MLEs
1. Introduction

The generalized extreme value (GEV) distribution is widely used in several areas to model data from extreme events that occur infrequently. For example, Lettenmainer et al. (1987), Hewa et al. (2007), Morrison and Smith (2002) used it in hydrology to treat return periods of flood frequency or high wind speeds. In finance, Embrechts et al. (1997), presents how to calculate the value at risk (VaR) of maximum financial returns, and in actuarial science how to calculate the probability of ruin as consequence of extreme events. Extreme events are more suitably modeled with heavy tails and the GEV distribution has this characteristic. However, there are extreme event data that do not follow GEV distribution, because they require a more asymmetric distribution or with a heavier tail than GEV distribution. Thus, new classes of probability distributions have been developed that are more general than the GEV distribution such as: dual gamma GEV distribution (GGEV), exponentiated GEV distribution (EGEV) studied by Nascimento et al. (2016), transmuted GEV (TGEV) distribution defined by Aryal and Tsokos (2009), and q-GEV given by Provost et al. (2018). The advantage of TGEV distribution in relation to other generalized distributions is that it has a heavier tail than GEV distribution as shown in Section 2. Moments, moment generating function, hazard rate function and order statistics have a simple closed form. Therefore, TGEV distribution becomes flexible to model extreme events in several areas. According to Jenkinson (1955), the GEV distribution is the limit distribution of properly normalized maximum (or minimum) of a sequence of independent and identically distributed (iid) random variables. That is, if X1, X2, . . . , Xn are iid random variables with cumulative distribution function (cdf) F(x) and if there are sequences of constants an > 0 and bn such that

P(X(n)-bnanx)G(x), 듼 듼 듼as n,

where X(1),X(2), . . . ,X(n) denote the order statistics and G(x) is a non-degenerate cdf, then G(x) belongs to one of the following three distribution families:

Gumbel:Λ(x)=exp {-exp [-(x-μσ)]}, 듼 듼 듼-<x<;Fre´Frechet:Φ(x)={0,xμ,exp{-(x-μσ)-α},xμ;Weibull:Ψ(x)={exp {-[-(x-μσ)α]},x<μ,1,x>μ,

where σ>0, μR, and α>0. Jenkinson (1955) introduced the GEV distribution that contemplates the three previous distributions. A random variable X follows the GEV distribution if its cdf is given by


G(x)={exp {-[1+γ(x-μσ)]-1γ},γ0,exp {-exp [-(x-μσ)]},γ=0,

where it is defined in the set {x : 1+γ((xμ)/σ) > 0}, μ is a location parameter, σ is a scale parameter and γ is a shape parameter. For γ = 0 the expression (1.1) is interpreted by taking the limit as γ > 0. The γ parameter governs the tail behavior with an important impact on the shape of the distribution that is called the tail index directly related to the shape parameter α. The Gumbel distribution is a special case for γ = 0. Also note that the cases for γ = 1/α and γ = −1/α we have that the GEV distribution is of the Fréchet type and the Weibull type respectively.

The transmutation map is a technique developed by Shaw and Buckley (2007) and consists of introducing skewness or kurtosis in a symmetric or other (asymmetrical) distribution. It is a relatively new technique; however, it has already been applied to several distribution functions. Some examples are the transmuted extreme value distribution introduced by Aryal and Tsokos (2009), Aryal and Tsokos (2011), Aryal (2013) obtained the transmuted Weibull distribution and the transmuted log-logistic distribution, and Merovci (2013) introduced the transmuted exponentiated exponential distribution. With respect to the Weibull distribution and some extensions, Khan and King (2013a, 2016) developed its transmutation. Khan and King (2013b, 2014) and Mahmoud and Mandouh (2013) developed the transmutation for the Weibull inverse distribution and some extensions. Khan and King also obtained the transmutation of inverse Rayleigh distributions (2015), Khan et al. (2016a, 2016b, 2017) obtained the transmutation of Kumaraswamy distribution, new generalized Weibull distribution, and new generalized inverse Weibull distribution. Elgarhy et al. (2017) introduced the transmuted generalized quasi Lindley distribution. Khan (2018) and Nassar et al. (2019) obtained the transmuted generalized power Weibull distribution and transmuted Weibull Logistic Distribution, respectively.

Aryal and Tsokos (2009) defined TGEV distribution and discussed some properties about the transmuted Gumbel distribution. Recently, Nascimento et al. (2016) applied the TGEV distribution to environmental data with the parameter estimation of this distribution was done under the Bayesian model. This work investigated the tail behavior and the main mathematical measures of the TGEV distribution. The parameters are also estimated by maximum likelihood that included an application to illustrate the model. This paper is organized as follows. Section 2 deals with some mathematical properties of the TGEV distribution, such as the tail behavior, the moments, the hazard rate function, and the order statistics. In Section 3, the inference procedure is performed by maximum likelihood and some simulations are used to test the efficiency of the estimators. An application for extreme data is presented in Section 4.

