Td-class
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 instantiationr
Object of class
"function"
: generates random numbers (calls functionrt
)d
Object of class
"function"
: density function (calls functiondt
)p
Object of class
"function"
: cumulative function (calls functionpt
)q
Object of class
"function"
: inverse of the cumulative function (calls functionqt
).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
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...
# }