```
## Example to define a new distribution:
## Students t-distribution with two parameters, df and mu:
## sub-Family for mu
## -> generate object of the class family from the package mboost
newStudentTMu <- function(mu, df){
# loss is negative log-Likelihood, f is the parameter to be fitted with
# id link -> f = mu
loss <- function(df, y, f) {
-1 * (lgamma((df + 1)/2) - lgamma(1/2) -
lgamma(df/2) - 0.5 * log(df) -
(df + 1)/2 * log(1 + (y - f)^2/(df )))
}
# risk is sum of loss
risk <- function(y, f, w = 1) {
sum(w * loss(y = y, f = f, df = df))
}
# ngradient is the negative derivate w.r.t. mu (=f)
ngradient <- function(y, f, w = 1) {
(df + 1) * (y - f)/(df + (y - f)^2)
}
# use the Family constructor of mboost
mboost::Family(ngradient = ngradient, risk = risk, loss = loss,
response = function(f) f,
name = "new Student's t-distribution: mu (id link)")
}
## sub-Family for df
newStudentTDf <- function(mu, df){
# loss is negative log-Likelihood, f is the parameter to be fitted with
# log-link: exp(f) = df
loss <- function( mu, y, f) {
-1 * (lgamma((exp(f) + 1)/2) - lgamma(1/2) -
lgamma(exp(f)/2) - 0.5 * f -
(exp(f) + 1)/2 * log(1 + (y - mu)^2/(exp(f) )))
}
# risk is sum of loss
risk <- function(y, f, w = 1) {
sum(w * loss(y = y, f = f, mu = mu))
}
# ngradient is the negative derivate of the loss w.r.t. f
# in this case, just the derivative of the log-likelihood
ngradient <- function(y, f, w = 1) {
exp(f)/2 * (digamma((exp(f) + 1)/2) - digamma(exp(f)/2)) -
0.5 - (exp(f)/2 * log(1 + (y - mu)^2 / (exp(f) )) -
(y - mu)^2 / (1 + (y - mu)^2 / exp(f)) * (exp(-f) + 1)/2)
}
# use the Family constructor of mboost
mboost::Family(ngradient = ngradient, risk = risk, loss = loss,
response = function(f) exp(f),
name = "Student's t-distribution: df (log link)")
}
## families object for new distribution
newStudentT <- Families(mu= newStudentTMu(mu=mu, df=df),
df=newStudentTDf(mu=mu, df=df))
### Do not test the following code per default on CRAN as it takes some time to run:
### usage of the new Student's t distribution:
library(gamlss) ## required for rTF
set.seed(1907)
n <- 5000
x1 <- runif(n)
x2 <- runif(n)
mu <- 2 -1*x1 - 3*x2
df <- exp(1 + 0.5*x1 )
y <- rTF(n = n, mu = mu, nu = df)
## model fitting
model <- glmboostLSS(y ~ x1 + x2, families = newStudentT,
control = boost_control(mstop = 100),
center = TRUE)
## shrinked effect estimates
coef(model, off2int = TRUE)
## compare to pre-defined three parametric t-distribution:
model2 <- glmboostLSS(y ~ x1 + x2, families = StudentTLSS(),
control = boost_control(mstop = 100),
center = TRUE)
coef(model2, off2int = TRUE)
## with effect on sigma:
sigma <- 3+ 1*x2
y <- rTF(n = n, mu = mu, nu = df, sigma=sigma)
model3 <- glmboostLSS(y ~ x1 + x2, families = StudentTLSS(),
control = boost_control(mstop = 100),
center = TRUE)
coef(model3, off2int = TRUE)
```

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