Moments of the sample covariance matrix.

Share:

Description

Calculates the moments of the sample covariance matrix. It assumes that the summands (the outer products of the samples' random data vector) that constitute the sample covariance matrix follow a Wishart-distribution with scale parameter \mathbf{Σ} and shape parameter ν. The latter is equal to the number of summands in the sample covariance estimate.

Usage

1
momentS(Sigma, shape, moment=1)

Arguments

Sigma

Positive-definite matrix, the scale parameter \mathbf{Σ} of the Wishart distribution.

shape

A numeric, the shape parameter ν of the Wishart distribution. Should exceed the number of variates (number of rows or columns of Sigma).

moment

An integer. Should be in the set \{-4, -3, -2, -1, 0, 1, 2, 3, 4\} (only those are explicitly specified in Lesac, Massam, 2004).

Value

The r-th moment of a sample covariance matrix: E(\mathbf{S}^r).

Author(s)

Wessel N. van Wieringen<w.vanwieringen@vumc.nl>.

References

Lesac, G., Massam, H. (2004), "All invariant moments of the Wishart distribution", Scandinavian Journal of Statistics, 31(2), 295-318.

Examples

1
2
3
4
5
6
7
# create scale parameter
Sigma <- matrix(c(1, 0.5, 0, 0.5, 1, 0, 0, 0, 1), byrow=TRUE, ncol=3)

# evaluate expectation of the square of a sample covariance matrix 
# that is assumed to Wishart-distributed random variable with the 
# above scale parameter Sigma and shape parameter equal to 40.
momentS(Sigma, 40, 2)

Want to suggest features or report bugs for rdrr.io? Use the GitHub issue tracker.