Distribution functions for the matrix variate t distribution.

Density and random generation for the matrix variate t distribution.

rmatrixt(
  n,
  df,
  mean,
  L = diag(dim(as.matrix(mean))[1]),
  R = diag(dim(as.matrix(mean))[2]),
  U = L %*% t(L),
  V = t(R) %*% R,
  list = FALSE,
  array = NULL,
  force = FALSE
)

dmatrixt(
  x,
  df,
  mean = matrix(0, p, n),
  L = diag(p),
  R = diag(n),
  U = L %*% t(L),
  V = t(R) %*% R,
  log = FALSE
)

Arguments

n	number of observations for generation
df	degrees of freedom (\(>0\), may be non-integer), `df = 0, Inf` is allowed and will return a normal distribution.
mean	\(p \times q\) This is really a 'shift' rather than a mean, though the expected value will be equal to this if \(df > 2\)
L	\(p \times p\) matrix specifying relations among the rows. By default, an identity matrix.
R	\(q \times q\) matrix specifying relations among the columns. By default, an identity matrix.
U	\(LL^T\) - \(p \times p\) positive definite matrix for rows, computed from \(L\) if not specified.
V	\(R^T R\) - \(q \times q\) positive definite matrix for columns, computed from \(R\) if not specified.
list	Defaults to `FALSE` . If this is `TRUE` , then the output will be a list of matrices.
array	If \(n = 1\) and this is not specified and `list` is `FALSE` , the function will return a matrix containing the one observation. If \(n > 1\) , should be the opposite of `list` . If `list` is `TRUE` , this will be ignored.
force	In `rmatrix`: if `TRUE`, will take the input of `R` directly - otherwise uses `V` and uses Cholesky decompositions. Useful for generating degenerate t-distributions. Will also override concerns about potentially singular matrices unless they are not, in fact, invertible.
x	quantile for density
log	logical; in `dmatrixt`, if `TRUE`, probabilities `p` are given as `log(p)`.

Value

rmatrixt returns either a list of \(n\) \(p \times q\) matrices or a \(p \times q \times n\) array.

dmatrixt returns the density at x.

Details

The matrix \(t\)-distribution is parameterized slightly differently from the univariate and multivariate \(t\)-distributions

the variance is scaled by a factor of 1/df. In this parameterization, the variance for a \(1 \times 1\) matrix variate \(t\)-distributed random variable with identity variance matrices is \(1/(df-2)\) instead of \(df/(df-2)\). A Central Limit Theorem for the matrix variate \(T\) is then that as df goes to infinity, \(MVT(0, df, I_p, df*I_q)\) converges to \(MVN(0,I_p,I_q)\).

References

Gupta, Arjun K, and Daya K Nagar. 1999. Matrix Variate Distributions. Vol. 104. CRC Press. ISBN:978-1584880462

Dickey, James M. 1967. “Matricvariate Generalizations of the Multivariate t Distribution and the Inverted Multivariate t Distribution.” Ann. Math. Statist. 38 (2): 511–18. doi: 10.1214/aoms/1177698967

Examples