2. Tail behavior and properties of the TGEV

In this section we present the behavior of the right tail of the TGEV distribution. We show that the right tail of the TGEV distribution is heavier than the right tail of the GEV distribution. Expressions of moments, moments gerating function, hazard rate function, quantile function, and order statistics were also obtained.

The transmutation map, proposed by Shaw and Buckley (2007), consists of a powerful technique that considers some perturbations of the symmetry and manage kurtosis adjustments. Given one base distribution function, say G(x), the transmuted distribution function F is defined by


F(x)=(1+λ)G(x)-λ[G(x)]2, 듼 듼 듼λ<1.

Aryal and Tsokos (2009) introduced the TGEV distribution and studied basic mathematical characteristics just of the transmuted Gumbel distribution. However, it is important to study moments, order statistics and other statistical properties in the modeling of extreme events because GEV distribution can occur with γ ≠ 0. A random variable X is said to be TGEV distributed, say X ~ FT ( · ; μ,σ, γ, λ) distribution, if its cdf can be expressed as


FT(x;μ,σ,γ,λ)={e{-[w]-1γ}[(1+λ)-λe{-[w]-1γ}],γ0,e-e-(x-μ)σ[(1+λ)-λe-e-(x-μ)σ],γ=0,

by replacing (1.1) in (2.1). The function (2.2) is well defined for x such that {x : w = 1+γ((xμ)/σ) > 0}, μ is a location parameter, σ is a scale parameter, γ is a shape parameter (tail index), and λ is the shape parameter. Note that to λ < 0 the model (2.2) corresponds to a mixture of a GEV distribution and a skew GEV distribution.

The probability density function (pdf) corresponding to (2.2) is given by


fT(x;μ,σ,γ,λ)={[(w)-1γ-11σe{-[w]-1γ}][(1+λ)-2λe{-[w]-1γ}], 듼 듼 듼γ0,1σe-(x-μ)σe-e-(x-μ)σ[(1+λ)-2λe-e-(x-μ)σ], 듼 듼 듼γ=0.

Figures 14 show how the TGEV density function is influenced by the parameter λ. Note that at λ = 0 we have the particular case of base GEV distribution. In the four figures μ = 0, σ = 1 are fixed and λ varies according to the values of the legend of the figures, but the tail index γ = 0, −0.5, 0.5, 0.5 in Figures 14, respectively.

Note that the parameter λ can modify the distribution according to the signal of γ. For γ < 0, the greater the absolute value of λ, the larger the maximum value of the pdf and the heavier its tail. For negative values of λ, the density curves are larger than those for the positive values of λ. For γ = 0, positive values of λ produce higher maximum values of the pdf. However, when γ > 0, we have that the greater the value of λ, the larger the maximum value of the pdf and its tail will be heavier. Extreme event data usually follows a heavy tail distribution and the GEV distribution has this property for γ > 0. Therefore, the study the tail behavior of the TGEV density for γ > 0 is important for modeling extreme events.

2.1. Tail behavior

By Embrechts et al. (1997) showed that a probability distribution is said to have a heavy tail if its reliability function is regularly varying. Thus, in this section we analyze the tail behavior of the TGEV distribution via regular variation property at infinity.


Definition 1

A positive measurable function f defined on some neighbourhood [xo,∞) is called regularly varying (at) with index αR if


limxf(tx)f(t)=xα.

If α = 0 f is said to be slowly varying (at).

From Equation (2.4) it is easy to see that every regularly varying function f of index α has representation


f(x)=xαL(x),

where L is some slowly varying function.


Proposition 1

Let X be a random variable with cdf FT(·;μ,σ,γ,λ),γ>0, then


F¯T(x)=2(1+λ)x-1γL1(x)+λx-2γL2(x), 듼 듼 듼x,

where F¯T=1-FT is the reliability function of the TGEV distribution and L1and L2are any two slowly varying functions (at).


Proof

The tail behavior of a TGEV distribution at infinity is determined by considering the reliability function of the GEV distribution as G¯=1-G and replacing it in (2.4). Therefore, we have

limxG¯(tx)G¯(t)=x-1γ,

where G is regularly varying with index −1/γ. Thus, by (2.5), the reliability function G can be represented by


G¯(x)=x-1γL1(x),

where L1(x) is slowly varying function (at ∞). We obtain (2.6) simply by replacing (2.7) in (2.1). Then from Equation (2.6), we can conclude that the tail behavior of the TGEV distribution is the same as that of a mixture of regular varying functions and, therefore, a mixture of heavy tail functions whose right tail weight is influenced by the parameters γ and λ.

2.2. Moments

The moments of a transmuted Gumbel random variable X ~ FT(·;μ,σ,γ,λ) have already been obtained by Aryal and Tsokos (2009). Therefore, in this section, we compute the moments of a random variable X ~ FT with γ ≠ 0.

