Power M-estimators of shape and location were recently suggested in
Frahm et al. (2020). They have a tuning parameter alpha taking values in
[0,1]. The extreme case alpha = 1 corresponds to Tyler's shape
matrix and alpha = 0 to the classical covariance matrix. These special
cases have their own, more efficient functions tylerShape and
classicShape, respectively.
If the true location is known, it should be supplied as center, otherwise
it is estimated simultaneously with the shape.
powerShape(x, alpha, center = NULL,
normalization = c("det", "trace", "one"), maxiter = 1e4, eps = 1e-6)tylerShape(x, center = NULL,
normalization = c("det", "trace", "one"), maxiter = 1e4, eps = 1e-6)
classicShape(x, center = NULL,
normalization = c("det", "trace", "one"), maxiter = 1e4, eps = 1e-6)
A numeric data matrix or data.frame without missing data.
Tail index, a numeric value in the interval [0, 1].
Determines the power function. For more information see 'Details'.
An optional vector of the data's center. If NULL the center
will be estimated simultaneously with the shape.
A string determining how the shape matrix is standardized. The possible values are
'det'such that the returned shape estimate has determinant 1.
'trace'such that the returned shape estimate has trace ncol(x).
'one'such that the returned shape estimate's top left entry
(S[1, 1]) is 1.
A positive integer, restricting the maximum number of iterations.
A numeric, specifying the tolerance level of when the iteration stops.
A list with class 'shapeNA' containing the following components:
The estimated shape matrix.
The scale with which the shape matrix may be scaled to obtain
a scatter estimate. If alpha = 1, then this value is NA, as Tyler's
shape matrix has no natural scale.
The location parameter, either provided by the user or estimated.
The tail index with which the Power M-estimator has been called.
NULL, since powerShape operates only on complete data.
Number of computed iterations before convergence.
The matched call.
These functions assume that the data were generated from an elliptical distribution, for Tyler's estimate this can be relaxed to generalized elliptical distributions.
For multivariate normally distributed data, classicShape is the maximum
likelihood estimator of location and scale. It is a special case of the
power M-estimator with tail index alpha = 0, which returns the empirical
covariance matrix and the empirical mean vector.
The function tylerShape maximizes the likelihood function after projecting
the observed data of each individual onto the unit hypersphere, in which case
we obtain an angular central Gaussian distribution. It is a special case of
the power M-estimator with tail index alpha = 1, which returns Tyler's
M-estimator of scatter and an affine equivariant multivariate median
according to Hettmansperger and Randles (2002).
The function powerShape requires an additional parameter, the so-called
tail index alpha. For heavy tailed data, the index should be chosen closer
to 1, whereas for light tailed data the index should be chosen closer to 0.
Tyler, D.E. (1987). A Distribution-Free M-Estimator of Multivariate Scatter. The Annals of Statistics, 15, 234.251. 10.1214/aos/1176350263.
Frahm, G., Nordhausen, K., & Oja, H. (2020). M-estimation with incomplete and dependent multivariate data. Journal of Multivariate Analysis, 176, 104569. 10.1016/j.jmva.2019.104569.
Hettmansperger, T. P., & Randles, R. H. (2002). A practical affine equivariant multivariate median. Biometrika, 89(4), 851-860. 10.1093/biomet/89.4.851
powerShapeNA, tylerShapeNA and classicShapeNA for the corresponding functions for data with missing values.
# NOT RUN {
## Generate example data
S <- toeplitz(c(1, 0.1))
x <- mvtnorm::rmvt(100, S)
## Compute some M-estimators
res0 <- classicShape(x, center = c(0, 0))
res1 <- powerShape(x, alpha = 0.67, normalization = 'one')
res2 <- tylerShape(x, normalization = 'trace')
## Get location estimates
res1$mu
res2$mu
## Get shape estimates
res0$S
res1$S
res2$S
## Print summary
summary(res0)
# }
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