Description Objects from the Class Slots Extends Methods Ad hoc methods Note Author(s) See Also Examples

The *t* distribution with `df`

*= n* degrees of
freedom has density

*f(x) = Gamma((n+1)/2) / (sqrt(n pi) Gamma(n/2)) (1 + x^2/n)^-((n+1)/2)*

for all real *x*.
It has mean *0* (for *n > 1*) and
variance *n/(n-2)* (for *n > 2*).
C.f. `rt`

Objects can be created by calls of the form `Td(df)`

.
This object is a *t* distribution.

`img`

Object of class

`"Reals"`

: The domain of this distribution has got dimension 1 and the name "Real Space".`param`

Object of class

`"TParameter"`

: the parameter of this distribution (df), declared at its instantiation`r`

Object of class

`"function"`

: generates random numbers (calls function`rt`

)`d`

Object of class

`"function"`

: density function (calls function`dt`

)`p`

Object of class

`"function"`

: cumulative function (calls function`pt`

)`q`

Object of class

`"function"`

: inverse of the cumulative function (calls function`qt`

)`.withArith`

logical: used internally to issue warnings as to interpretation of arithmetics

`.withSim`

logical: used internally to issue warnings as to accuracy

`.logExact`

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

`.lowerExact`

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

`Symmetry`

object of class

`"DistributionSymmetry"`

; used internally to avoid unnecessary calculations.

Class `"AbscontDistribution"`

, directly.

Class `"UnivariateDistribution"`

, by class `"AbscontDistribution"`

.

Class `"Distribution"`

, by class `"AbscontDistribution"`

.

- initialize
`signature(.Object = "Td")`

: initialize method- df
`signature(object = "Td")`

: returns the slot df of the parameter of the distribution- df<-
`signature(object = "Td")`

: modifies the slot df of the parameter of the distribution- ncp
`signature(object = "Td")`

: returns the slot ncp of the parameter of the distribution- ncp<-
`signature(object = "Td")`

: modifies the slot ncp of the parameter of the distribution

For R Version `<2.3.0`

ad hoc methods are provided for slots `q`

, `r`

if `ncp!=0`

;
for R Version `>=2.3.0`

the methods from package stats are used.

The general *non-central* *t*
with parameters *(df,Del)* `= (df, ncp)`

is defined as a the distribution of
*
T(df,Del) := (U + Del) / (Chi(df) / sqrt(df)) *
where *U* and *Chi(df)* are independent random
variables, *U \~ N(0,1)*, and
*Chi(df)^2*
is chi-squared, see `rchisq`

.

The most used applications are power calculations for *t*-tests:

Let *T= (mX - m0) / (S/sqrt(n))*
where
*mX* is the `mean`

and *S* the sample standard
deviation (`sd`

) of *X_1,X_2,…,X_n* which are i.i.d.
*N(mu,sigma^2)*.
Then *T* is distributed as non-centrally *t* with
`df`

*= n-1*
degrees of freedom and **n**on-**c**entrality **p**arameter
`ncp`

*= (mu - m0) * sqrt(n)/sigma*.

Thomas Stabla [email protected],

Florian Camphausen [email protected],

Peter Ruckdeschel [email protected],

Matthias Kohl [email protected]

`TParameter-class`

,
`AbscontDistribution-class`

,
`Reals-class`

,
`rt`

1 2 3 4 5 6 7 8 9 10 11 | ```
T <- Td(df = 1) # T is a t distribution with df = 1.
r(T)(1) # one random number generated from this distribution, e.g. -0.09697573
d(T)(1) # Density of this distribution is 0.1591549 for x = 1.
p(T)(1) # Probability that x < 1 is 0.75.
q(T)(.1) # Probability that x < -3.077684 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
df(T) # df of this distribution is 1.
df(T) <- 2 # df of this distribution is now 2.
Tn <- Td(df = 1, ncp = 5)
# T is a noncentral t distribution with df = 1 and ncp = 5.
d(Tn)(1) ## from R 2.3.0 on ncp no longer ignored...
``` |

distr documentation built on July 9, 2018, 3 a.m.

Embedding an R snippet on your website

Add the following code to your website.

For more information on customizing the embed code, read Embedding Snippets.