mt | R Documentation |
The probability density function, the distribution function and random number
generation for a d
-dimensional Student's t random variable.
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) ptriv.nt(df, x, mean, S)
x |
either a vector of length |
mean |
either a vector of length |
S |
a symmetric positive definite matrix with dimensions |
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
and
sadmvt
. If this threshold is exceeded, NA
is returned.
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,
or a vector in case n=1
or d=1
.
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 77 code of SADMVT
, MVTDSTPACK
, TVPACK
and many auxiliary functions by Alan Genz;
some additional auxiliary functions by people referred to within his
programs; 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
https://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
,
plot_fxy
for plotting examples
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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