PE {gamlss.dist}

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)

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

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)

Value

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

Note

\mu is the mean and \sigma is the standard deviation of the Power Exponential distribution

Author(s)

Mikis Stasinopoulos, Bob Rigby

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, 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, 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. doi:10.1201/b21973

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

See Also

gamlss.family, BCPE

Examples

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")