momentS: Moments of the sample covariance matrix.

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 $Σ$ and shape parameter $ν$ . The latter is equal to the number of summands in the sample covariance estimate.

Usage

momentS(Sigma, shape, moment = 1)

Value

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

Arguments

Sigma: Positive-definite matrix, the scale parameter $Σ$ 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).

Author

Wessel N. van Wieringen.

References

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

Examples

Run this code


# 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)

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