Compute a multivariate location and scale estimate with a high
breakdown point -- this can be thought of as estimating the mean and
covariance of the `good`

part of the data. `cov.mve`

and
`cov.mcd`

are compatibility wrappers.

```
cov.rob(x, cor = FALSE, quantile.used = floor((n + p + 1)/2),
method = c("mve", "mcd", "classical"),
nsamp = "best", seed)
```cov.mve(…)
cov.mcd(…)

x

a matrix or data frame.

cor

should the returned result include a correlation matrix?

quantile.used

the minimum number of the data points regarded as `good`

points.

method

the method to be used -- minimum volume ellipsoid, minimum
covariance determinant or classical product-moment. Using
`cov.mve`

or `cov.mcd`

forces `mve`

or `mcd`

respectively.

nsamp

the number of samples or `"best"`

or `"exact"`

or
`"sample"`

. The limit
If `"sample"`

the number chosen is `min(5*p, 3000)`

, taken
from Rousseeuw and Hubert (1997). If `"best"`

exhaustive
enumeration is done up to 5000 samples: if `"exact"`

exhaustive enumeration will be attempted.

seed

the seed to be used for random sampling: see `RNGkind`

. The
current value of `.Random.seed`

will be preserved if it is set.

…

arguments to `cov.rob`

other than `method`

.

A list with components

the final estimate of location.

the final estimate of scatter.

(only is `cor = TRUE`

) the estimate of the correlation
matrix.

message giving number of singular samples out of total

the value of the criterion on log scale. For MCD this is the determinant, and for MVE it is proportional to the volume.

the subset used. For MVE the best sample, for MCD the best
set of size `quantile.used`

.

total number of observations.

For method `"mve"`

, an approximate search is made of a subset of
size `quantile.used`

with an enclosing ellipsoid of smallest volume; in
method `"mcd"`

it is the volume of the Gaussian confidence
ellipsoid, equivalently the determinant of the classical covariance
matrix, that is minimized. The mean of the subset provides a first
estimate of the location, and the rescaled covariance matrix a first
estimate of scatter. The Mahalanobis distances of all the points from
the location estimate for this covariance matrix are calculated, and
those points within the 97.5% point under Gaussian assumptions are
declared to be `good`

. The final estimates are the mean and rescaled
covariance of the `good`

points.

The rescaling is by the appropriate percentile under Gaussian data; in
addition the first covariance matrix has an *ad hoc* finite-sample
correction given by Marazzi.

For method `"mve"`

the search is made over ellipsoids determined
by the covariance matrix of `p`

of the data points. For method
`"mcd"`

an additional improvement step suggested by Rousseeuw and
van Driessen (1999) is used, in which once a subset of size
`quantile.used`

is selected, an ellipsoid based on its covariance
is tested (as this will have no larger a determinant, and may be smaller).

There is a hard limit on the allowed number of samples, \(2^{31} -
1\). However, practical limits are likely to be much lower
and one might check the number of samples used for exhaustive
enumeration, `combn(NROW(x), NCOL(x) + 1)`

, before attempting it.

P. J. Rousseeuw and A. M. Leroy (1987)
*Robust Regression and Outlier Detection.*
Wiley.

A. Marazzi (1993)
*Algorithms, Routines and S Functions for Robust Statistics.*
Wadsworth and Brooks/Cole.

P. J. Rousseeuw and B. C. van Zomeren (1990) Unmasking
multivariate outliers and leverage points,
*Journal of the American Statistical Association*, **85**, 633--639.

P. J. Rousseeuw and K. van Driessen (1999) A fast algorithm for the
minimum covariance determinant estimator. *Technometrics*
**41**, 212--223.

P. Rousseeuw and M. Hubert (1997) Recent developments in PROGRESS. In
*L1-Statistical Procedures and Related Topics *
ed Y. Dodge, IMS Lecture Notes volume **31**, pp. 201--214.

# NOT RUN { set.seed(123) cov.rob(stackloss) cov.rob(stack.x, method = "mcd", nsamp = "exact") # }

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