This function defines the Power Exponential t distribution (NET), a four parameter distribution, for a gamlss.family object to be used for a
GAMLSS fitting using the function gamlss(). The functions dNET,
pNET define the density and distribution function the NET distribution.
NET(mu.link = "identity", sigma.link = "log", nu.link ="identity",
tau.link = "identity")
pNET(q, mu=0, sigma=1, nu=1.5, tau=2, lower.tail = TRUE, log.p = FALSE)
dNET(x, mu=0, sigma=1, nu=1.5, tau=2, log=FALSE)
qNET(p, mu=0, sigma=1, nu=1.5, tau=2, lower.tail = TRUE, log.p = FALSE)
rNET(n, mu=0, sigma=1, nu=1.5, tau=2)NET() returns a gamlss.family object which can be used to fit a Box Cox Power Exponential distribution in the gamlss() function.
dNET() gives the density, pNET() gives the distribution
function.
Defines the mu.link, with "identity" link as the default for the mu parameter. Other links are "inverse", "log" and "own"
Defines the sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse", "identity" and "own"
Defines the nu.link, and because nu is fixed we use "identity" link
Defines the tau.link, and because tau is fixed we use "identity" link
vector of quantiles
vector of probabilities
number of observations.
vector of location parameter values
vector of scale parameter values
vector of nu parameter values
vector of tau parameter values
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou
The NET distribution was introduced by Rigby and Stasinopoulos (1994) as a robust distribution for a response
variable with heavier tails than the normal. The NET
distribution is the abbreviation of the Normal Exponential Student t distribution.
The NET distribution is a four parameter continuous distribution, although in the GAMLSS implementation only
the two parameters, mu and sigma, of the distribution are modelled with
nu and tau fixed.
The distribution takes its names because it is normal up to
nu, Exponential from nu to tau (hence abs(nu)<=abs(tau)) and Student-t with
nu*tau-1 degrees of freedom after tau. Maximum
likelihood estimator of the third and forth parameter can be
obtained, using the GAMLSS functions, find.hyper or prof.dev.
For more details about the NET distribution please refer to pp. 393-396 of of Rigby et al. (2019).
Rigby, R. A. and Stasinopoulos, D. M. (1994), Robust fitting of an additive model for variance heterogeneity, COMPSTAT : Proceedings in Computational Statistics, editors:R. Dutter and W. Grossmann, pp 263-268, Physica, Heidelberg.
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").
SStasinopoulos 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/).
gamlss.family, BCPE
NET() #
data(abdom)
plot(function(x)dNET(x, mu=0,sigma=1,nu=2, tau=3), -5, 5)
plot(function(x)pNET(x, mu=0,sigma=1,nu=2, tau=3), -5, 5)
# fit NET with nu=1 and tau=3
# library(gamlss)
#h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=NET,
# data=abdom, nu.start=2, tau.start=3)
#plot(h)
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