rcmvt {CondMVT}

Conditional Multivariate t Density and Random Deviates

Description

This function provides the random number generator for the conditional multivariate t distribution, [Y given X], where Z = (X,Y) is the fully-joint multivariate t distribution with location vector equal to mean and scatter matrix sigma.

Usage

rcmvt(n, mean, sigma, df,dependent.ind, given.ind, X.given,
check.sigma = TRUE,type = c("Kshirsagar", "shifted"),
method = c("eigen", "svd", "chol"))

Arguments

n

number of random deviates.

mean

location vector, which must be specified.

sigma

a symmetric, positive-definte matrix of dimension n x n, which must be specified.

df

degrees of freedom, which must be specified

dependent.ind

a vector of integers denoting the indices of dependent variable Y.

given.ind

a vector of integers denoting the indices of conditoning variable X.

X.given

a vector of reals denoting the conditioning value of X. When both given.ind and X.given are missing, the distribution of Y becomes Z[dependent.ind]

check.sigma

logical; if TRUE, the scatter matrix is checked for appropriateness (symmetry, positive-definiteness). This could be set to FALSE if the user knows it is appropriate.

type

type of the noncentral multivariate t-distribution. 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.

method

string specifying the matrix decomposition used to determine the matrix root of sigma. Possible methods are eigenvalue decomposition ("eigen", default), singular value decomposition ("svd"), and Cholesky decomposition ("chol"). The Cholesky is typically fastest, not by much though.

Value

A 'vector' of length n, equal to the length of 'mean'

Examples

# 10-dimensional multivariate t distribution
n <- 10
df=3
A <- matrix(rt(n^2,df), n, n)
A <- tcrossprod(A,A) #A %*% t(A)

# density of Z[c(2,5)] given Z[c(1,4,7,9)]=c(1,1,0,-1)
dcmvt(x=c(1.2,-1), mean=rep(1,n), sigma=A, df=df,
         dependent.ind=c(2,5), given.ind=c(1,4,7,9),
         X.given=c(1,1,0,-1))

dcmvt(x=-1, mean=rep(1,n), sigma=A,df=df, dep=3, given=c(1,4,7,9,10), X=c(1,1,0,0,-1))

dcmvt(x=c(1.2,-1), mean=rep(1,n), sigma=A,df=df, dep=c(2,5))

# gives an error since `x' and `dep' are incompatibe
#dcmvt(x=-1, mean=rep(1,n), sigma=A,df=df, dep=c(2,3),
#given=c(1,4,7,9,10), X=c(1,1,0,0,-1))

rcmvt(n=10, mean=rep(1,n), sigma=A,df=df, dep=c(2,5),
         given=c(1,4,7,9,10), X=c(1,1,0,0,-1),type="shifted",
         method="eigen")

rcmvt(n=10, mean=rep(1,n), sigma=A,df=df, dep=3,
         given=c(1,4,7,9,10), X=c(1,1,0,0,-1),type="Kshirsagar",
         method="chol")