cdfgno: Generalized normal distribution

Description

Distribution function and quantile function of the generalized normal distribution.

Usage

cdfgno(x, para = c(0, 1, 0))
quagno(f, para = c(0, 1, 0))

Value

cdfgno gives the distribution function;

quagno gives the quantile function.

Arguments

x: Vector of quantiles.
f: Vector of probabilities.
para: Numeric vector containing the parameters of the distribution, in the order $\xi, \alpha, k$ (location, scale, shape).

Details

The generalized normal distribution with location parameter $\xi$, scale parameter $\alpha$ and shape parameter $k$ has distribution function $$F(x)=\Phi(y)$$ where $$y=-k^{-1}\log\lbrace1-k(x-\xi)/\alpha\rbrace$$ and $\Phi(y)$ is the distribution function of the standard normal distribution, with $x$ bounded by $\xi+\alpha/k$ from below if $k<0$ and from above if $k>0$.

The generalized normal distribution contains as special cases the usual three-parameter lognormal distribution, corresponding to $k<0$, with a finite lower bound and positive skewness; the normal distribution, corresponding to $k=0$; and the reverse lognormal distribution, corresponding to $k>0$, with a finite upper bound and negative skewness. The two-parameter lognormal distribution, with a lower bound of zero and positive skewness, is obtained when $k<0$ and $\xi+\alpha/k=0$.

Examples

Run this code

# Random sample from the generalized normal distribution
# with parameters xi=0, alpha=1, k=-0.5.
quagno(runif(100), c(0,1,-0.5))

# The generalized normal distribution with parameters xi=1, alpha=1, k=-1,
# is the standard lognormal distribution.  An illustration:
fval<-seq(0.1,0.9,by=0.1)
cbind(fval, lognormal=qlnorm(fval), g.normal=quagno(fval, c(1,1,-1)))