lmom (version 2.8)

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))

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).

Value

cdfgno gives the distribution function; quagno gives the quantile function.

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 \(\xi+\alpha/k=0\).

See Also

cdfln3 for the lmom package's version of the three-parameter lognormal distribution.

cdfnor for the lmom package's version of the normal distribution.

pnorm for the standard R version of the normal distribution.

plnorm for the standard R version of the two-parameter lognormal distribution.

Examples

# NOT RUN {
# 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)))
# }