Hannu Oja 
Department of Mathematic and Statistics
University of Jyväskylä
P.O.Box 35 (MaD)
FIN-40351 Jyväskylä
Finland
1. Introduction \
Classical multivariate methods (principal component analysis, multivariate regression, canonical correlation, discriminant analysis, Mahalanobis distance, Mahalanobis angle, etc.) are based on the sample mean vector and sample covariance matrix. Mean vector and covariance matrix are optimal if the data come from a multivariate normal distribution but they are very sensitive to outlying observations and loose in efficiency in the case of heavy tailed distrutions. In this talk, robust and nonparametric competitors of the mean vector and covariance matrix and their use in multivariate inference are considered.
2. Location vector, scatter matrix, shape matrix
2.1 Definitions
We assume that 
is a
random sample from a k-variate elliptically symmetric
distribution with cumulative distribution function (cdf) F,
symmetry center 
and covariance matrix
(if they exist). The aim is to consider and compare
the location vector, scatter matrix and shape matrix
functionals. The location, scatter and shape functionals
are then denoted by T(F), C(F) and V(F), or alternatively
by T( x), C( x) and V( x)
if x is a random vector with cdf F.
To be specific, a k-vector valued
functional T=T(F) is a location vector
if it is affine equivariant, that is,
if T(A z+ b)=AT( z)+ b for any
nonsingular matrix A and k-vector b.
A matrix valued functional C=C(F) is a scatter matrix if it is
PDS(k) (a positive definite symmetric matrix) and affine
equivariant, which in this case means that
.
Finally, functional V=V(F) is a shape matrix
if it is PDS(k), Tr(V)=k and it is affine equivariant in the sense that
Note that if C(F) is a scatter matrix then the
related shape matrix is given by
The affine equivariance property implies that, if the distribution
of z is a spherically symmetric distribution with cdf 
, mean vector 0
and covariance matrix 
,
then, for all location, scatter and shape functionals T, C and V,
where constant 
depends of both functional C and distribution .
Note that, for elliptic models, location vectors and shape
matrices are directly comparable without any modifications.
2.2 Influence functions and efficiency
The influence function is a tool to describe the robustness properties of an estimator; it also often serves a way to consider the asymptotic properties. The influence function (IF) of a functional T at F measures the effect of an infinitesimal contamination located at a single point x as follows. We consider the contaminated distribution
where 
is the cumulative distribution function
of a distribution with probability mass one
at x. The
influence function is defined as
The influence functions of
location scatter and shape functionals
T(F), C(F) and V(F) at spherical
are then given by
and
for a contamination point x,
r=|| x|| and 
.
See Croux and Haesbroeck (1999).
If V=(k/Tr(C))C then 
.
Note that the regular estimates (mean vector, covariance matrix)
use weight functions
,
and 
.
For robust functionals, the influence functions are
continuous and bounded.
The constants
are then used in efficiency comparisons.
It is easy to see for example that,
under general assumptions in the spherical case ,
and
where
3.Robust and nonparametric alternatives
In this talk we consider and compare multivariate location, scatter and shape estimates of three kinds, namely M-estimates, S-estimates and R-estimates.
Let again is a
random sample from a k-variate elliptically symmetric
distribution with cumulative distribution function (cdf) F.
The location and scatter estimates are then constructed
as follows. Let 
be a PDF(k) matrix and 
a k-vector.
Consider transformed observations
, i=1,...,n,
with symmetric 
. Write
and 
,
i=1,...,n.
The multivariate location and scatter M-estimates
are the choices and
for which
for some weight functions 
, 
and 
.
See Maronna (1976) and Huber(1981).
Next we define S-estimates.
The multivariate location and scatter S-estimates
are the choices and
which minimize 
subject to
for some function 
. See Rousseeuw and Leroy(1987)
and Davies(1987).
For the relation between M- and S-estimates, see
Lopuhaä (1989).
Finally, Ollila, Hettmansperger and Oja (2002) introduced
estimates based on multivariate sign vectors.
In their approach
location and scatter estimates based on signs
are the choices and
for which
where S( z) is a multivariate sign function. Multivariate rank vectors may be used similarly as well and the resulting family of estimates can be called multivariate location and scatter R-estimates.
4. Applications
4.1 Subspace estimation
In our first example we consider the problem
of subspace estimation.
Let be a
random sample from a k-variate elliptically symmetric
distribution with covariance matrix
where P is an orthogonal matrix with the eigenvectors
of in its columns and 
the diagonal matrix with the corresponding distinct
eigenvalues 
as diagonal entries. Write
where the r columns on 
and s columns
of 
, r+s=k, are supposed to span the
signal and noise subspaces, respectively.
Any shape matrix estimate 
may now be used to
estimate the signal space. If
is the estimate of 
obtained from then
where 
is the so called Frobenius
matrix norm, measures the distance between
the estimated and true signal subspace. See Crone and Crosby (1995).
If we then compare the accuracy of the estimates based
on and 
, a natural measure is
as 
.
4.2 Mahalanobis distance, Mahalanobis angle
Let again be a
random sample from a k-variate elliptically symmetric
distribution and let and
be location and scatter estimates.
Mahalanobis distance is sometimes used to
measure a distance of an observation from the
center of the data
The so called Mahalanobis angles
measure angular distances between vectors
and 
.
Finally, the Mahalanobis distance between two observations
and 
is given by
All the measures are naturally affine invariant. Again, mean vector and regular covariance matrix give measures which are sensitive to outlying observations; we end this talk with a discussion on the robustified versions of these measures.