Bimodal Gumbel Distribution.

bgumbel

We propose a new model with three parameters called bimodal Gumbel (BG) as a generalization of the Gumbel distribution. The advantage of our model in comparison to other generalizations of the Gumbel distribution is the number of parameters and the fact that it can be used to model extreme data with one or two modes.

Installation

You can install the released version of bgumbel from CRAN with:

install.packages("bgumbel")

or using devtools

library(devtools)
install_github('pcbrom/bgumbel')

Load package

library(bgumbel)

Density Function

dbgumbel(x = 0, mu = -2, sigma = 1, delta = -1)
curve(dbgumbel(x, mu = -2, sigma = 1, delta = -1), xlim = c(-5, 10))
integrate(dbgumbel, mu = -2, sigma = 1, delta = -1, lower = -5, upper = 0)

Distribution Function

pbgumbel(0, mu = -2, sigma = 1, delta = -1)
integrate(dbgumbel, mu = -2, sigma = 1, delta = -1, lower = -Inf, upper = 0)
pbgumbel(0, mu = -2, sigma = 1, delta = -1, lower.tail = FALSE)
curve(pbgumbel(x, mu = -2, sigma = 1, delta = -1), xlim = c(-5, 10))

Quantile Function

It is recommended to set up a pbgumbel graph to see the starting and ending range of the desired quantile.

curve(pbgumbel(x, mu = -2, sigma = 1, delta = -1), xlim = c(-5, 5))
(value <- qbgumbel(.25, mu = -2, sigma = 1, delta = -1, initial = -4, final = -2))
pbgumbel(value, mu = -2, sigma = 1, delta = -1)

Pseudo-Random Numbers Generator

x <- rbgumbel(100000, mu = -2, sigma = 1, delta = -1)
hist(x, probability = TRUE)
curve(dbgumbel(x, mu = -2, sigma = 1, delta = -1), add = TRUE, col = 'blue')
lines(density(x), col = 'red')

Theoretical E(X) and empirical first moment

(EX <- m1bgumbel(mu = -2, sigma = 1, delta = -1))
x <- rbgumbel(100000, mu = -2, sigma = 1, delta = -1)
mean(x)
abs(EX - mean(x))/abs(EX) # relative error

# grid 1
mu <- seq(-5, 5, length.out = 100)
delta <- seq(-5, 5, length.out = 100)
z <- outer(X <- mu, Y <- delta, FUN = function(x, y) m1bgumbel(mu = x, sigma = 1, delta = y))
persp(x = mu, y = delta, z = z, theta = -60, ticktype = 'detailed')

# grid 2
mu <- seq(-5, 5, length.out = 100)
delta <- seq(-5, 5, length.out = 100)
sigmas <- seq(.1, 10, length.out = 20)
for (sigma in sigmas) {
  z <- outer(X <- mu, Y <- delta, FUN = function(x, y) m1bgumbel(mu = x, sigma = sigma, delta = y))
  persp(x = mu, y = delta, z = z, theta = -60, zlab = 'E(X)')
  Sys.sleep(.5)
}

Theoretical E(X²) and empirical second moment

(EX2 <- m2bgumbel(mu = -2, sigma = 1, delta = -1))
x <- rbgumbel(100000, mu = -2, sigma = 1, delta = -1)
mean(x^2)
abs(EX2 - mean(x))/abs(EX2) # relative error

# Variance
EX <- m1bgumbel(mu = -2, sigma = 1, delta = -1)
EX2 - EX^2
var(x)
abs(EX2 - EX^2 - var(x))/abs(EX2 - EX^2) # relative error

# grid 1
mu <- seq(-5, 5, length.out = 100)
delta <- seq(-5, 5, length.out = 100)
z <- outer(X <- mu, Y <- delta, FUN = function(x, y) m2bgumbel(mu = x, sigma = 1, delta = y))
persp(x = mu, y = delta, z = z, theta = -30, ticktype = 'detailed')

# grid 2
mu <- seq(-5, 5, length.out = 100)
delta <- seq(-5, 5, length.out = 100)
sigmas <- seq(.1, 10, length.out = 20)
for (sigma in sigmas) {
  z <- outer(X <- mu, Y <- delta, FUN = function(x, y) m2bgumbel(mu = x, sigma = sigma, delta = y))
  persp(x = mu, y = delta, z = z, theta = -45, zlab = 'E(X^2)')
  Sys.sleep(.5)
}

Maximum Likelihood Estimation

# Let's generate some values
set.seed(123)
x <- rbgumbel(1000, mu = -2, sigma = 1, delta = -1)

# Look for these references in the figure:
hist(x, probability = TRUE)
lines(density(x), col = 'blue')
abline(v = c(-2.5, -.5), col = 'red')
text(x = c(c(-2.5, -.5)), y = c(.05, .05), c('mu\nnear here', 'delta\nnear here'))

# If argument auto = FALSE
fit <- mlebgumbel(
  data = x,
  # try some values near the region. Format: theta = c(mu, sigma, delta)
  theta = c(-3, 2, -2),
  auto = FALSE
)
print(fit)

# If argument auto = TRUE
fit <- mlebgumbel(
  data = x,
  auto = TRUE
)
print(fit)

# Kolmogorov-Smirnov Tests
mu.sigma.delta <- fit$estimate$estimate
ks.test(
  x, 
  y = 'pbgumbel', 
  mu = mu.sigma.delta[[1]],
  sigma = mu.sigma.delta[[2]],
  delta = mu.sigma.delta[[3]]
)

Issues

Please, send to: https://github.com/pcbrom/bgumbel/issues

To cite package ‘bgumbel’ in publications use

citation("bgumbel")