# Joint M-estimation of Location and Scatter for a Multivariate t-distribution

### Description

The functions implements three EM algorithms to M-estimate the location vector and scatter matrix of a multivariate t-distribution.

### Usage

1 2 3 |

### Arguments

`X` |
numeric data matrix or dataframe. |

`df` |
assumed degrees of freedom of the t-distribution. Default is |

`alg` |
specifies which algorithm to use. Options are |

`mu.init` |
initial value for the location vector if available. |

`V.init` |
initial value for the scatter matrix if available. |

`gamma.init` |
initial value for gamma if available. Only needed for |

`eps` |
convergence tolerance. |

`maxiter` |
maximum number of iterations. |

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

### Details

This implements the EM algorithms described in Kent et al. (1994). The norm used to define convergence is as in Arslan et al. (1995).

Algorithm 1 is valid for all degrees of freedom `df`

> 0. Algorithm 2 is well defined only for degrees of freedom `df`

> 1.
Algorithm 3 is the limiting case of Algorithm 2 with degrees of freedom `df`

= 1.

The performance of the algorithms are compared in Arslan et al. (1995).

Note that `cov.trob`

in the MASS package implements also a covariance estimate for a multivariate t-distribution.
That function provides for example also the possibility to fix the location. It requires however that the degrees of freedom exceeds 2.

### Value

A list containing:

`mu ` |
vector with the estimated loaction. |

`V ` |
matrix of the estimated scatter. |

`gam ` |
estimated value of gamma. Only present when |

`iter ` |
number of iterations. |

### Author(s)

Klaus Nordhausen

### References

Kent, J.T., Tyler, D.E. and Vardi, Y. (1994), A curious likelihood identity for the multivariate t-distribution, *Communications in Statistics, Simulation and Computation*, **23**, 441–453.
<doi:10.1080/03610919408813180>.

Arslan, O., Constable, P.D.L. and Kent, J.T. (1995), Convergence behaviour of the EM algorithm for the multivariate t-distribution, *Communications in Statistics, Theory and Methods*, **24**, 2981–3000. <doi:10.1080/03610929508831664>.

### See Also

`cov.trob`

### Examples

1 2 3 4 5 |