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.
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)
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.
Defines the mu.link, with "identity" link as the default for the mu parameter
Defines the sigma.link, with "log" link as the default for the sigma parameter
Defines the nu.link, with "log" link as the default for the nu parameter
vector of quantiles
vector of location parameter values
vector of scale parameter values
vector of the degrees of freedom 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]
vector of probabilities.
number of observations. If length(n) > 1, the length is
taken to be the number required
Mikis Stasinopoulos, Bob Rigby and Kalliope Akantziliotou
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).
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, 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. tools:::Rd_expr_doi("10.1201/b21973")
gamlss.family
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)
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