dgnd: The Generalized Normal Distribution (GND)

In cmgnd: Constrained Mixture of Generalized Normal Distributions

The Generalized Normal Distribution (GND)

Description

Density function for the GND with location parameter mu, scale parameter sigma and shape parameter nu.

Usage

dgnd(x, mu = 0, sigma = 1, nu = 2)

Arguments

x	A numeric vector of observations.
mu	A numeric value indicating the location parameter \mu.
sigma	A numeric value indicating the scale parameter \sigma.
nu	A numeric value indicating the shape parameter \nu.

Details

If mu, sigma and nu are not specified they assume the default values of 0, 1 and 2, respectively. The GND distribution has density

f_{GND}(x|\mu,\sigma,\nu)=\frac{\nu}{2\sigma\Gamma(1\mathbin{/}\nu)}\exp\Biggr\{-\Biggr|\frac{x-\mu}{\sigma}\Biggr|^\nu\Biggr\}.

The shape parameter \nu controls both the peakedness and tail weights. If \nu=1 the GND reduces to the Laplace distribution and if \nu=2 it coincides with the normal distribution. It is noticed that 1<\nu<2 yields an intermediate distribution between the normal and the Laplace distribution. As limit cases, for \nu\rightarrow\infty the distribution tends to a uniform distribution, while for \nu\rightarrow0 it will be impulsive.

Value

dgnd returns the density.

References

Nadarajah, S. (2005). A generalized normal distribution. Journal of Applied Statistics, 32(7):685–694.

cmgnd documentation built on Aug. 8, 2025, 6:34 p.m.