bgev-package {bgev}

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

y

a unidimensional vector containing the points to compute the density (dbgev) or the probability (pbgev) froma bimodal GEV distribution with parameters mu, sigma, xi and delta.

p

a unidimensional vector containing the probabilities used to compute the quantiles

n

an integer describing the number of observations to generate random bimodal GEV observations

mu

location parameter

sigma

shape parameter

xi

shape parameter

delta

bimodality parameter

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 <[e-mail]>

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)