IGAMMA: Inverse Gamma distribution for fitting a GAMLSS

Description

The function IGAMMA() defines the Inverse Gamma distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(), with parameters mu (the mode) and sigma. The functions dIGAMMA, pIGAMMA, qIGAMMA and rIGAMMA define the density, distribution function, quantile function and random generation for the IGAMMA parameterization of the Inverse Gamma distribution.

Usage

IGAMMA(mu.link = "log", sigma.link="log")
dIGAMMA(x, mu = 1, sigma = .5, log = FALSE)
pIGAMMA(q, mu = 1, sigma = .5, lower.tail = TRUE, log.p = FALSE)
qIGAMMA(p, mu = 1, sigma = .5, lower.tail = TRUE, log.p = FALSE)
rIGAMMA(n, mu = 1, sigma = .5)

Arguments

mu.link

Defines the mu.link, with log link as 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

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]

vector of probabilities

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 an Inverse Gamma distribution in the gamlss() function.

Details

The parameterization of the Inverse Gamma distribution in the function IGAMMA is $$f(y|\mu, \sigma) = \frac{\left[\mu\,(\alpha+1)\right]^{\alpha}}{\Gamma(\alpha)} \,y^{-(\alpha+1)}\, \exp{\left[-\frac{\mu\,(\alpha+1)}{y}\right]}$$ where $alpha = 1/(sigma^2)$ for $y>0$, $mu>0$ and $sigma>0$.

References

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 http://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, 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.

Examples

Run this code

# NOT RUN {
par(mfrow=c(2,2))
y<-seq(0.2,20,0.2)
plot(y, dIGAMMA(y), type="l")
q <- seq(0.2, 20, 0.2)
plot(q, pIGAMMA(q), type="l")
p<-seq(0.0001,0.999,0.05)
plot(p , qIGAMMA(p), type="l")
dat <- rIGAMMA(50)
hist(dat)
#summary(gamlss(dat~1, family="IGAMMA"))
# }

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