# Multivariate Huber's M-estimator and its symmetrized version

### Description

Iterative algorithms to estimate M-estimators of location and scatter as well as symmetrized M-estimator using Huber's weight functions.

### Usage

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### Arguments

`X` |
a matrix or a data frame |

`qg` |
a tuning parameter. The default is 0.9, see details |

`fixed.loc` |
a logical, see details |

`location` |
an optional vector giving the location of the data or the initial value for the location if it is estimated |

`init` |
an optional starting value for scatter |

`steps` |
fixed number of iteration steps to take, if |

`eps` |
tolerance for convergence |

`maxiter` |
maximum number of iteration steps. Ignored if |

`na.action` |
a function which indicates what should happen when the data contain 'NA's. Default is to fail. |

### Details

`mvhuberM`

computes multivariate M-estimators of location and scatter
using Huber's weight functions. The tuning parameter `qg`

defines cutoff-point c for weight functions so that *c=F^{-1}(q)*, where F is the cdf of *chi^2*-distribution with p degrees of freedom. The estimators with maximal breakdown point are obtained with the choice qg=F(p+1). If `fixed.loc`

is set TRUE, scatter estimator is computed with fixed location given by
`location`

(default is column means).

`symmhuber`

computes Huber's M-estimator of scatter using pairwise
differences of the data therefore avoiding location estimation.

### Value

`mvhuberM`

returns a list with components

`location ` |
a vector |

`scatter ` |
a matrix |

`symmhuber`

returns a matrix.

### Author(s)

Klaus Nordhausen, klaus.nordhausen@uta.fi,

Jari Miettinen, jari.p.miettinen@jyu.fi

### References

Huber, P.J. (1981), Robust Statistics, Wiley, New York.

Lopuhaa, H.P. (1989). On the relation between S-estimators and M-estimators of multivariate location and covariance. *Annals of Statistics*, 17, 1662-1683.

Sirkia, S., Taskinen, S., Oja, H. (2007) Symmetrised M-estimators of scatter. *Journal of Multivariate Analysis*, 98, 1611-1629.

### Examples

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