dFG {GUD}

The Flexible Gumbel Distribution

Description

The Flexible Gumbel Distribution

Usage

dFG(x, w, loc, sigma1, sigma2)

rFG(n, w, loc, sigma1, sigma2)

Arguments

x

vector of quantiles.

w

vector of weight parameters.

loc

vector of the location parameters.

sigma1

vector of the scale parameters of the left skewed part.

sigma2

vector of the scale parameters of the right skewed part.

n

number of observations.

Details

The Gumbel distribution has the density

f_{\text {Gumbel }}(y \mid \theta, \sigma)=\frac{1}{\sigma} \exp \left\{-\frac{y-\theta}{\sigma}-\exp \left(-\frac{y-\theta}{\sigma}\right)\right\},

where \theta \in \mathbb{R} is the mode as the location parameter, \sigma > 0 is the scale parameter.

The flexible Gumbel distribution has the density

f_{\mathrm{FG}}\left(y \mid w, \theta, \sigma_1, \sigma_2\right)=w f_{\text {Gumbel }}\left(-y \mid-\theta, \sigma_1\right)+(1-w) f_{\text {Gumbel }}\left(y \mid \theta, \sigma_2\right) .

where w \in [0,1] is the weight parameter, \sigma_{1} > 0 is the scale parameter of the left skewed part and \sigma_{2} > 0 is the scale parameter of the right skewed part.

Value

dFG gives the density. rFG generates random deviates.

References

Liu Q, Huang X, Bai R (2024). “Bayesian Modal Regression Based on Mixture Distributions.” Computational Statistics & Data Analysis, 108012. doi:10.1016/j.csda.2024.108012.

Examples

set.seed(100)
require(graphics)

# Random Number Generation
X <- rFG(n = 1e5, w = 0.3, loc = 0, sigma1 = 1, sigma2 = 2)

# Plot the histogram
hist(X, breaks = 100, freq = FALSE)

# The red dashed line should match the underlining histogram
points(x = seq(-10,20,length.out = 1000),
       y = dFG(x = seq(-10,20,length.out = 1000),
               w = 0.3, loc = 0, sigma1 = 1, sigma2 = 2),
       type = "l",
       col = "red",
       lwd = 3,
       lty = 2)