simEd (version 2.0.0)

vt: Variate Generation for Student T Distribution

Description

Variate Generation for Student T Distribution

Usage

vt(n, df, ncp = 0, stream = NULL, antithetic = FALSE, asList = FALSE)

Arguments

n

number of observations

df

Degrees of freedom > 0

ncp

Non-centrality parameter delta (default NULL)

stream

if NULL (default), uses stats::runif to generate uniform variates to invert via stats::qt; otherwise, an integer in 1:25 indicates the rstream stream from which to generate uniform variates to invert via stats::qt;

antithetic

if FALSE (default), inverts \(u\) = uniform(0,1) variate(s) generated via either stats::runif or rstream::rstream.sample; otherwise, uses \(1 - u\)

asList

if FALSE (default), output only the generated random variates; otherwise, return a list with components suitable for visualizing inversion. See return for details

Value

If asList is FALSE (default), return a vector of random variates.

Otherwise, return a list with components suitable for visualizing inversion, specifically:

u

A vector of generated U(0,1) variates

x

A vector of Student t random variates

quantile

Parameterized quantile function

text

Parameterized title of distribution

Details

Generates random variates from the Student t distribution.

Student T variates are generated by inverting uniform(0,1) variates produced either by stats::runif (if stream is NULL) or by rstream::rstream.sample (if stream is not NULL). In either case, stats::qt is used to invert the uniform(0,1) variate(s). In this way, using vt provides a monotone and synchronized binomial variate generator, although not particularly fast.

The stream indicated must be an integer between 1 and 25 inclusive.

The t-distribution with df \(= v\) degrees of freedom has density

$$f(x) = \frac{\Gamma((v+1)/2)}{\sqrt{v\pi} \ \Gamma(v/2)} \ (1 + x^2/v)^{-(v+1)/2}$$

for all real \(x\). It has mean 0 (for \(v > 1\)) and variance \(v/(v-2)\) (for \(v > 2\)).

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

See Also

rstream, set.seed, stats::runif

stats::rt

Examples

# NOT RUN {
 set.seed(8675309)
 # NOTE: following inverts rstream::rstream.sample using stats::qt
 vt(3, df = 3, ncp = 2)

 set.seed(8675309)
 # NOTE: following inverts rstream::rstream.sample using stats::qt
 vt(3, 2, stream = 1)
 vt(3, 2, stream = 2)

 set.seed(8675309)
 # NOTE: following inverts rstream::rstream.sample using stats::qt
 vt(1, 2, stream = 1)
 vt(1, 2, stream = 2)
 vt(1, 2, stream = 1)
 vt(1, 2, stream = 2)
 vt(1, 2, stream = 1)
 vt(1, 2, stream = 2)

 set.seed(8675309)
 variates <- vt(1000, 2, stream = 1)
 set.seed(8675309)
 variates <- vt(1000, 2, stream = 1, antithetic = TRUE)

# }