PE: Power Exponential distribution for fitting a GAMLSS

Description Usage Arguments Details Value Note Author(s) References See Also Examples

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

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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)=(1/sigma)*(nu*exp(-0.5*|z/c|^nu))/(c*2^(1+1/nu)*Gamma(1/nu))

where c=[2^(-2/nu)Gamma(1/nu)/Gamma(3/nu)]^0.5, for y=(-Inf,+Inf), μ=(-Inf,+Inf), σ>0 and ν>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 rages above)

Thw Power Exponential distribution (PE2) is defined as

f(y|mu,sigma,nu)=(nu *exp(-abs(z)^2))/(2*sigma*Gamma(1/nu))

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 d.stasinopoulos@londonmet.ac.uk, Bob Rigby r.rigby@londonmet.ac.uk

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.

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.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.

See Also

gamlss.family, BCPE

Examples

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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") 

Example output

Loading required package: MASS

GAMLSS Family: PE Power Exponential 
Link function for mu   : identity 
Link function for sigma: log 
Link function for nu   : log 

gamlss.dist documentation built on May 2, 2019, 5:20 p.m.

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