R: Joint M-estimation of Location and Scatter for a Multivariate...

tM {ICS} | R Documentation

## 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

    tM(X, df = 1, alg = "alg3", mu.init = NULL, V.init = NULL,
            gamma.init = NULL, eps = 1e-06, maxiter = 100,
            na.action = na.fail)

### Arguments

`X` | numeric data matrix or dataframe.
`df` | assumed degrees of freedom of the t-distribution. Default is `1` which corresponds to the Cauchy distribution.
`alg` | specifies which algorithm to use. Options are `alg1`, `alg2` or `alg3`. `alg3` is the default.
`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 `alg2`.
`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 `alg2` is used.
`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

    set.seed(654321)
    cov.matrix <- matrix(c(3,2,1,2,4,-0.5,1,-0.5,2), ncol=3)
    X <- rmvt(100, cov.matrix, 1)
    tM(X)
    rm(.Random.seed)

[Package ICS version 1.3-0 Index]