Multivariate t Distribution

Description

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.

Usage

1
2
3
pmvt(lower=-Inf, upper=Inf, delta=rep(0, length(lower)),
     df=1, corr=NULL, sigma=NULL, algorithm = GenzBretz(),
     type = c("Kshirsagar", "shifted"), ...)

Arguments

lower

the vector of lower limits of length n.

upper

the vector of upper limits of length n.

delta

the vector of noncentrality parameters of length n, for type = "shifted" delta specifies the mode.

df

degree of freedom as integer. Normal probabilities are computed for df=0.

corr

the correlation matrix of dimension n.

sigma

the scale matrix of dimension n. Either corr or sigma can be specified. If sigma is given, the problem is standardized. If neither corr nor sigma is given, the identity matrix is used for sigma.

algorithm

an object of class GenzBretz or TVPACK defining the hyper parameters of this algorithm.

type

type of the noncentral multivariate t distribution to be computed. type = "Kshirsagar" corresponds to formula (1.4) in Genz and Bretz (2009) (see also Chapter 5.1 in Kotz and Nadarajah (2004)). This is the noncentral t-distribution needed for calculating the power of multiple contrast tests under a normality assumption. type = "shifted" corresponds to the formula right before formula (1.4) in Genz and Bretz (2009) (see also formula (1.1) in Kotz and Nadarajah (2004)). It is a location shifted version of the central t-distribution. This noncentral multivariate t distribution appears for example as the Bayesian posterior distribution for the regression coefficients in a linear regression. In the central case both types coincide.

...

additional parameters (currently given to GenzBretz for backward compatibility issues).

Details

This function 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 (for default algorithm = GenzBretz()) is based on randomized quasi Monte Carlo methods and described in Genz and Bretz (1999, 2002).
Because of the randomization, the result for this algorithm (slightly) depends on .Random.seed.

For 2- and 3-dimensional problems one can also use the TVPACK routines described by Genz (2004), which only handles semi-infinite integration regions (and for type = "Kshirsagar" only central problems).

For type = "Kshirsagar" and a given correlation matrix corr, for short A, say, (which has to be positive semi-definite) and degrees of freedom ν the following values are numerically evaluated

I = 2^{1-ν/2} / Γ(ν/2) \int_0^∞ s^{ν-1} \exp(-s^2/2) Φ(s \cdot lower/√{ν} - δ, s \cdot upper/√{ν} - δ) \, ds

where

Φ(a,b) = (det(A)(2π)^m)^{-1/2} \int_a^b \exp(-x^\prime Ax/2) \, dx

is the multivariate normal distribution and m is the number of rows of A.

For type = "shifted", a positive definite symmetric matrix S (which might be the correlation or the scale matrix), mode (vector) δ and degrees of freedom ν the following integral is evaluated:

c\int_{lower_1}^{upper_1}...\int_{lower_m}^{upper_m} (1+(x-δ)'S^{-1}(x-δ)/ν)^{-(ν+m)/2}\, dx_1 ... dx_m,

where

c = Γ((ν+m)/2)/((π ν)^{m/2}Γ(ν/2)|S|^{1/2}),

and m is the number of rows of S.

Note that both -Inf and +Inf may be specified in the lower and upper integral limits in order to compute one-sided probabilities.

Univariate problems are passed to pt. If df = 0, normal probabilities are returned.

Value

The evaluated distribution function is returned with attributes

error

estimated absolute error and

msg

status message (a character string).

Source

http://www.sci.wsu.edu/math/faculty/genz/homepage

References

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, 361–378.

Genz, A. and Bretz, F. (2002), Methods for the computation of multivariate t-probabilities. Journal of Computational and Graphical Statistics, 11, 950–971.

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

Genz, A. and Bretz, F. (2009), Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics, Vol. 195. Springer-Verlag, Heidelberg.

S. Kotz and S. Nadarajah (2004), Multivariate t Distributions and Their Applications. Cambridge University Press. Cambridge.

Edwards D. and Berry, Jack J. (1987), The efficiency of simulation-based multiple comparisons. Biometrics, 43, 913–928.

See Also

qmvt

Examples

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
n <- 5
lower <- -1
upper <- 3
df <- 4
corr <- diag(5)
corr[lower.tri(corr)] <- 0.5
delta <- rep(0, 5)
prob <- pmvt(lower=lower, upper=upper, delta=delta, df=df, corr=corr)
print(prob)

pmvt(lower=-Inf, upper=3, df = 3, sigma = 1) == pt(3, 3)

# Example from R News paper (original by Edwards and Berry, 1987)

n <- c(26, 24, 20, 33, 32)
V <- diag(1/n)
df <- 130
C <- c(1,1,1,0,0,-1,0,0,1,0,0,-1,0,0,1,0,0,0,-1,-1,0,0,-1,0,0)
C <- matrix(C, ncol=5)
### scale matrix
cv <- C %*% V %*% t(C)
### correlation matrix
dv <- t(1/sqrt(diag(cv)))
cr <- cv * (t(dv) %*% dv)
delta <- rep(0,5)

myfct <- function(q, alpha) {
  lower <- rep(-q, ncol(cv))
  upper <- rep(q, ncol(cv))
  pmvt(lower=lower, upper=upper, delta=delta, df=df,
       corr=cr, abseps=0.0001) - alpha
}

### uniroot for this simple problem
round(uniroot(myfct, lower=1, upper=5, alpha=0.95)$root, 3)

# compare pmvt and pmvnorm for large df:

a <- pmvnorm(lower=-Inf, upper=1, mean=rep(0, 5), corr=diag(5))
b <- pmvt(lower=-Inf, upper=1, delta=rep(0, 5), df=rep(300,5),
          corr=diag(5))
a
b

stopifnot(round(a, 2) == round(b, 2))

# correlation and scale matrix

a <- pmvt(lower=-Inf, upper=2, delta=rep(0,5), df=3,
          sigma = diag(5)*2)
b <- pmvt(lower=-Inf, upper=2/sqrt(2), delta=rep(0,5),
          df=3, corr=diag(5))
attributes(a) <- NULL
attributes(b) <- NULL
a
b
stopifnot(all.equal(round(a,3) , round(b, 3)))

a <- pmvt(0, 1,df=10)
attributes(a) <- NULL
b <- pt(1, df=10) - pt(0, df=10)
stopifnot(all.equal(round(a,10) , round(b, 10)))