PearsonI: The Pearson Type I (aka Beta) Distribution

In PearsonDS: Pearson Distribution System

The Pearson Type I (aka Beta) Distribution

Description

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

Usage

dpearsonI(x, a, b, location, scale, params, log = FALSE)

ppearsonI(q, a, b, location, scale, params, lower.tail = TRUE, 
          log.p = FALSE)

qpearsonI(p, a, b, location, scale, params, lower.tail = TRUE, 
          log.p = FALSE)

rpearsonI(n, a, b, location, scale, params)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations.
a	first shape parameter of Pearson type I distribution.
b	second shape parameter of Pearson type I distribution.
location	location parameter of Pearson type I distribution.
scale	scale parameter of Pearson type I distribution.
params	vector/list of length 4 containing parameters a, b, location, scale for Pearson type I 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, Pearson type I distributions are (location-scale transformations of) Beta distributions, the above functions are thus simple wrappers for dbeta, pbeta, qbeta and rbeta contained in package stats. The probability density function with parameters a, b, scale=s and location=\lambda is given by

f(x)=\frac{1}{|s|}\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\left(\frac{x-\lambda}{s} \right)^{a-1}\left(1-\frac{x-\lambda}{s}\right)^{b-1}

for a>0, b>0, s\ne 0, 0<\frac{x-\lambda}{s}<1.

Value

dpearsonI gives the density, ppearsonI gives the distribution function, qpearsonI gives the quantile function, and rpearsonI generates random deviates.

Note

Negative values for scale are not excluded, albeit negative skewness is usually obtained by switching a and b (such that a>b) and not by using negative values for scale (and a<b).

Author(s)

Martin Becker [e-mail]

References

See the references in Beta.

See Also

Beta, PearsonDS-package, Pearson

Examples

## define Pearson type I parameter set with a=2, b=3, location=1, scale=2
pIpars <- list(a=2, b=3, location=1, scale=2)
## calculate probability density function
dpearsonI(seq(1,3,by=0.5),params=pIpars)
## calculate cumulative distribution function
ppearsonI(seq(1,3,by=0.5),params=pIpars)
## calculate quantile function
qpearsonI(seq(0.1,0.9,by=0.2),params=pIpars)
## generate random numbers
rpearsonI(5,params=pIpars)

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