Title:	Bimodal Gumbel Distribution
Version:	0.0.3
Description:	Bimodal Gumbel distribution. General functions for performing extreme value analysis.
Imports:	MCMCpack, MASS, quantreg, SparseM, coda, stats
License:	MIT + file LICENSE
SystemRequirements:	gcc (>= 4.0), gfortran, clang++
NeedsCompilation:	yes
URL:	https://CRAN.R-project.org/package=bgumbel
Language:	en-US
Encoding:	UTF-8
LazyData:	false
Date:	2021-03-31
RoxygenNote:	7.1.1
Packaged:	2021-03-31 21:14:59 UTC; pcbrom
Author:	Pedro C. Brom [aut, cre, cph] (Lattes: http://lattes.cnpq.br/0154064396756002), Cira E. G. Otiniano [aut, cph] (Lattes: http://lattes.cnpq.br/0307717595727716), Roberto Vila [aut, cph] (Lattes: http://lattes.cnpq.br/4978745622057574), Marcelo B. Pereira [aut, cph] (Lattes: https://lattes.cnpq.br/9358366674842900)
Maintainer:	Pedro C. Brom <pcbrom@gmail.com>
Repository:	CRAN
Date/Publication:	2021-03-31 22:10:07 UTC

Bimodal Gumbel: Density Function

Description

Bimodal Gumbel: Density Function

Usage

dbgumbel(x, mu, sigma, delta)

Arguments

Value

Vector.

Examples

dbgumbel(x = 0, mu = -2, sigma = 1, delta = -1)
curve(dbgumbel(x, mu = -2, sigma = 1, delta = -1), xlim = c(-5, 10), ylim= c(0, .4))
integrate(dbgumbel, mu = -2, sigma = 1, delta = -1, lower = -5, upper = 0)

Bimodal Gumbel: Theoretical E(X)

Description

Bimodal Gumbel: Theoretical E(X)

Usage

m1bgumbel(mu, sigma, delta)

Arguments

Value

Vector.

Examples

(EX <- m1bgumbel(mu = -2, sigma = 1, delta = -1))


# Comparison: Theoretical E(X) and empirical mean

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)
}

Bimodal Gumbel: Theoretical E(X^2)

Description

Bimodal Gumbel: Theoretical E(X^2)

Usage

m2bgumbel(mu, sigma, delta)

Arguments

Value

Vector.

Examples

(EX2 <- m2bgumbel(mu = -2, sigma = 1, delta = -1))


# Comparison: Theoretical E(X^2) and empirical second moment

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)
}

Bimodal Gumbel: Maximum Likelihood Estimation

Description

Bimodal Gumbel: Maximum Likelihood Estimation

Usage

mlebgumbel(data, theta, auto = TRUE)

Arguments

Value

List.

Examples

# 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'))

# Time to fit!

# 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]]
)

Bimodal Gumbel: Distribution Function

Description

Bimodal Gumbel: Distribution Function

Usage

pbgumbel(q, mu, sigma, delta, lower.tail = TRUE)

Arguments

Value

Vector.

Examples

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))

Bimodal Gumbel: Quantile Function

Description

Bimodal Gumbel: Quantile Function

Usage

qbgumbel(p, mu, sigma, delta, initial = -10, final = 10)

Arguments

Value

Vector.

Examples

# 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)

Bimodal Gumbel: Pseudo-Random Numbers Generator

Description

Bimodal Gumbel: Pseudo-Random Numbers Generator

Usage

rbgumbel(n, mu, sigma, delta)

Arguments

Value

A matrix nx1.

Examples

x <- rbgumbel(40000, 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')