Description Usage Arguments Details Value Author(s) References See Also Examples

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

1 | ```
DUEMBGENshape(X, nmax = 500, eps = 1e-06, maxiter = 100, perm = FALSE)
``` |

`X` |
numeric data matrix or dataframe. Missing values are not allowed. |

`nmax` |
integer, if the sample size n (number of rows of |

`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. |

`perm` |
logical. If TRUE the rows of |

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

The function does not check if there are several identical observations. In that case the function will fail.

To get a good initial value for the algorithm, the estimator is first computed based on the pairwise differences of
successive observations. Therefore the order of the rows of `X`

is supposed to be random. If this is not the case, the data
should be first permuted using the argument `perm`

.

In case `maxiter`

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

A list containing:

`Sigma` |
Estimated shape matrix. |

`iter` |
Number of iterations of the algorithm. |

Lutz Duembgen and Klaus Nordhausen

Duembgen, L. (1998), On Tyler's M-functional of scatter in high dimension, *Annals of Institute of Statistical Mathematics*, **50**, 471–491.

Duembgen, L., Nordhausen, K. and Schuhmacher, H. (2016), New algorithms for M-estimation of multivariate location and scatter, *Journal of Multivariate Analysis*, **144**, 200–217. doi: 10.1016/j.jmva.2015.11.009

1 2 3 4 5 | ```
DUEMBGENshape(longley)
DUEMBGENshape(longley, nmax=10)
# compare to
# library(ICSNP)
# duembgen.shape(longley)
``` |

```
$Sigma
GNP.deflator GNP Unemployed Armed.Forces Population
GNP.deflator 9.632427 87.01088 49.96269 28.545623 5.947350
GNP 87.010882 798.73276 439.31650 252.870216 54.724575
Unemployed 49.962692 439.31650 707.87227 -97.815603 34.792811
Armed.Forces 28.545623 252.87022 -97.81560 398.037813 14.403952
Population 5.947350 54.72458 34.79281 14.403952 3.816897
Year 4.165543 38.06977 23.39288 11.356145 2.629198
Employed 3.039588 28.06546 12.79177 9.237134 1.894036
Year Employed
GNP.deflator 4.165543 3.039588
GNP 38.069769 28.065464
Unemployed 23.392881 12.791773
Armed.Forces 11.356145 9.237134
Population 2.629198 1.894036
Year 1.831520 1.325913
Employed 1.325913 1.019916
$iter
[1] 10
$Sigma
GNP.deflator GNP Unemployed Armed.Forces Population
GNP.deflator 9.632427 87.01088 49.96269 28.545623 5.947350
GNP 87.010882 798.73276 439.31650 252.870216 54.724575
Unemployed 49.962692 439.31650 707.87227 -97.815603 34.792811
Armed.Forces 28.545623 252.87022 -97.81560 398.037813 14.403952
Population 5.947350 54.72458 34.79281 14.403952 3.816897
Year 4.165543 38.06977 23.39288 11.356145 2.629198
Employed 3.039588 28.06546 12.79177 9.237134 1.894036
Year Employed
GNP.deflator 4.165543 3.039588
GNP 38.069769 28.065464
Unemployed 23.392881 12.791773
Armed.Forces 11.356145 9.237134
Population 2.629198 1.894036
Year 1.831520 1.325913
Employed 1.325913 1.019916
$iter
[1] 10
```

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