OpVaR (version 1.2)

gh: Tukey's gh Distribution

Description

Density, distribution function, quantile function, and random generation for Tukey's gh distribution.

Usage

dgh(x, A, B, g, h, log = FALSE)
pgh(q, A, B, g, h, log.p = FALSE)
qgh(q, A, B, g, h, log.p =FALSE)
rgh(n, A, B, g, h)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations.

A

location parameter.

B

scale parameter, has to be positive.

g

skewness parameter.

h

kurtosis parameter.

log, log.p

logical; if TRUE, probabilities p are given as log(p)

Value

dgh gives the density, pgh gives the distribution function, qgh gives the quantile function, and rgh generates random deviates.

The length of the result is determined by n for rgh, and is the length of the numerical arguments for the other function.

Details

Tukey's gh distribution with location parameter A, scale parameter B, skewness parameter g, and kurtosis parameter h is obtained by transforming a standard normal variable X by

$$T(X)=A+B exp(h/2 X^2)(exp(gX)-1)/g$$

if g is not equal to zero, and else by

$$T(X)=A+B exp(h/2 X^2) X.$$

References

Tukey, J. W. (1960): The Practical Relationship between the Common Transformations of Counts of Amounts. Technical Report 36, Princeton University Statistical Techniques Research Group, Princeton.

Klein, I. and Fischer, M. (2002): Symmetrical gh-transformed Distributions. in S. Mittnek and I. Klein: Contributions to Modern Econometrics, Kluwer Academic Publishers.

Pfaelzner, F. (2017): Einsatz von Tukey-type Verteilungen bei der Quantifizierung von operationellen Risiken. MMasterthesis Friedrich-Alexander-University Erlangen-Nueremberg.

Examples

# NOT RUN {
##Parameters  for a gh distribution
  A=500 
  B=3
  g=0.2
  h=0.5 
  
  hist(rgh(n=1000,A,B,g,h))
  curve(dgh(x,A,B,g,h),480,520)
  curve(pgh(x,A,B,g,h),480,520)
  curve(qgh(x,A,B,g,h),0,1)
# }