R: Pareto distributions for fitting in GAMLSS

PARETO2 {gamlss.dist} | R Documentation

## 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)

### 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 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 2 distribution in the function `PARETO2` 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.

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. 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, https://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.

### 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"))

[Package gamlss.dist version 5.3-2 Index]