TF: t family distribution for fitting a GAMLSS

In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

t family distribution for fitting a GAMLSS

Description

The function TF defines the t-family distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dTF, pTF, qTF and rTF define the density, distribution function, quantile function and random generation for the specific parameterization of the t distribution given in details below, with mean equal to \mu and standard deviation equal to \sigma (\frac{\nu}{\nu-2})^{0.5} with the degrees of freedom \nu The function TF2 is a different parametrization where sigma is the standard deviation.

Usage

TF(mu.link = "identity", sigma.link = "log", nu.link = "log")
dTF(x, mu = 0, sigma = 1, nu = 10, log = FALSE)
pTF(q, mu = 0, sigma = 1, nu = 10, lower.tail = TRUE, log.p = FALSE)
qTF(p, mu = 0, sigma = 1, nu = 10, lower.tail = TRUE, log.p = FALSE) 
rTF(n, mu = 0, sigma = 1, nu = 10)

TF2(mu.link = "identity", sigma.link = "log", nu.link = "logshiftto2")
dTF2(x, mu = 0, sigma = 1, nu = 10, log = FALSE)
pTF2(q, mu = 0, sigma = 1, nu = 10, lower.tail = TRUE, log.p = FALSE)
qTF2(p, mu = 0, sigma = 1, nu = 10, lower.tail = TRUE, log.p = FALSE)
rTF2(n, mu = 0, sigma = 1, nu = 10)

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 the degrees of freedom parameter values
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

Definition file for t family distribution TF():

f(y|\mu,\sigma, \nu)=\frac{1}{\sigma B(1/2,\nu/2)) \nu^{0.5}} \left[1+\frac{(y-\mu)^2}{\nu \sigma^2}\right]^{-(\nu+1)/2}

for -\infty<y<+\infty, -\infty<\mu<+\infty, \sigma>0 and \nu>0 see pp. 382-383 of Rigby et al. (2019). Note that z=(y-\mu)/\sigma has a standard t distribution with degrees of freedom \nu see pp. 382-383 of Rigby et al. (2019).

Definition file for t family distribution TF2():

f(y|\mu,\sigma, \nu)=\frac{1}{\sigma B(1/2, \nu/2) (\nu-2)^{0.5}} \left[1+\frac{(y-\mu)^2}{(\nu-2) \sigma^2}\right]^{-(\nu+1)/2}

for -\infty<y<+\infty, -\infty<\mu<+\infty, \sigma>0 and \nu>2 see pp. 382-383 of Rigby et al. (2019). Note that z=(y-\mu)/\sigma has a standard t distribution with degrees of freedom \nu see pp. 383-384 of Rigby et al. (2019).

Value

TF() returns a gamlss.family object which can be used to fit a t distribution in the gamlss() function. dTF() gives the density, pTF() gives the distribution function, qTF() gives the quantile function, and rTF() generates random deviates. The latest functions are based on the equivalent R functions for gamma distribution.

Note

\mu is the mean and \sigma [\nu/(\nu-2)]^{0.5} is the standard deviation of the t family distribution. \nu>0 is a positive real valued parameter.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Kalliope Akantziliotou

References

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. 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, \Sexpr[results=rd]{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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/b21973")}

See Also

gamlss.family

Examples

TF()# gives information about the default links for the t-family distribution 
# library(gamlss)
#data(abdom)
#h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=TF, data=abdom) # fits 
#plot(h)
newdata<-rTF(1000,mu=0,sigma=1,nu=5) # generates 1000 random observations
hist(newdata) 

gamlss.dist documentation built on Aug. 24, 2023, 1:06 a.m.