lmom (version 1.1)

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 $$y=-k^{-1}\log(1-k(x-\xi)/\alpha)$$ 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="" $k="0$;" reverse="" lognormal="">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

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 Rversion of the normal distribution. plnorm for the standard Rversion of the two-parameter lognormal distribution.

Examples

# 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)))