R: Class "Td"

Td-class {distr} | R Documentation

## Class "Td"

### Description

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 from the Class

Objects can be created by calls of the form `Td(df)`. This object is a t distribution.

### Slots

`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.

### Extends

Class `"AbscontDistribution"`, directly.
Class `"UnivariateDistribution"`, by class `"AbscontDistribution"`.
Class `"Distribution"`, by class `"AbscontDistribution"`.

### Methods

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

### Ad hoc methods

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.

### Note

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 non-centrality parameter `ncp`= (mu - m0) * sqrt(n)/sigma.

### Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

### See Also

`TParameter-class`, `AbscontDistribution-class`, `Reals-class`, `rt`

### Examples

    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...

[Package distr version 2.8.0 Index]