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

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@itwm.fraunhofer.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.
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 2.1.3 Index]