Proposition 2

If X ~ FT(·;μ,σ,γ,λ) with γ ≠ 0, then the kth moment (non-negative integer) of X is given by


E(Xk)=i=0kCki(μ-σγ)k-i(σγ)i[Γ(1-γi)(1+λ-2γiλ)],

where μσ/γ > 0.


Proof

For γ > 0, from (2.2) we obtain


E(Xk)=μ-σγxkdF(x)=(1+λ)σμ-σγxk[(1+γ(x-μ)σ)-1γ-1][e{-[1+γ(x-μ)σ]-1γ}]dx-2λσμ-σγxk[(1+γ(x-μ)σ)-1γ-1][(e{-[1+γ(x-μ)σ]-1γ})2]dx.

In order to solve the integrals in (2.9), we first replace w by 1 + γ(xμ)/σ and then use the Newton’s formula,

xk=i=0kCki(μ-σγ)k-i(σγy)i,

then


E(Xk)=(1+λ)σ0i=0kCki(μ-σγ)k-i(σγw)iw-1γ-1e-w-1γσγdw-2λσ0i=0kCki(μ-σγ)k-i(σγw)iw-1γ-1e-2w-1γσγdw=(1+λ)σi=0kCki(μ-σγ)k-i(σγ)i+10wi-1γ-1e-w-1γdw-2λσi=0kCki(μ-σγ)k-i(σγ)i+10wi-1γ-1e-2w-1γdw.

Now, taking u = w−1 in (2.10), we have


E(Xk)=[(1+λ)σi=0kCki(μ-σγ)k-i(σγ)i+1]0u-i+1γ-1e-u1γdu-[2λσi=0kCki(μ-σγ)k-i(σγ)i+1]0u-i+1γ-1e-2u1γdu.

Note that the integrals in (2.11) are Gamma functions, then (2.8) is obtained.

Analogously, we compute E(Xk) for γ < 0, where the integration domain in this case is [−∞, μσ/γ], and we obtain the same result (2.8).

The mean and variance of the TGEV distribution, also obtained by Nascimento et al. (2016), can be deduced directly from Equation (2.8).

For k = 1,

E(X)=(μ-σγ)+(σγ)[Γ(1-γ)(1+λ-2γλ)]

and for k = 2,

E(X2)=(μ-σγ)2+2(μ-σγ)σγ[Γ(1-γ)(1+λ-2γλ)]+(σγ)2[Γ(1-2γ)(1+λ-22γλ)],

then

Var(X)=(σγ)2[Γ(1-2γ)(1+λ-22γλ)-Γ2(1-γ)(1+λ-2γλ)2].

In the case of the transmuted Gumbel distribution, X ~ FT(·;μ,σ,γ,λ), the moment generating function of X, say M(t) = E(Xk), is of great importance to obtain moments of distribution, since a general expression of E(Xk) is not simple to calculate.

Proposition 3

If X ~ FT(·;μ,σ,γ,λ), then


MX(t)=E(etX)=etμΓ(1-tσ)[(1+λ)-2tσλ], 듼 듼 듼t<1σ.
Proof

By definition,


MX(t)=-etxe-(x-μ)σe-e-(x-μ)σσ[(1+λ)-2λe-e-(x-μ)σ]dx.

Setting u = e−(xμ)/σ, we can rewrite (2.13) as


MX(t)=0et(μ-σlog(u))ue-uσ[1+λ-2λe-u]-σudu=0etμelog(u-tσ)e-u[(1+λ)-2λe-u]du=etμ[(1+λ)0u-tσe-udu-2λ0u-tσe-2udu].

The expression (2.12) is obtained by solving the integrals in (2.14), considering t < 1/σ and using the Gamma function.

The same expressions of E(X) and Var(X), obtained by Aryal and Tsokos (2009), we now obtain with the first and second derivatives of MX(t) at t = 0,


E(X)=μ+σC-λσlog(2)Var(X)=σ2[π26-λ(1+λ)log2(2)].

However, the moment generating function (2.12) can also be useful for analyzing data on the sum of TGEV random variables.

2.3. Reliability measures

If we consider a random variable T ~ FT(·;μ,σ,0,λ) with μ > 0 whose distribution support is a subset of non-negative real numbers, T can represent the failure time of an event of interest. In this sense, this random variable can be characterized by the survival function, R(t)=F¯T(t)=1-FT(t; μ, σ, 0, λ). The survival and hazard rate functions for T are given, respectively, by


R(t)=1-[e{-[1+γ(t-μ)σ]-1γ}(1+λ-λe{-[1+γ(t-μ)σ]-1γ})]

and


h(t)=[(1+γ(t-μ)σ)-1γ-1e{-[1+γ(t-μ)σ]-1γ}σ][(1+λ)-2λe{-[1+γ(t-μ)σ]-1γ}]1-[e{-[1+γ(t-μ)σ]-1γ}(1+λ-λe{-[1+γ(t-μ)σ]-1γ})].

