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

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

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

`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. |

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.

A list containing:

`mu` |
Estimated location if |

`Sigma` |
Estimated shape matrix. |

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

Lutz Duembgen and Klaus Nordhausen

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

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 6 7 8 9 | ```
TYLERshape(longley)
# compare to
# library(ICSNP)
# tyler.shape(longley)
TYLERshape(longley, location=FALSE)
# compare to
# library(ICSNP)
# tyler.shape(longley, location=0)
``` |

```
$mu
GNP.deflator GNP Unemployed Armed.Forces Population Year
101.6813 387.6984 319.3313 260.6687 117.4240 1954.5000
Employed
65.3170
$Sigma
GNP.deflator GNP Unemployed Armed.Forces Population
GNP.deflator 12.793292 114.58414 61.42678 30.940787 7.687998
GNP 114.584138 1037.89856 534.48419 271.330207 69.699551
Unemployed 61.426780 534.48419 793.90844 -130.391943 41.014834
Armed.Forces 30.940787 271.33021 -130.39194 440.456188 15.089037
Population 7.687998 69.69955 41.01483 15.089037 4.749266
Year 5.465601 49.32485 28.13933 12.057188 3.336538
Employed 4.090532 37.26874 16.44562 9.886402 2.475810
Year Employed
GNP.deflator 5.465601 4.090532
GNP 49.324849 37.268743
Unemployed 28.139334 16.445616
Armed.Forces 12.057188 9.886402
Population 3.336538 2.475810
Year 2.362310 1.757463
Employed 1.757463 1.371000
$iter
[1] 18
$mu
GNP.deflator GNP Unemployed Armed.Forces Population Year
0 0 0 0 0 0
Employed
0
$Sigma
GNP.deflator GNP Unemployed Armed.Forces Population
GNP.deflator 67.40170 263.0191 208.6951 174.6547 76.89560
GNP 263.01911 1043.7347 823.0857 677.9078 298.43753
Unemployed 208.69509 823.0857 682.5417 519.0968 237.60184
Armed.Forces 174.65475 677.9078 519.0968 475.5066 199.24861
Population 76.89560 298.4375 237.6018 199.2486 87.90447
Year 1269.55096 4894.1264 3900.4637 3299.6967 1454.38492
Employed 42.83573 166.1661 131.9765 111.1626 48.97212
Year Employed
GNP.deflator 1269.5510 42.83573
GNP 4894.1264 166.16609
Unemployed 3900.4637 131.97651
Armed.Forces 3299.6967 111.16260
Population 1454.3849 48.97212
Year 24127.1568 810.47337
Employed 810.4734 27.28807
$iter
[1] 22
```

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