PE: Power Exponential distribution for fitting a GAMLSS

Description

The functions define the Power Exponential distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dPE, pPE, qPE and rPE define the density, distribution function, quantile function and random generation for the specific parameterization of the power exponential distribution showing below. The functions dPE2, pPE2, qPE2 and rPE2 define the density, distribution function, quantile function and random generation of a standard parameterization of the power exponential distribution.

Usage

PE(mu.link = "identity", sigma.link = "log", nu.link = "log")
dPE(x, mu = 0, sigma = 1, nu = 2, log = FALSE)
pPE(q, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)
qPE(p, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)
rPE(n, mu = 0, sigma = 1, nu = 2)
PE2(mu.link = "identity", sigma.link = "log", nu.link = "log")
dPE2(x, mu = 0, sigma = 1, nu = 2, log = FALSE)
pPE2(q, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)
qPE2(p, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)
rPE2(n, mu = 0, sigma = 1, nu = 2)

Value

returns a gamlss.family object which can be used to fit a Power Exponential distribution in the gamlss() function.

Arguments

mu.link: Defines the mu.link, with "identity" link as the default for the mu parameter
sigma.link: Defines the sigma.link, with "log" link as the default for the sigma parameter
nu.link: Defines the nu.link, with "log" link as the default for the nu parameter
x,q: vector of quantiles
mu: vector of location parameter values
sigma: vector of scale parameter values
nu: vector of kurtosis parameter
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

Mikis Stasinopoulos, Bob Rigby

Details

Power Exponential distribution (PE) is defined as:

$$f(y|\mu,\sigma,\nu)=\frac{\nu \exp[- |z|^{\nu}]}{2 c \sigma \Gamma(\frac{1}{\nu})}$$ where $z=(y-\mu)/ c \sigma$ and $c^2=\Gamma(1/\nu)\left[/\Gamma(3/\nu) \right]^{-1}$, for $y=(-\infty,+\infty)$, $\mu=(-\infty,+\infty)$, $\sigma>0$ and $\nu>0$. This parametrization was used by Nelson (1991) and ensures $\mu$ is the mean and $\sigma$ is the standard deviation of y (for all parameter values of $\mu$, $\sigma$ and $\nu$ within the ranges above), see p. 374 of Rigby et al. (2019)

Thw Power Exponential distribution (PE2) is defined as $$f(y|\mu,\sigma,\nu)=\frac{\nu \exp[-\left|z\right|^\nu]} {2\sigma \Gamma\left(\frac{1}{\nu}\right)}$$ see p. 376 of Rigby et al. (2019)

References

Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: a new approach. Econometrica, 57, 347-370.

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

PE()# gives information about the default links for the Power Exponential distribution  
# library(gamlss)
# data(abdom)
# h1<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=PE, data=abdom) # fit
# h2<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=PE2, data=abdom) # fit 
# plot(h1)
# plot(h2)
# leptokurtotic
plot(function(x) dPE(x, mu=10,sigma=2,nu=1), 0.0, 20, 
 main = "The PE  density mu=10,sigma=2,nu=1")
# platykurtotic
plot(function(x) dPE(x, mu=10,sigma=2,nu=4), 0.0, 20, 
 main = "The PE  density mu=10,sigma=2,nu=4")

Run the code above in your browser using DataLab