Figure 5 shows the behavior of the reliability function (2.17) for λ taking values from −1 to 1. Note that for smaller values of λ the function R decays more slowly, i.e., the parameters λ and γ influence the tail weight of the TGEV distribution. This is illustrated by Proposition 1. Also notice the hazard rate function (2.18), shown in Figure 6, presents unimodal behavior. We observe that the mode changes as the parameter λ varies.

2.4. Order statistics

Let X(1), X(2), X(3), . . . , X(n) denote the order statistics of a random sample X1, X2, X3, . . . , Xn from a population X ~ FT(·;μ,σ,γ,λ). Then we have that the pdf of the jth order statistic X(j), for γ ≠ 0, is given by


fX(j)(x)=n!(j-1)!(n-j)![(w-1γ-1e-w-1γσ)(1+λ-2λe-w-1γ)]×[e-w-1γ(1+λ-λe-w-1γ)]j-1[1-e-w-1γ(1+λ-λe-w-1γ)]n-j,

where w = 1 + γ(xμ)/σ. Therefore, the pfd of the nth order statistic and the pdf of the 1st order statistic are given, respectively, by


fX(n)(x)=n[(w-1γ-1e-w-1γσ)(1+λ-2λe-w-1γ)][e-w-1γ(1+λ-λe-w-1γ)]n-1

and


fX(1)(x)=n[(w-1γ-1e-w-1γσ)(1+λ-2λe-w-1γ)][1-e-w-1γ(1+λ-λe-w-1γ)]n-1.

For γ = 0, the pdfs of the jth, nth, and 1st order statistics are given, respectively, by


fX(j)(x)=n!(j-1)!(n-j)!ve-vσ[(1+λ)-2λe-v][(1+λ-λe-v)j-1][1-(e-v(1+λ-λe-v))]n-j,

where v = e−(xμ)/σ,


fX(n)(x)=nve-nvσ[(1+λ)-2λe-v][(1+λ-λe-v)n-1]

and


fX(1)(x)=nve-vσ[(1+λ)-2λe-v].
3. Estimation and results

Nascimento et al. (2016) used Bayesian inference to obtain parameter estimators of the TGEV distribution. In this section we present the system to be solved to obtain the MLEs estimators, then simulation experiments were run in order to test the performance of these estimators.

3.1. Estimation

The parameters of the TGEV distribution are estimated by the method of maximum likelihood. Let X1, X2, . . . , Xn be a sample of size n from X ~ FT(·;μ,σ,γ,λ). Let θ = (μ,σ, γ, λ) be the parametric vector. The log-likelihood function for θ with γ = 0 and γ ≠ 0 can be expressed, respectively, as


l(θ)=-i=1n[log(σ)+xi-μσ+e-(xi-μ)σ]+i=1nlog (1+λ-2λe-e-(xi-μ)σ)

and


l(θ)=-nlog(σ)-(1γ+1)i=1nlog (1+γσ(xi-μ))-i=1n(1+γσ(xi-μ))-1γ+i=1nlog (1+λ-2λe{-(1+γσ(xi-μ))-1γ}).

Thus, for γ = 0, the MLEs of μ,σ, λ which maximize l(θ)), given by (3.1), must satisfy the equations


lμ=n-i=1ne-y+2λi=1ne-ye-e-y1+λ-2λe-e-y=0,lσ=-n+i=1n(xi-μ)[1-e-y]+2λσi=1n(xi-μ)e-ye-e-y1+λ-2λe-e-y=0,lλ=i=1n1-2e-e-y1+λ-2λe-e-y=0,

where y = (xiμ)/σ and for γ ≠ 0 the MLEs of μ,σ, μ, λ which maximize l(θ), given by (3.2), must satisfy the equations


lμ=i=1n1+γσ+γ(xi-μ)-i=1nw-1γ-1σ+i=1nw-1γ-12λe-w-1γσ(1+λ-2λe-w-1γ)=0,lσ=-nσ+i=1n(1+γ)(xi-μ)σ[σ+γ(xi-μ)]-i=1n(xi-μ)w-1γ-1σ2+i=1n(xi-μ)w-1γ-12λe-w-1γσ2[1+λ-2λe-w-1γ]=0,lγ=i=1n[log(w)γ2+(1γ)(xi-μ)σw]-i=1nw-1γ[log(w)γ2-(xi-μ)γσw]+i=1nw-1γ[log(w)γ2-(xi-μ)γσw][2λe-w-1γ1+λ-2λe-w-1γ]=0,lλ=i=1n1-2e-w-1γ1+λ-2λe-w-1γ=0,

where w = 1 + (γ/σ)(xiμ).

3.2. Simulation and results

In order to investigate the performance of the MLE θ^=(μ^,σ^,γ^,λ^) of θ = (μ,σ, γ, λ), random samples of size n = 500 and 1,000 of the random variable X ~ FT(·;μ,σ,γ,λ) were simulated for 32 combinations of μ, σ, γ and λ. These parameters are presented in Tables 1 and 3 that can be seen in two different configurations. In configuration 1, the values of the parametric vector corresponding to γ > 0, μ = 0, σ = 1, and λ = −0.9, −0.5, 0.5, 0.9 are presented in Table 1. In configuration 2, the values of the parametric vector corresponding to γ ≤ 0, μ = 0, σ = 1, and λ = −0.9, −0.5, 0.5, 0.9 are seen in Table 1.

