PearsonIII: The Pearson Type III (aka Gamma) Distribution

In PearsonDS: Pearson Distribution System

The Pearson Type III (aka Gamma) Distribution

Description

Density, distribution function, quantile function and random generation for the Pearson type III (aka Gamma) distribution.

Usage

dpearsonIII(x, shape, location, scale, params, log = FALSE)

ppearsonIII(q, shape, location, scale, params, lower.tail = TRUE, 
            log.p = FALSE)

qpearsonIII(p, shape, location, scale, params, lower.tail = TRUE, 
            log.p = FALSE)

rpearsonIII(n, shape, location, scale, params)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations.
shape	shape parameter of Pearson type III distribution.
location	location parameter of Pearson type III distribution.
scale	scale parameter of Pearson type III distribution.
params	vector/list of length 3 containing parameters shape, location, scale for Pearson type III distribution (in this order!).
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE, probabilities are P[X\le x], otherwise, P[X>x].

Details

Essentially, the above functions are wrappers for dgamma, pgamma, qgamma and rgamma contained in package stats. As a minor (but important) extension, negative scale parameters (which reflect the distribution at location) are permitted to allow for negative skewness. The probability density function with parameters shape=a, scale=s and location=\lambda is thus given by

f(x)= \frac{1}{|s|^a \Gamma(a)} |x-\lambda|^{a-1} e^{-\frac{x-\lambda}{s}}

for s\ne 0, a>0 and \frac{x-\lambda}{s}\ge 0.

Value

dpearsonIII gives the density, ppearsonIII gives the distribution function, qpearsonIII gives the quantile function, and rpearsonIII generates random deviates.

Author(s)

Martin Becker [e-mail]

References

See the references in GammaDist.

See Also

GammaDist, PearsonDS-package, Pearson

Examples

## define Pearson type III parameter set with shape=3, location=1, scale=-2
pIIIpars <- list(shape=3, location=1, scale=-0.5)
## calculate probability density function
dpearsonIII(-4:1,params=pIIIpars)
## calculate cumulative distribution function
ppearsonIII(-4:1,params=pIIIpars)
## calculate quantile function
qpearsonIII(seq(0.1,0.9,by=0.2),params=pIIIpars)
## generate random numbers
rpearsonIII(5,params=pIIIpars)

PearsonDS documentation built on April 3, 2025, 5:37 p.m.