PARETO2: Pareto distributions for fitting in GAMLSS

Description

The functions PARETO() defines the one parameter Pareto distribution for y>1.

The functions PARETO1() defines the one parameter Pareto distribution for y>0.

The functions PARETOo1() defines the one parameter Pareto distribution for y>mu therefor requires mu to be fixed.

The functions PARETO2() and PARETO2o() define the Pareto Type 2 distribution, for y>0, 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

PARETO(mu.link = "log")
dPARETO(x, mu = 1, log = FALSE)
pPARETO(q, mu = 1, lower.tail = TRUE, log.p = FALSE)
qPARETO(p, mu = 1, lower.tail = TRUE, log.p = FALSE)
rPARETO(n, mu = 1)
PARETO1(mu.link = "log")
dPARETO1(x, mu = 1, log = FALSE)
pPARETO1(q, mu = 1, lower.tail = TRUE, log.p = FALSE)
qPARETO1(p, mu = 1, lower.tail = TRUE, log.p = FALSE)
rPARETO1(n, mu = 1)
PARETO1o(mu.link = "log", sigma.link = "log")
dPARETO1o(x, mu = 1, sigma = 0.5, log = FALSE)
pPARETO1o(q, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
qPARETO1o(p, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
rPARETO1o(n, mu = 1, sigma = 0.5)
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)

Value

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

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

Author

Fiona McElduff, Bob Rigby and Mikis Stasinopoulos

Details

The parameterization of the one parameter Pareto distribution in the function PARETO is:

$$f(y|\mu) = \mu y^{\mu+1}$$

for $y>1$ and $\mu>0$.

The parameterization of the Pareto Type 1 original distribution in the function PARETO1o is:

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

for $y>=0$, $\mu>0$ and $\sigma>0$ see pp. 430-431 of Rigby et al. (2019).

The parameterization of the Pareto Type 2 original distribution in the function PARETO2o is:

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

for $y>=0$, $\mu>0$ and $\sigma>0$ see pp. 432-433 of Rigby et al. (2019).

The parameterization of the Pareto Type 2 distribution in the function PARETO2 is:

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

for $y>=0$, $\mu>0$ and $\sigma>0$ see pp.433-434 The parameterization of the Pareto Type 1 original distribution in the function PARETO1o is:

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

for $y>=0$, $\mu>0$ and $\sigma>0$ see pp. 430-431 of Rigby et al. (2019).

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.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, tools:::Rd_expr_doi("10.1201/9780429298547"). An older version can be found in https://www.gamlss.com/.

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, tools:::Rd_expr_doi("10.18637/jss.v023.i07").

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. tools:::Rd_expr_doi("10.1201/b21973")

(see also https://www.gamlss.com/).

Examples

Run this code

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"))

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