dmt {mnormt} | R Documentation |
The probability density function, the distribution function and random number generation for the multivariate Student's t distribution
dmt(x, mean = rep(0, d), S, df=Inf, log = FALSE) pmt(x, mean = rep(0, d), S, df=Inf, ...) rmt(n = 1, mean = rep(0, d), S, df=Inf, sqrt=NULL) sadmvt(df, lower, upper, mean, S, maxpts = 2000*d, abseps = 1e-06, releps = 0) biv.nt.prob(df, lower, upper, mean, S)
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 |
degrees of freedom; it must be a positive integer for |
log |
a logical value(default value is |
sqrt |
if not |
... |
parameters 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 functions sadmvt
and biv.nt.prob
are interfaces to
Fortran-77 routines by Alan Genz, and available from his web page;
they makes uses of some auxiliary functions whose authors are
documented in the Fortran code. The routine sadmvt
uses an adaptive
integration method. The routine biv.nt.prob
is specific for the
bivariate case; if df<1
or df=Inf
, it computes the bivariate
normal distribution function using a non-iterative method described in a
reference given below.
If pmt
is called with d>2
, this is converted into
a suitable call to sadmvt
; if d=2
, a call to
biv.nt.prob
is used; if d=1
, then pt
is used.
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 by Adelchi Azzalini.
Genz, A.: Fortran code in files mvt.f
and mvtdstpack.f
available at http://www.math.wsu.edu/math/faculty/genz/software/
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
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)