Bimodal GEV Distribution with Location Parameter

Description

Density, distribution function, quantile function random generation and estimation of bimodal GEV distribution given in Otiniano et al. (2023) <doi:10.1007/s10651-023-00566-7>. This new generalization of the well-known GEV (Generalized Extreme Value) distribution is useful for modeling heterogeneous bimodal data from different areas.

Usage

dbgev(y, mu = 1, sigma = 1, xi = 0.3, delta = 2)
pbgev(y, mu = 1, sigma = 1, xi = 0.3, delta = 2)
qbgev(p, mu = 1, sigma = 1, xi = 0.3, delta = 2)
rbgev(n, mu = 1, sigma = 1, xi = 0.3, delta = 2)

Arguments

Details

Density, distribution function, quantile function and random generation of bimodal GEV distribution with location parameter. In addition, maximum likelihood estimation based on real data is also provided.

Value

dbgev gives the density, pbgev gives the distribution function, qbgev gives the quantile function, and rbgev generates random bimodal GEV observations.

Note

The probability density of a GEV random variable; Y \sim F_{\xi, \sigma, \mu} is given by:

f_{\xi, \mu, \sigma}(y)= \begin{cases} \dfrac{1}{\sigma} \left[ 1+ \xi \left(\dfrac{y-\mu}{\sigma}\right) \right]^{(-1/\xi) -1} \exp\left\{- \left[1+\xi\left(\dfrac{y-\mu}{\sigma}\right)\right]^{-1/\xi}\right\} ,& \text{if } \xi \ne 0 \\ \dfrac{1}{\sigma} \exp \left\{ - \left( \dfrac{y-\mu}{\sigma}\right) - \exp \left[ - \left( \dfrac{y-\mu}{\sigma}\right) \right] \right\}, & \text{if } \xi = 0 , \end{cases}

where \xi and \sigma are the shape parameters and \mu is the location parameter.

The bimodal Generalized Extreme Value (GEV) model, denoted as BGEV, consists of composing the distribution of a random variable following the GEV distribution with a location parameter \mu=0, i.e., Y \sim F_{\xi, 0, \sigma}, with the transformation T_{\mu, \delta} defined below. Thus, the cumulative distribution function (CDF) of a random variable BGEV, denoted as X \sim F_{BG_{\xi, \mu, \sigma, \delta}}, is given by:

F_{BG_{\xi,\mu,\sigma, \delta}}(x) = F_{\xi, 0, \sigma}(T_{\mu, \delta}(x)),

where the function T_{\mu, \delta} is defined as:

T_{\mu, \delta}(x)=\left( x - \mu \right) \left| x -\mu \right| ^{\delta}, \quad \delta > -1, \quad \mu \in \mathbb{R}.

Moreover, the function T is invertible, with the inverse given by:

T^{-1}_{\mu, \delta}(x) = \text{sng}(x) |x|^{\dfrac{1}{\left( \delta +1 \right) }} + \mu.

Additionally, it is differentiable, and its derivative has the following form:

T'_{\mu, \delta}(x) = (\delta + 1 ) |x - \mu|^{\delta}.

Its probability density function X\sim F_{BG_{\xi,\mu,\sigma, \delta}} is given by

f_{BG_{\xi,\mu,\sigma, \delta}} (x)= \begin{cases} \dfrac{1}{\sigma} \left[ 1+ \xi \left(\dfrac{T_{\mu, \delta}(x)}{\sigma}\right) \right]^{(-1/\xi) -1} \exp\left[- \left[1+\xi\left(\dfrac{T_{\mu, \delta}(x)} {\sigma}\right)\right]^{-1/\xi}\right] T'_{\mu, \delta}(x) , & \xi \neq 0 \\ \dfrac{1}{\sigma} \exp \left( - \dfrac{T_{\mu, \delta}(x)}{\sigma} \right) \exp \left[- \exp \left( - \dfrac{T_{\mu, \delta}(x)}{\sigma}\right) \right] T'_{\mu, \delta}(x), & \xi=0. \end{cases}

Author(s)

Cira Otiniano Author [aut], Yasmin Lirio Author [aut], Thiago Sousa Developer [cre]

Maintainer: Thiago Sousa Developer <thiagoestatistico@gmail.com>

References

Otiniano, Cira EG, et al. (2023). A bimodal model for extremes data. Environmental and Ecological Statistics, 1-28. http://dx.doi.org/10.1007/s10651-023-00566-7

Examples

# generate 100 values distributed according to a bimodal GEV
x = rbgev(50, mu = 0.2, sigma = 1, xi = 0.5, delta = 0.2) 
# estimate the bimodal GEV parameters using the generated data
bgev.mle(x)

Parameter estimation of bimodal GEV distribution based on real data that appears to be bimodal.

Description

Finds the maximum likelihood estimators of the bimodal GEV distribution parameters by minimizing the log-likelihood function of the data.

