pmvt {mvtnorm} | R Documentation |
Computes the the distribution function of the multivariate t distribution for arbitrary limits, degrees of freedom and correlation matrices based on algorithms by Genz and Bretz.
pmvt(lower, upper, df, corr, delta, maxpts=25000, abseps=0.001, releps=0)
lower |
the vector of lower limits of length n. |
upper |
the vector of upper limits of length n. |
df |
degree of freedom as integer. |
corr |
the correlation matrix of dimension n. |
delta |
the vector of noncentrality parameters of length n. |
maxpts |
maximum number of function values as integer. |
abseps |
absolute error tolerance as double. |
releps |
relative error tolerance as double. |
This program involves the computation of central and noncentral multivariate t-probabilities with arbitrary correlation matrices. It involves both the computation of singular and nonsingular probabilities. The methodology is described in Genz and Bretz (1999, 2001).
For a given correlation matrix corr
, for short A, say,
(which has to be positive semi-definite) and
degrees of freedom df
the following
values are numerically evaluated
I = K int s^{df-1} exp(-s^2/2) Phi(s cdot lower/sqrt{df}-delta, s cdot upper/sqrt{df}-delta) ds
where Phi(a,b) = K^prime int_a^b exp(-x^prime Ax/2) dx is the multivariate normal distribution, K^prime = 1/sqrt{det(A)(2π)^m} and K = 2^{1-df/2} / Gamma(df/2) are constants and the (single) integral of I goes from 0 to +Inf.
Note that both -Inf
and +Inf
may be specified in the lower and upper integral
limits. Randomized quasi-Monte Carlo methods are used for the computations.
Univariate problems are passed to pt
.
Further information can be obtained from the quoted articles, which can be downloaded (together with additional material and additional codes) from the websites http://www.bioinf.uni-hannover.de/~bretz/ and http://www.sci.wsu.edu/math/faculty/genz/homepage.
A list with the following components:
value |
estimated integral value. |
error |
estimated absolute error. |
msg |
status messages. |
Fortran Code by Alan Genz <AlanGenz@wsu.edu> and Frank Bretz <bretz@ifgb.uni-hannover.de>, R port by Torsten Hothorn <Torsten.Hothorn@rzmail.uni-erlangen.de>
Genz, A. and Bretz, F. (1999), Numerical computation of multivariate t-probabilities with application to power calculation of multiple contrasts. Journal of Statistical Computation and Simulation, 63, 361378.
Genz, A. and Bretz, F. (2001), Methods for the computation of multivariate t-probabilities. (submitted)
n <- 5 lower <- rep(-1, 5) upper <- rep(3, 5) df <- 4 corr <- diag(5) corr[lower.tri(corr)] <- 0.5 delta <- rep(0, 5) prob <- pmvt(lower, upper, df, corr , delta) print(prob) pmvt(-Inf, 3, df = 3, corr = 0)$value == pt(3, 3)