TYLERshape function

Tyler's Shape Matrix

Iterative algorithm to estimate Tyler's shape matrix using a partial Newton-Raphson approach.

TYLERshape(X, location = TRUE, eps = 1e-06, maxiter = 100)

Arguments

• X: numeric data matrix or dataframe. Missing values are not allowed.
• location: logical or numeric. If FALSE, it is assumed that the scatter should be computed wrt to the origin. If TRUE the location will be estimated as the mean vector and if it is a numeric vector it will be computed wrt to the given vector.
• eps: convergence tolerance, which means that the algorithm stops when the Frobenius norm of the gradient is smaller than eps.
• maxiter: maximum number of iterations.

Details

The estimate is based on the new fast algorithm described in Duembgen et al. (2016). Note that Tyler's shape matrix is standardized such that it has determinant 1.

The function does not check if there are observations equal to the mean (if location=TRUE), to the provided location vector or to the origin (if location=FALSE). In these cases the function will fail.

In case maxiter is reached before convergence, the estimate at that iteration is returned and a warning is given.

Returns

A list containing: - mu: Estimated location if location=TRUE, otherwise the user specified location.

• Sigma: Estimated shape matrix.

• iter: Number of iterations of the algorithm.

References

Tyler, D.E. (1987), A distribution-free M-estimator of scatter, Annalsof Statistics, 15, 234-251.

Duembgen, L., Nordhausen, K. and Schuhmacher, H. (2016), New algorithmsfor M-estimation of multivariate location and scatter, Journal ofMultivariate Analysis, 144, 200-217.\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.jmva.2015.11.009")}

Author(s)

Lutz Duembgen and Klaus Nordhausen

See Also

tyler.shape

Examples

TYLERshape(longley)
# compare to
# library(ICSNP)
# tyler.shape(longley)

TYLERshape(longley, location=FALSE)
# compare to
# library(ICSNP)
# tyler.shape(longley, location=0)