Generalized normal distribution
Distribution function and quantile function of the generalized normal distribution.
cdfgno(x, para = c(0, 1, 0)) quagno(f, para = c(0, 1, 0))
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). |
The generalized normal distribution with location parameter xi, scale parameter alpha and shape parameter k has distribution function
F(x) = Phi(y)
where
y = (-1/k) 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; 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.
cdfgno gives the distribution function;
quagno gives the quantile function.
The functions expect the distribution parameters in a vector,
rather than as separate arguments as in the standard R
distribution functions pnorm, qnorm, etc.
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.
# 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)))
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