PearsonVI {PearsonDS}

The Pearson Type VI (aka Beta Prime) Distribution

Description

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

Usage

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

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

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

rpearsonVI(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 VI distribution.

b

second shape parameter of Pearson type VI distribution.

location

location parameter of Pearson type VI distribution.

scale

scale parameter of Pearson type VI distribution.

params

vector/list of length 4 containing parameters a, b, location, scale for Pearson type VI 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

Pearson type VI distributions are (location-scale transformations of) Beta prime distributions, and Beta prime distributions are scaled F-distributions. The above functions are thus implemented via calls to df, pf, qf and rf (contained in package stats). The probability density function with parameters a, b, scale=s and location=\lambda is given by

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

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

Value

dpearsonVI gives the density, ppearsonVI gives the distribution function, qpearsonVI gives the quantile function, and rpearsonVI generates random deviates.

Note

Negative values for scale are permitted to allow for negative skewness.

Author(s)

Martin Becker [e-mail]

References

See the references in FDist.

See Also

FDist, PearsonDS-package, Pearson

Examples

## define Pearson type VI parameter set with a=2, b=3, location=1, scale=2
pVIpars <- list(a=2, b=3, location=1, scale=2)
## calculate probability density function
dpearsonVI(seq(1,6,by=1),params=pVIpars)
## calculate cumulative distribution function
ppearsonVI(seq(1,6,by=1),params=pVIpars)
## calculate quantile function
qpearsonVI(seq(0.1,0.9,by=0.2),params=pVIpars)
## generate random numbers
rpearsonVI(5,params=pVIpars)