yJohnsonDistribution: Standard normal (Z) to Johnson variable (Y) transformation

In JohnsonDistribution: Johnson Distribution

Description

A normal variable with mean zero and variance one is transformed to a Johnson distribution variate specified by the Johnson distribution parameters. This is useful in simulating random variables from a specified Johnson distribution and in computing the quantiles for a Johnson distribution.

Usage

yJohnsonDistribution(z, ITYPE, GAMMA, DELTA, XLAM, XI)

Arguments

z	vector of standard normal variables
ITYPE	is 1, SL; 2 for SU, 3 for SB and 4 for Normal
GAMMA	parameter in Johnson distribution
DELTA	parameter in Johnson distribution
XLAM	parameter in Johnson distribution
XI	parameter in Johnson distribution

Details

Our function provides an interface to the Fortran algorithm AS 100 (Hill, 1976).

Value

Corresponding vector of Johnson distribution variables.

Note

The input parameters ITYPE, GAMMA, DELTA, XLAM, XI must all be scalars. An error is given if they are not.

Author(s)

A. I. McLeod and Leanna King

References

I. D. Hill, Algorithm AS 100. Normal-Johnson and Johnson-normal transformations, Appl. Statist.,25, No. 2, 190-192 (1976).

See Also

zJohnsonDistribution, FitJohnsonDistribution

Examples

#Example 1.
#fit SL with mean 1, variance 1, skewness 2 then find corresponding variate to Z=2
ans <- FitJohnsonDistribution(1, 1, 2, -1)
GAMMA <- ans["GAMMA"]
DELTA <- ans["DELTA"]
XLAM <- ans["XLAM"]
XI <- ans["XI"]
ITYPE <- 1
z <- 2
yJohnsonDistribution(z, ITYPE, GAMMA, DELTA, XLAM, XI)
#Example 2: find quantiles of SL distribution
#The 0.01, 0.05, 0.1, 0.25, 0.5, 0.75, 0.9, 0.95, 0.99
#quantiles for an SL distribution are found and a qq plot is produced.
#SL distribution parameters is determined
#    with mean 1, standard deviation 1, skewness 3
ans <- FitJohnsonDistribution(1, 1, 3, -1)
GAMMA <- ans["GAMMA"]
DELTA <- ans["DELTA"]
XLAM <- ans["XLAM"]
XI <- ans["XI"]
ITYPE <- 1
p<-c(0.01, 0.05, 0.1, 0.25, 0.5, 0.75, 0.9, 0.95, 0.99)
z <- qnorm(p)
y<-yJohnsonDistribution(z, ITYPE, GAMMA, DELTA, XLAM, XI)
plot(z,y,xlab="normal quantiles", ylab="SL quantiles")
#
#Example 3: simulate SL distribution
#with mean 1, sd 1 and skewness 3
#plot estimated pdf
ans <- FitJohnsonDistribution(1, 1, 3, -1)
GAMMA <- ans["GAMMA"]
DELTA <- ans["DELTA"]
XLAM <- ans["XLAM"]
XI <- ans["XI"]
ITYPE <- 1
z <- rnorm(1000)
y <- yJohnsonDistribution(z, ITYPE, GAMMA, DELTA, XLAM, XI)
pdf <- density(y, bw = "sj")
plot(pdf, main="Estimated pdf of SL with mean 1, sd 1, g1 3", xlab="x", ylab="est.pdf(x)" )

Example output

[1] 3.663545

JohnsonDistribution documentation built on May 29, 2017, 1:38 p.m.