Estimating the parameters of a Student t distribution.

```
studentt (ldf = "loglog", idf = NULL, tol1 = 0.1, imethod = 1)
studentt2(df = Inf, llocation = "identitylink", lscale = "loge",
ilocation = NULL, iscale = NULL, imethod = 1, zero = "scale")
studentt3(llocation = "identitylink", lscale = "loge", ldf = "loglog",
ilocation = NULL, iscale = NULL, idf = NULL,
imethod = 1, zero = c("scale", "df"))
```

llocation, lscale, ldf

ilocation, iscale, idf

Optional initial values. If given, the values must be in range. The default is to compute an initial value internally.

tol1

A positive value, the tolerance for testing whether an initial value is 1. Best to leave this argument alone.

df

Numeric, user-specified degrees of freedom. It may be of length equal to the number of columns of a response matrix.

imethod, zero

An object of class `"vglmff"`

(see `vglmff-class`

).
The object is used by modelling functions such as `vglm`

,
and `vgam`

.

The Student t density function is
$$f(y;\nu) = \frac{\Gamma((\nu+1)/2)}{\sqrt{\nu \pi} \Gamma(\nu/2)}
\left(1 + \frac{y^2}{\nu} \right)^{-(\nu+1)/2}$$
for all real \(y\).
Then \(E(Y)=0\) if \(\nu>1\) (returned as the fitted values),
and \(Var(Y)= \nu/(\nu-2)\)
for \(\nu > 2\).
When \(\nu=1\) then the Student \(t\)-distribution
corresponds to the standard Cauchy distribution,
`cauchy1`

.
When \(\nu=2\) with a scale parameter of `sqrt(2)`

then
the Student \(t\)-distribution
corresponds to the standard (Koenker) distribution,
`sc.studentt2`

.
The degrees of freedom can be treated as a parameter to be estimated,
and as a real and not an integer.
The Student t distribution is used for a variety of reasons
in statistics, including robust regression.

Let \(Y = (T - \mu) / \sigma\) where
\(\mu\) and \(\sigma\) are the location
and scale parameters respectively.
Then `studentt3`

estimates the location, scale and
degrees of freedom parameters.
And `studentt2`

estimates the location, scale parameters
for a user-specified degrees of freedom, `df`

.
And `studentt`

estimates the degrees of freedom parameter only.
The fitted values are the location parameters.
By default the linear/additive predictors are
\((\mu, \log(\sigma), \log\log(\nu))^T\)
or subsets thereof.

In general convergence can be slow, especially when there are covariates.

Student (1908)
The probable error of a mean.
*Biometrika*, **6**, 1--25.

Zhu, D. and Galbraith, J. W. (2010)
A generalized asymmetric Student-*t* distribution with
application to financial econometrics.
*Journal of Econometrics*, **157**, 297--305.

`uninormal`

,
`cauchy1`

,
`logistic`

,
`huber2`

,
`sc.studentt2`

,
`TDist`

,
`simulate.vlm`

.

# NOT RUN { tdata <- data.frame(x2 = runif(nn <- 1000)) tdata <- transform(tdata, y1 = rt(nn, df = exp(exp(0.5 - x2))), y2 = rt(nn, df = exp(exp(0.5 - x2)))) fit1 <- vglm(y1 ~ x2, studentt, data = tdata, trace = TRUE) coef(fit1, matrix = TRUE) fit2 <- vglm(y1 ~ x2, studentt2(df = exp(exp(0.5))), data = tdata) coef(fit2, matrix = TRUE) # df inputted into studentt2() not quite right fit3 <- vglm(cbind(y1, y2) ~ x2, studentt3, data = tdata, trace = TRUE) coef(fit3, matrix = TRUE) # }

Run the code above in your browser using DataCamp Workspace