The procedure to investigate the performance of the MLEs consists of:

  • (1) Generating M = 100 random samples of size n = 500 and 1,000 from the TGEV distribution by the method of inversion using the quantiles

    xQ={σ{-1+[-ln (1+λ-(1+λ)2-4λQ2λ)]-γ}γ+μ,for γ0,xQ=σ{-ln [-ln (1+λ-(1+λ)2-4λQ2λ)]}+μ,for γ=0.

  • (2) Obtaining the maximum likelihood estimates of the parameters μ,σ, γ, λ by maximizing the log-likelihood function, (3.1) or (3.2), through of the “optim function” of the R Core Team software 2015. In this function the method for optimization is a derivative-free optimization routine called the Nelder-Mead simplex algorithm. This method already provides the Hessian matrix.

Results about mean estimates of each parameter and their corresponding mean square errors (MSEs) were calculated via Monte Carlo simulation with M = 100 samples of size n = 500 and 1,000. The results are presented in Tables 14.

In Section 2.2 we have that for γ > 0 the TGEV moments are defined in [1/2, 1], so the values chosen for γ > 0 (Configuration 1) were 0.5, 0.75, 0.9, 1. Table 2 shows that the bias and MSE of the mean estimates are small. The algorithm has obtained good estimates. In configuration 2, Table 4 shows estimates of the TGEV for γ ≤ 0. We have the convergence of the algorithm for γ in [−0.5, 0], the bias is significant in some cases, but the MSE values are low.

Illustrations graphical of the fitted density f(x,θ^) together the theoretical density f (x, θ) for several cases of θ1 to θ32 are shown in Appendix in Figures A.1, A.2, and A.3. These figures also show that the method yields satisfactory results.

4. Application

In order to apply the model TGEV, we use three sets of real financial data: Ibovespa, S&P 500, and Dow Jones. The data were obtained from the website http://br.investing.com/indices. For each data set we use daily log-returns from June 3, 2006 to October 31, 2016. The log-returns are given by rt = log(Pt) − log(Pt−1), where Pt is the opening price at day t. For the Ibovespa were used 2,707 observations, for the S&P 500 were 2,768 and for the Dow Jones 2,795. The data modeled by the TGEV distribution correspond to the maximum values of the returns in blocks of size 7. The histograms for these values are shown in Figure 7.

Table 5 shows the main descriptive statistics for the three data sets in question. The MLEs of the parameters μ,σ, γ, λ are obtained from Equations (3.3) and (3.4). Table 6 shows the estimates obtained for the three data sets.

Figures 810 present the histogram of the returns versus the fitted density for each data set, whose parameters are shown in Table 6 along with QQplot that compares the theoretical quantiles on the vertical axis with the empirical quantiles on the horizontal axis. From the results, we can observe that the estimates of λ are different from zero. This indicates that TGEV distribution is more appropriate for these databases than the GEV distribution often considered by many authors.

Analyzing the QQplots, we can see that the TGEV distribution fits very well as an estimated distribution for the data used. The Kolmogorov-Smirnov (KS) test and the Anderson Darling (AD) test were performed to verify the adjustment of the estimated distribution to the data. Both test the following hypotheses:

{H0:The data follow the TGEV distribution;HA:The data does not follow the TGEV distribution.

Table 8 therefore presents the results for the KS test statistic and its p-value, as well as the AD test statistic and its p-value.

It is possible to conclude from the KS and AD tests and the analysis of the p-values obtained that there is no statistical evidence against the hypothesis that the data follow the TGEV distributions estimated here.

5. Conclusion

In this paper, we present important properties of the TGEV distribution. TGEV distribution is a more flexible model than GEV distribution to model extreme event data. The estimation of the parameters is approached by the maximum likelihood method. Applications of TGEV for three data sets show that the new distribution can be used to effectively provide better adjustments than GEV distribution.

Appendix

We added the Figures A.1. A.3 in order to illustrate the fit of the simulated data in Section 3. In almost all cases, the adjusted densities were close to the theoretical densities.

Fitted density f(x,θ^) (blue) and the theoretical density f (x, θ) (red) for θ1θ4 and θ13θ16, according to Table 1.

Fitted density f(x,θ^) (blue) and the theoretical density f (x, θ) (red) for θ17θ20 and θ25θ30, according to Table 3.

Fitted density f(x,θ^) (blue) and the theoretical density f (x, θ) (red) for θ10θ14, according to Table 3.

