Description Usage Arguments Details Value Note Author(s) References See Also Examples

The probability density function, the distribution function and random
number generation for the multivariate Student's *t* distribution

1 2 3 4 5 6 |

`x` |
either a vector of length |

`mean` |
either a vector of length |

`S` |
a symmetric positive-definite matrix representing the
scale matrix of the distribution, such that |

`df` |
the degrees of freedom.
For |

`log` |
a logical value(default value is |

`sqrt` |
if not |

`...` |
arguments passed to |

`n` |
the number of random vectors to be generated |

`lower` |
a numeric vector of lower integration limits of
the density function; must be of maximal length |

`upper` |
a numeric vector of upper integration limits
of the density function; must be of maximal length |

`maxpts` |
the maximum number of function evaluations
(default value: |

`abseps` |
absolute error tolerance (default value: |

`releps` |
relative error tolerance (default value: |

The dimension `d`

cannot exceed `20`

for `pmt`

.

The functions `sadmvt`

, `ptriv.mt`

and `biv.nt.prob`

are
interfaces to Fortran-77 routines by Alan Genz, available from his web page;
they makes use of some auxiliary functions whose authors are indicated
in the Fortran code itself.
The routine `sadmvt`

uses an adaptive integration method.
If `df=3`

, a call to `pmt`

activates a call to `ptriv.nt`

which is specific for the trivariate case, and uses Genz's Fortran
code `tvpack.f`

; see Genz (2004) for the background methodology.
A similar fact takes place when `df=2`

with function `biv.nt.prob`

;
note however that the underlying Fortran code is taken from
`mvtdstpack.f`

, not from `tvpack.f`

.
If `pmt`

is called with `d>3`

, this is converted into
a suitable call to `sadmvt`

.

If `sqrt=NULL`

(default value), the working of `rmt`

involves
computation of a square root of `S`

via the Cholesky decomposition.
If a non-`NULL`

value of `sqrt`

is supplied, it is assumed that
it represents a square root of the scale matrix,
otherwise represented by `S`

, whose value is ignored in this case.
This mechanism is intended primarily for use in a sequence of calls to
`rmt`

, all sampling from a distribution with fixed scale matrix;
a suitable matrix `sqrt`

can then be computed only once beforehand,
avoiding that the same operation is repeated multiple times along the
sequence of calls. For examples of use of this argument, see those in the
documentation of `rmnorm`

.
Another use of `sqrt`

is to supply a different form of square root
of the scale matrix, in place of the Cholesky factor.

For efficiency reasons, `rmt`

does not perform checks on the supplied
arguments.

`dmt`

returns a vector of density values (possibly log-transformed);
`pmt`

and `sadmvt`

return a single probability with
attributes giving details on the achieved accuracy, provided `x`

of `pmnorm`

is a vector;
`rmt`

returns a matrix of `n`

rows of random vectors

The attributes `error`

and `status`

of the probability returned
by `sadmvt`

and by `pmt`

(the latter only if `x`

is a vector
and `d>2`

) indicate whether the function
had a normal termination, achieving the required accuracy.
If this is not the case, re-run the function with a higher value of
`maxpts`

.

Fortran code of `SADMVT`

and most auxiliary functions by Alan Genz;
some additional auxiliary functions by people referred to within his
program; interface to **R** and additional **R** code (for `dmt`

, `rmt`

etc.) by Adelchi Azzalini.

Genz, A.: Fortran-77 code in files `mvt.f`

, `mvtdstpack.f`

and codetvpack, downloaded in 2005 and again in 2007 from his webpage,
whose URL as of 2020-06-01 is
http://www.math.wsu.edu/faculty/genz/software/software.html

Genz, A. (2004).
Numerical computation of rectangular bivariate and trivariate normal
and *t* probabilities.
*Statistics and Computing* 14, 251-260.

Dunnett, C.W. and Sobel, M. (1954).
A bivariate generalization of Student's *t*-distribution with tables
for certain special cases. *Biometrika* 41, 153–169.

`dt`

,
`rmnorm`

for use of argument `sqrt`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
x <- seq(-2,4,length=21)
y <- 2*x+10
z <- x+cos(y)
mu <- c(1,12,2)
Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
df <- 4
f <- dmt(cbind(x,y,z), mu, Sigma,df)
p1 <- pmt(c(2,11,3), mu, Sigma, df)
p2 <- pmt(c(2,11,3), mu, Sigma, df, maxpts=10000, abseps=1e-8)
x <- rmt(10, mu, Sigma, df)
p <- sadmvt(df, lower=c(2,11,3), upper=rep(Inf,3), mu, Sigma) # upper tail
#
p0 <- pmt(c(2,11), mu[1:2], Sigma[1:2,1:2], df=5)
p1 <- biv.nt.prob(5, lower=rep(-Inf,2), upper=c(2, 11), mu[1:2], Sigma[1:2,1:2])
p2 <- sadmvt(5, lower=rep(-Inf,2), upper=c(2, 11), mu[1:2], Sigma[1:2,1:2])
c(p0, p1, p2, p0-p1, p0-p2)
``` |

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