| PE {gamlss.dist} | R Documentation |
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.
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)
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 |
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 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 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))
returns a gamlss.family object which can be used to fit a Power Exponential distribution in the gamlss() function.
mu is the mean and sigma is the standard deviation of the Power Exponential distribution
Mikis Stasinopoulos d.stasinopoulos@londonmet.ac.uk, Bob Rigby r.rigby@londonmet.ac.uk
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.
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")