Figures
Fig. 1. Plots for the transmuted generalized extreme value density μ = 0, σ = 1, γ = 0, and λ varying as shown in the caption.
Fig. 2. Plots for the transmuted generalized extreme value density for μ = 0, σ = 1, γ = −0.5, and λ varying as shown in the caption.
Fig. 3. Plots for the transmuted generalized extreme value density for μ = 0, σ = 1, γ = 0.5, and λ varying as show in the caption.
Fig. 4. Plots for the transmuted generalized extreme value density for μ = 2, σ = 1, γ = 0.5, and λ varying as show in the caption.
Fig. 5. Reliability function of TGEV distribution for μ = 2, σ = 1, γ = 0.5, and λ varying in [−1, 1].
Fig. 6. Hazard rate function of transmuted generalized extreme value distribution for μ = 2, σ = 1, γ = 0.5, and λ varying in [−1, 1].
Fig. 7. Histogram of maximums of blocks of size 7 of the returns.
Fig. 8. Fitted Ibovespa data by TGEV (left) and QQplot empirical versus theoretical (right). TGEV = transmuted generalized extreme value.
Fig. 9. Fitted S&P 500 data by TGEV (left) and QQplot empirical versus theoretical (right).
Fig. 10. Fitted Dow Jones data by TGEV (left) and QQplot empirical versus theoretical (right).
TABLES

Table 1

Parameters and mean estimates for γ > 0

θ n μ σ γ λ μ^ σ^ γ^ λ^
θ1 500 0 1 0.50 −0.50 0.001 1.026 0.508 −0.500
1000 0 1 0.50 −0.50 −0.025 1.006 0.504 −0.531

θ2 500 0 1 0.50 −0.90 0.229 1.111 0.506 −0.614
1000 0 1 0.50 −0.90 0.183 1.078 0.505 −0.681

θ3 500 0 1 0.50 0.50 −0.368 0.704 0.469 −0.269
1000 0 1 0.50 0.50 −0.388 0.697 0.462 −0.314

θ4 500 0 1 0.50 0.90 −0.427 0.597 0.322 −0.145
1000 0 1 0.50 0.90 −0.433 0.596 0.326 −0.140

θ5 500 0 1 0.75 −0.50 0.020 1.027 0.756 −0.481
1000 0 1 0.75 −0.50 0.003 1.023 0.754 −0.506

θ6 500 0 1 0.75 −0.90 0.187 1.117 0.754 −0.677
1000 0 1 0.75 −0.90 0.179 1.120 0.744 −0.701

θ7 500 0 1 0.75 0.50 −0.271 0.730 0.700 −0.111
1000 0 1 0.75 0.50 −0.306 0.685 0.681 −0.173

θ8 500 0 1 0.75 0.90 −0.391 0.533 0.507 −0.091
1000 0 1 0.75 0.90 −0.379 0.553 0.515 −0.063

θ9 500 0 1 0.90 −0.50 0.034 1.044 0.919 −0.451
1000 0 1 0.90 −0.50 0.002 1.011 0.908 −0.501

θ10 500 0 1 0.90 −0.90 0.184 1.135 0.897 −0.677
1000 0 1 0.90 −0.90 0.167 1.129 0.898 −0.701

θ11 500 0 1 0.90 0.50 −0.252 0.708 0.832 −0.084
1000 0 1 0.90 0.50 −0.270 0.692 0.832 −0.134

θ12 500 0 1 0.90 0.90 −0.330 0.556 0.643 0.021
1000 0 1 0.90 0.90 −0.314 0.576 0.650 0.084

θ13 500 0 1 1.00 −0.50 −0.023 0.981 1.014 −0.549
1000 0 1 1.00 −0.50 −0.004 1.011 1.013 −0.516

θ14 500 0 1 1.00 −0.90 0.149 1.126 1.004 −0.728
1000 0 1 1.00 −0.90 0.135 1.118 0.990 −0.751

θ15 500 0 1 1.00 0.50 −0.248 0.701 0.936 −0.094
1000 0 1 1.00 0.50 −0.235 0.707 0.918 −0.062

θ16 500 0 1 1.00 0.90 −0.329 0.529 0.720 0.021
1000 0 1 1.00 0.90 −0.352 0.511 0.719 −0.063

Table 2

Bias and MSE of θ^ for γ > 0

θ n Bias MSE


μ^ σ^ γ^ λ^ μ^ σ^ γ^ λ^
θ1 500 −0.001 −0.026 −0.008 0.000 0.070 0.034 0.002 0.120
1000 0.025 −0.006 −0.004 0.031 0.046 0.015 0.001 0.085

θ2 500 −0.229 −0.111 −0.006 −0.286 0.121 0.058 0.002 0.165
1000 −0.183 −0.078 −0.005 −0.219 0.065 0.032 0.002 0.091

θ3 500 0.368 0.296 0.031 0.769 0.059 0.035 0.002 0.245
1000 0.388 0.303 0.038 0.814 0.056 0.032 0.002 0.251

θ4 500 0.427 0.403 0.178 1.045 0.032 0.019 0.003 0.216
1000 0.433 0.404 0.174 1.040 0.040 0.020 0.002 0.263

