# PARETO2: Pareto Type 2 distribution for fitting a GAMLSS In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

## 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

 ``` 1 2 3 4 5 6 7 8 9 10``` ```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

## Details

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

f(y|mu, sigma) = (1/sigma) mu^(1/sigma) (y+mu)^(-(1/sigma+1))

for y>=0, mu>0 and sigma>0.

## Value

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

## Author(s)

Fiona McElduff, Bob Rigby and Mikis Stasinopoulos

## 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.

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.

`gamlss.family`
 ``` 1 2 3 4 5 6 7 8 9 10``` ```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")) ```