# momentS: Moments of the sample covariance matrix.

Description Usage Arguments Value Author(s) References Examples

### 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) 

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