θ5 500 −0.020 −0.027 −0.006 −0.019 0.044 0.033 0.003 0.077
1000 −0.003 −0.023 −0.004 0.006 0.045 0.039 0.001 0.077

θ6 500 −0.187 −0.117 −0.004 −0.223 0.048 0.027 0.002 0.062
1000 −0.179 −0.120 0.006 −0.199 0.047 0.027 0.001 0.058

θ7 500 0.271 0.270 0.050 0.611 0.066 0.077 0.007 0.298
1000 0.306 0.315 0.069 0.673 0.047 0.050 0.004 0.238

θ8 500 0.391 0.467 0.243 0.991 0.023 0.020 0.006 0.195
1000 0.379 0.447 0.235 0.963 0.030 0.032 0.008 0.229

θ9 500 −0.034 −0.044 −0.019 −0.049 0.038 0.039 0.002 0.064
1000 −0.002 −0.011 −0.008 0.001 0.029 0.027 0.002 0.053

θ10 500 −0.184 −0.135 0.003 −0.223 0.036 0.027 0.003 0.049
1000 −0.167 −0.129 0.002 −0.199 0.047 0.036 0.001 0.060

θ11 500 0.252 0.292 0.068 0.584 0.052 0.075 0.010 0.290
1000 0.270 0.308 0.068 0.634 0.053 0.070 0.007 0.311

θ12 500 0.330 0.444 0.257 0.879 0.029 0.042 0.014 0.235
1000 0.314 0.424 0.250 0.816 0.031 0.046 0.015 0.263

θ13 500 0.023 0.019 −0.014 0.049 0.033 0.039 0.003 0.055
1000 0.004 −0.011 −0.013 0.016 0.039 0.068 0.004 0.061

θ14 500 −0.149 −0.126 −0.004 −0.172 0.037 0.033 0.003 0.047
1000 −0.135 −0.118 0.010 −0.149 0.030 0.023 0.002 0.036

θ15 500 0.248 0.299 0.064 0.594 0.050 0.089 0.013 0.306
1000 0.235 0.293 0.082 0.562 0.050 0.079 0.008 0.304

θ16 500 0.329 0.471 0.280 0.879 0.025 0.041 0.017 0.251
1000 0.352 0.489 0.281 0.963 0.030 0.047 0.015 0.254

Table 3

Parameters and mean estimates for γ ≤ 0

θ n μ σ γ λ μ^ σ^ γ^ λ^
θ17 500 0 1 −0.10 0.50 −0.052 1.024 −0.105 −0.546
1000 0 1 −0.10 0.50 −0.165 1.047 −0.112 −0.700

θ18 500 0 1 −0.10 −0.90 0.164 0.992 −0.094 −0.673
1000 0 1 −0.10 −0.90 0.173 0.978 −0.101 −0.649

θ19 500 0 1 −0.10 0.50 −0.650 0.861 −0.051 −0.632
1000 0 1 −0.10 0.50 −0.691 0.867 −0.049 −0.691

θ20 500 0 1 −0.10 0.90 −0.757 0.763 −0.085 −0.598
1000 0 1 −0.10 0.90 −0.781 0.761 −0.083 −0.644

θ21 500 0 1 −0.25 −0.50 −0.110 1.059 −0.263 −0.625
1000 0 1 −0.25 −0.50 −0.150 1.066 −0.262 −0.668

θ22 500 0 1 −0.25 −0.90 0.156 0.966 −0.256 −0.652
1000 0 1 −0.25 −0.90 0.109 0.965 −0.245 −0.744

θ23 500 0 1 −0.25 0.50 −0.704 0.956 −0.179 −0.656
1000 0 1 −0.25 0.50 −0.784 0.980 −0.186 −0.771

θ24 500 0 1 −0.25 0.90 −0.791 0.850 −0.188 −0.579
1000 0 1 −0.25 0.90 −0.873 0.859 −0.184 −0.713

θ25 500 0 1 −0.50 −0.50 −0.217 1.142 −0.519 −0.748
1000 0 1 −0.50 −0.50 −0.210 1.140 −0.518 −0.743

θ26 500 0 1 −0.50 −0.90 0.140 0.941 −0.510 −0.663
1000 0 1 −0.50 −0.90 0.095 0.945 −0.497 −0.754

θ27 500 0 1 −0.50 0.50 −0.793 1.165 −0.411 −0.679
1000 0 1 −0.50 0.50 −0.754 1.135 −0.402 −0.639

θ28 500 0 1 −0.50 0.90 −0.957 1.045 −0.362 −0.680
1000 0 1 −0.50 0.90 −0.934 1.030 −0.356 −0.665

θ29 500 0 1 0.00 −0.50 0.440 1.187 0.000 0.135
1000 0 1 0.00 −0.50 0.412 1.170 0.000 0.098

θ30 500 0 1 0.00 −0.90 0.729 1.137 0.000 0.167
1000 0 1 0.00 −0.90 0.715 1.148 0.000 0.121

θ31 500 0 1 0.00 0.50 −0.084 0.962 0.000 0.340
1000 0 1 0.00 0.50 −0.095 0.963 0.000 0.324

