# tt

##### The Truncated Student-t Distribution

Density, distribution function, quantile function, and random generation
for the left and/or right truncated student-t distribution with `df`

degrees of freedom.

- Keywords
- distribution

##### Usage

`dtt(x, location = 0, scale = 1, df, left = -Inf, right = Inf, log = FALSE)`ptt(q, location = 0, scale = 1, df, left = -Inf, right = Inf,
lower.tail = TRUE, log.p = FALSE)

rtt(n, location = 0, scale = 1, df, left = -Inf, right = Inf)

qtt(p, location = 0, scale = 1, df, left = -Inf, right = Inf,
lower.tail = TRUE, log.p = FALSE)

##### Arguments

- x, q
vector of quantiles.

- p
vector of probabilities.

- n
number of observations. If

`length(n) > 1`

, the length is taken to be the number required.- location
location parameter.

- scale
scale parameter.

- df
degrees of freedom (> 0, maybe non-integer).

`df = Inf`

is allowed.- left
left censoring point.

- right
right censoring point.

- 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].

##### Details

If `location`

or `scale`

are not specified they assume the default values
of `0`

and `1`

, respectively. `left`

and `right`

have the defaults `-Inf`

and `Inf`

respectively.

The truncated student-t distribution has density $$f(x) = 1/\sigma \tau((x - \mu)/\sigma) / (T((right - \mu)/\sigma) - T((left - \mu)/\sigma))$$ for \(left \le x \le right\), and 0 otherwise.

where \(T\) and \(\tau\) are the cumulative distribution function
and probability density function of the student-t distribution with
`df`

degrees of freedom respectively, \(\mu\) is the location of the
distribution, and \(\sigma\) the scale.

##### Value

`dtt`

gives the density, `ptt`

gives the distribution
function, `qtt`

gives the quantile function, and `rtt`

generates random deviates.

##### See Also

*Documentation reproduced from package crch, version 1.0-4, License: GPL-2 | GPL-3*