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 \(\mathbf{\Sigma}\) and shape parameter \(\nu\). The latter is equal to the number of summands in the sample covariance estimate.
momentS(Sigma, shape, moment = 1)Positive-definite matrix, the scale parameter
\(\mathbf{\Sigma}\) of the Wishart distribution.
A numeric, the shape parameter \(\nu\) of the Wishart
distribution. Should exceed the number of variates (number of rows or
columns of Sigma).
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).
The \(r\)-th moment of a sample covariance matrix: \(E(\mathbf{S}^r)\).
Lesac, G., Massam, H. (2004), "All invariant moments of the Wishart distribution", Scandinavian Journal of Statistics, 31(2), 295-318.
# 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)
#> [,1] [,2] [,3]
#> [1,] 1.35625 1.06250 0.0
#> [2,] 1.06250 1.35625 0.0
#> [3,] 0.00000 0.00000 1.1