Usage

bgev.mle(x, lower = c(-3, 0.1, -3, -0.9), upper = c(3, 3, 3,3), itermax = 50)

Arguments

Value

A list with components returned by the optim R function for which the main values are

Author(s)

Cira Otiniano Author [aut], Yasmin Lirio Author [aut], Thiago Sousa Developer [cre]

References

OTINIANO, Cira EG, et al. (2023). A bimodal model for extremes data. Environmental and Ecological Statistics, 1-28. http://dx.doi.org/10.1007/s10651-023-00566-7

Examples

# generate 100 values distributed according to a bimodal GEV
x = rbgev(50, mu = 0.2, sigma = 1, xi = 0.5, delta = 0.2) 
# estimate the bimodal GEV parameters using the generated data
bgev.mle(x)

Support of the bimodal GEV distribution

Description

When the shape parameter xi is different from zero, the support is truncated either at the left or at the right side of the real. Considering the support is particularly useful to estimating momoments and to compute the likelihood function.

Usage

bgev.support(mu, sigma, xi, delta)

Arguments

Value

It returns a vector representing the interval for which the density function is not null.

Note

The Support of the bimodal GEV distribution is given by the following equation:

\begin{cases} x \in \mathbb {R} : x \geq \mathbf{sng} \left(-\frac{\sigma}{\xi}\right) | \frac{\sigma}{\xi}| ^{\frac{1}{\delta+1}}+\mu, & \xi \neq 0 \\ x \in \mathbb {R}, & \xi =0. \end{cases}

Author(s)

Cira Otiniano Author [aut], Yasmin Lirio Author [aut], Thiago Sousa Developer [cre]

References

Otiniano, Cira EG, et al. (2023). A bimodal model for extremes data. Environmental and Ecological Statistics, 1-28. http://dx.doi.org/10.1007/s10651-023-00566-7

Examples

# Computes the support of a specific bimodal GEV distribution
support = bgev.support(mu=1, sigma=10, xi=0.3, delta=2)

Check implementation of a distribution in R.

Description

This is just a refactorization of the original R function distCheck to allow customization for heavy tailed distributions and distribution with constrained support. Assumes one has implemented density, probability, quantile and random generation for a new distribution. Originally implemented to be used with the bimodal GEV distribution

Usage

distCheck(fun = "norm", n = 1000, robust = FALSE, subdivisions = 1500, 
support.lower = -Inf, support.upper = Inf, 
var.exists = TRUE, print.result = TRUE, ...)

Arguments

Details

All tests results are organized into a list to make it easier to find where results went wrong and to make it scalable for testing distributions for several parameters.

Value

A list of lists with test results

list( functionTested = fun, functionParam = list(...),
test1.density = list(computed = NULL, expected = NULL, error.check = NULL),
test2.quantile.cdf = list(computed = NULL, expected = NULL, error.check = NULL),
test3.mean.var = list(computed = list(mean = NULL, var = NULL, log = NULL),
expected = list(mean = NULL, var = NULL, log = NULL),
error.check = NULL,
condition.is.var.finite = TRUE))

Author(s)

Cira Otiniano Author [aut], Yasmin Lirio Author [aut], Thiago Sousa Developer [cre]

References

Otiniano, Cira EG, et al. (2023). A bimodal model for extremes data. Environmental and Ecological Statistics, 1-28. http://dx.doi.org/10.1007/s10651-023-00566-7

Examples

# generate random values for the parameters and test the 
# bimodal gev distribution implementation
set.seed(1)
mu = runif(1,-2,2)
sigma = runif(1,0.1,2)
xi = runif(1,0.3,0.9) * sign(runif(1,-1,1))
delta = 1#runif(1,-0.6,2)
support = bgev.support(mu, sigma, xi, delta)
var.exists = ( xi != 0) & (xi < (delta + 1)/2)
ret = distCheck(fun="bgev", n = 2000, 
      support.lower = support[1], support.upper = support[2], 
      subdivisions = 5000, mu = mu, sigma = sigma, xi = xi, 
      delta = delta, var.exists = var.exists, print.result = TRUE)

Log likelihood function for the bimodal GEV distribution.

Description

Uses the density function to evaluate the likelihood. This is useful for the 'bgev.mle' function.

Usage

likbgev(y, theta = c(1, 1, 0.3, 2))

Arguments

Value

a unidimensional vector containing the computed log likelihood for y.

Author(s)

Cira Otiniano Author [aut], Yasmin Lirio Author [aut], Thiago Sousa Developer [cre]

References

Otiniano, Cira EG, et al. (2023). A bimodal model for extremes data. Environmental and Ecological Statistics, 1-28. http://dx.doi.org/10.1007/s10651-023-00566-7

Examples

# get random points from bimodal GEV
y = rbgev(100, mu = 1, sigma = 1, xi = 0.3, delta = 2)

# compute log-likelihood
likbgev (y, theta = c(1, 1, 0.3, 2))