Description Usage Arguments Details Value References See Also Examples

`evShape2evSSCM`

transforms the eigenvalues of the shape matrix of an elliptical distribution into that of the spatial sign covariance matrix.

1 | ```
evShape2evSSCM(evShape)
``` |

`evShape` |
(required) p-dimensional numeric, representing the eigenvalues of the shape matrix. |

The eigenvalues of the SSCM can be calculated from the eigenvalues of the shape matrix by numerical evaluation of onedimensional integrals, see Proposition 3 of Dürre, Tyler, Vogel (2016). We use the substitution

*x=(1+t)/(1-t)*

and Gaussian quadrature with Jacobi polynomials up to order 500 and *beta=0* as well as *alpha=p/2-1*, see chapter 2.4 (iv) of Gautschi (1997) for details.

The nodes and weights of the Gauss-Jacobi-quadrature are originally computed by the `gaussquad`

package and saved in the file `jacobiquad`

for faster computation.

p-dimensional numeric, representing the eigenvalues of the corresponding spatial sign covariance matrix.

Dürre, A., Vogel, D., Fried, R. (2015): Spatial sign correlation, *Journal of Multivariate Analyis*, vol. 135, 89–105.
arvix 1403.7635

Dürre, A., Tyler, D. E., Vogel, D. (2016): On the eigenvalues of the spatial sign covariance matrix in more than two dimensions, to appear in: *Statistics and Probability Letters*. arvix 1512.02863

Gautschi, W. (1997): *Numerical Analysis - An Introduction*, Birkhäuser, Basel.

Novomestky, F. (2013): gaussquad: Collection of functions for Gaussian quadrature. R package version 1.0-2.

Calculating the theoretical SSCM from the theoretical shape matrix `Shape2SSCM`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
# defining eigenvalues of the shape matrix
evShape <- seq(from=0,to=1,by=0.1)
# standardized to have sum 1
evShape <- evShape/sum(evShape)
# calculating the related eigenvalues of the SSCM
evSSCM <- evShape2evSSCM(evShape)
plot(evShape,evSSCM)
# recalculate the eigenvalues of the shape matrix
evShape2 <- evSSCM2evShape(evSSCM)
# error is negligible
sum(abs(evShape-evShape2))
``` |

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