Td-class

0th

Percentile

Class "Td"

The \(t\) distribution with df \(= \nu\) degrees of freedom has density $$ f(x) = \frac{\Gamma ((\nu+1)/2)}{\sqrt{\pi \nu} \Gamma (\nu/2)} (1 + x^2/\nu)^{-(\nu+1)/2}% $$ for all real \(x\). It has mean \(0\) (for \(\nu > 1\)) and variance \(\frac{\nu}{\nu-2}\) (for \(\nu > 2\)). C.f. rt

Keywords
distribution
Note

The general non-central \(t\) with parameters \((\nu,\delta)\) = (df, ncp) is defined as a the distribution of \(T_{\nu}(\delta) := \frac{U + \delta}{\chi_{\nu}/\sqrt{\nu}}\) where \(U\) and \(\chi_{\nu}\) are independent random variables, \(U \sim {\cal N}(0,1)\), and \(\chi^2_\nu\) is chi-squared, see rchisq.

The most used applications are power calculations for \(t\)-tests: Let \(T= \frac{\bar{X} - \mu_0}{S/\sqrt{n}}\) where \(\bar{X}\) is the mean and \(S\) the sample standard deviation (sd) of \(X_1,X_2,\dots,X_n\) 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\(= (\mu - \mu_0) \sqrt{n}/\sigma\).

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.

See Also

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

Aliases
  • Td-class
  • Td
  • initialize,Td-method
Examples
# NOT RUN {
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...
# }
Documentation reproduced from package distr, version 2.8.0, License: LGPL-3

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