θ32 500 0 1 0.00 0.90 −0.360 0.853 0.000 0.157
1000 0 1 0.00 0.90 −0.362 0.846 0.000 0.163

Table 4

Bias and MSE of θ^ for γ ≤ 0

θ n Bias MSE


μ^ σ^ γ^ λ^ μ^ σ^ γ^ λ^
θ17 500 0.052 −0.024 0.005 0.046 0.057 0.004 0.001 0.107
1000 0.165 −0.047 0.012 0.200 0.046 0.004 0.001 0.076

θ18 500 −0.164 0.008 −0.006 −0.227 0.077 0.007 0.004 0.168
1000 −0.173 0.022 0.001 −0.251 0.060 0.002 0.001 0.126

θ19 500 0.650 0.139 −0.049 1.132 0.039 0.002 0.001 0.096
1000 0.691 0.133 −0.051 1.191 0.027 0.002 0.000 0.071

θ20 500 0.757 0.237 −0.015 1.498 0.031 0.003 0.001 0.105
1000 0.781 0.239 −0.017 1.544 0.015 0.001 0.001 0.046

θ21 500 0.110 −0.059 0.013 0.125 0.058 0.007 0.001 0.096
1000 0.150 −0.066 0.012 0.168 0.049 0.006 0.000 0.083

θ22 500 −0.156 0.034 0.006 −0.248 0.070 0.010 0.001 0.128
1000 −0.109 0.035 −0.005 −0.156 0.035 0.005 0.001 0.071

θ23 500 0.704 0.044 −0.071 1.156 0.044 0.004 0.001 0.092
1000 0.784 0.020 −0.064 1.271 0.029 0.004 0.000 0.055

θ24 500 0.791 0.150 −0.062 1.479 0.031 0.003 0.002 0.079
1000 0.873 0.141 −0.066 1.613 0.016 0.002 0.001 0.032

θ25 500 0.217 −0.142 0.019 0.248 0.038 0.016 0.001 0.048
1000 0.210 −0.140 0.018 0.243 0.043 0.015 0.001 0.069

θ26 500 −0.140 0.059 0.010 −0.237 0.056 0.014 0.002 0.149
1000 −0.095 0.055 −0.003 −0.146 0.029 0.010 0.001 0.069

θ27 500 0.793 −0.165 −0.089 1.179 0.068 0.014 0.001 0.116
1000 0.754 −0.135 −0.098 1.139 0.068 0.011 0.001 0.114

θ28 500 0.957 −0.045 −0.138 1.580 0.030 0.008 0.001 0.049
1000 0.934 −0.030 −0.144 1.565 0.022 0.005 0.000 0.038

θ29 500 −0.440 −0.187 0.000 −0.635 0.200 0.041 0.000 0.393
1000 −0.412 −0.170 0.000 −0.598 0.191 0.036 0.000 0.390

θ30 500 −0.729 −0.137 0.000 −1.067 0.142 0.026 0.000 0.307
1000 −0.715 −0.148 0.000 −1.021 0.191 0.027 0.000 0.410

θ31 500 0.084 0.038 0.000 0.160 0.041 0.008 0.000 0.117
1000 0.095 0.037 0.000 0.176 0.043 0.007 0.000 0.135

θ32 500 0.360 0.147 0.000 0.743 0.139 0.022 0.000 0.581
1000 0.362 0.154 0.000 0.737 0.134 0.022 0.000 0.571

Table 5

Descriptive statistics of data

Index E(X) Var(X) max(X) min(X) median(X)
Ibovespa 0.0214 0.0002 0.1208 3 × 10−6 0.0186
S&P 500 0.0144 0.0001 0.1055 −7 × 10−4 0.0112
Dow Jones 0.0130 0.0001 0.0966 −0.0011 0.0099

Table 6

Parameter estimates

Index μ^ σ^ γ^ λ^
Ibovespa 0.02098 0.01500 0.28645 0.95497
S&P 500 0.01222 0.01080 0.43309 0.80260
Dow Jones 0.00782 0.00651 0.30470 0.14741

Table 7

Model selection

Dados Model log-likelihood AIC BIC
Bovespa GEV −1159.070 2326.141 2341.964
GEVT −1159.584 2327.168 2342.991
S&P500 GEV −1284.209 2576.417 2592.333
GEVT −1283.709 2575.418 2591.333
DowJones GEV −1336.870 2681.739 2697.695
GEVT −1336.921 2681.843 2697.799

AIC = Akaike information criteria; BIC = Bayesian information criteria; GEV = generalized extreme value.


Table 8

Hypothesis test result

Index KS p-valor AD p-valor
Ibovespa 0.03363 0.7752 0.4488 0.7993
S&P 500 0.03613 0.6810 0.5901 0.6573
Dow Jones 0.03558 0.6935 0.7058 0.5539

KS = Kolmogorov-Smirnov test; AD = Anderson Darling text.


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