gamlss.dist (version 5.0-0)

PARETO2: Pareto Type 2 distribution for fitting a GAMLSS

Description

The functions PARETO2() and PARETO2o() define the Pareto Type 2 distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The parameters are mu and sigma in both functions but the parameterasation different. The mu is identical for both PARETO2() and PARETO2o(). The sigma in PARETO2o() is the inverse of the sigma in codePARETO2() and coresponse to the usual parameter alpha of the Patreto distribution. The functions dPARETO2, pPARETO2, qPARETO2 and rPARETO2 define the density, distribution function, quantile function and random generation for the PARETO2 parameterization of the Pareto type 2 distribution while the functions dPARETO2o, pPARETO2o, qPARETO2o and rPARETO2o define the density, distribution function, quantile function and random generation for the original PARETO2o parameterization of the Pareto type 2 distribution

Usage

PARETO2(mu.link = "log", sigma.link = "log")
dPARETO2(x, mu = 1, sigma = 0.5, log = FALSE)
pPARETO2(q, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
qPARETO2(p, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
rPARETO2(n, mu = 1, sigma = 0.5)
PARETO2o(mu.link = "log", sigma.link = "log")
dPARETO2o(x, mu = 1, sigma = 0.5, log = FALSE)
pPARETO2o(q, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
qPARETO2o(p, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
rPARETO2o(n, mu = 1, sigma = 0.5)

Arguments

mu.link

Defines the mu.link, with "`"' link sa the default for the mu parameter

sigma.link

Defines the sigma.link, with "`log"' as the default for the sigma parameter

x, q

vector of quantiles

mu

vector of location parameter values

sigma

vector of scale parameter values

log, log.p

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

lower.tail

logical; if TRUE (default), probabilities are P[X <= x], otherwise P[X > x]

p

vector of probabilities

n

number of observations. If length(n) > 1, the length is taken to be the number required

Value

returns a gamlss.family object which can be used to fit a Pareto type 2 distribution in the gamlss() function.

Details

The parameterization of the Pareto Type 2 distribution in the function PA2 is:

$$f(y|\mu, \sigma) = \frac{1}{\sigma} \mu^{\frac{1}{\sigma}} \, (y+mu)^{-\frac{1 }{sigma+1}}$$

for \(y>=0\), \(mu>0\) and \(sigma>0\).

References

Johnson, N., Kotz, S., and Balakrishnan, N. (1997). Discrete Multivariate Distributions. Wiley-Interscience, NY, USA.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also http://www.gamlss.org/).

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, http://www.jstatsoft.org/v23/i07.

See Also

gamlss.family

Examples

par(mfrow=c(2,2))
y<-seq(0.2,20,0.2)
plot(y, dPARETO2(y), type="l" , lwd=2)
q<-seq(0,20,0.2)
plot(q, pPARETO2(q), ylim=c(0,1), type="l", lwd=2) 
p<-seq(0.0001,0.999,0.05)
plot(p, qPARETO2(p), type="l", lwd=2)
dat <- rPARETO2(100)
hist(rPARETO2(100), nclass=30)
#summary(gamlss(a~1, family="PARETO2"))