Class "Td"
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.
imgObject of class "Reals": The domain of this distribution has got dimension 1
and the name "Real Space".
paramObject of class "TParameter": the parameter of this distribution (df),
declared at its instantiation
rObject of class "function": generates random numbers (calls function rt)
dObject of class "function": density function (calls function dt)
pObject of class "function": cumulative function (calls function pt)
qObject of class "function": inverse of the cumulative function (calls function qt)
.withArithlogical: used internally to issue warnings as to interpretation of arithmetics
.withSimlogical: used internally to issue warnings as to accuracy
.logExactlogical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function
.lowerExactlogical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function
Symmetryobject of class "DistributionSymmetry";
used internally to avoid unnecessary calculations.
Class "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".
signature(.Object = "Td"): initialize method
signature(object = "Td"): returns the slot df of the parameter of the distribution
signature(object = "Td"): modifies the slot df of the parameter of the distribution
signature(object = "Td"): returns the slot ncp of the parameter of the distribution
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 non-centrality parameter
ncp= (mu - m0) * sqrt(n)/sigma.
Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de
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...
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