4,500 research outputs found
Multidimensional trimming based on projection depth
As estimators of location parameters, univariate trimmed means are well known
for their robustness and efficiency. They can serve as robust alternatives to
the sample mean while possessing high efficiencies at normal as well as
heavy-tailed models. This paper introduces multidimensional trimmed means based
on projection depth induced regions. Robustness of these depth trimmed means is
investigated in terms of the influence function and finite sample breakdown
point. The influence function captures the local robustness whereas the
breakdown point measures the global robustness of estimators. It is found that
the projection depth trimmed means are highly robust locally as well as
globally. Asymptotics of the depth trimmed means are investigated via those of
the directional radius of the depth induced regions. The strong consistency,
asymptotic representation and limiting distribution of the depth trimmed means
are obtained. Relative to the mean and other leading competitors, the depth
trimmed means are highly efficient at normal or symmetric models and
overwhelmingly more efficient when these models are contaminated. Simulation
studies confirm the validity of the asymptotic efficiency results at finite
samples.Comment: Published at http://dx.doi.org/10.1214/009053606000000713 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Outlier robust corner-preserving methods for reconstructing noisy images
The ability to remove a large amount of noise and the ability to preserve
most structure are desirable properties of an image smoother. Unfortunately,
they usually seem to be at odds with each other; one can only improve one
property at the cost of the other. By combining M-smoothing and
least-squares-trimming, the TM-smoother is introduced as a means to unify
corner-preserving properties and outlier robustness. To identify edge- and
corner-preserving properties, a new theory based on differential geometry is
developed. Further, robustness concepts are transferred to image processing. In
two examples, the TM-smoother outperforms other corner-preserving smoothers. A
software package containing both the TM- and the M-smoother can be downloaded
from the Internet.Comment: Published at http://dx.doi.org/10.1214/009053606000001109 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On robustness properties of bootstrap approximations
Bootstrap approximations to the sampling distribution can be seen as generalized statistics taking values in a space of probability measures. We first analyze qualitative robustness [in Hampel's (1971) sense] of these statistics when the initial estimators {Tn } (whose distributions we want to approximate using bootstrap resampling) are obtained by restriction from a statistical functional T defined for all probability distributions. Whereas continuity of T turns out to be the natural condition to ensure qualitative robustness of {Tn }, we show that the uniform continuity of T is a sufficient condition for robustness of the bootstrap. This result applies to M-estimators.
Next, we study asymptotic properties of the bootstrap estimator for the infiuence function T'(F; x) of T at a distribution F and we prove that continuous Hadamard differentiability of the operator F_ T'(F;.) with respect to F is a natural condition to establish the validity of bootstrap confidence bands for this estimator
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
Tyler shape depth
In many problems from multivariate analysis, the parameter of interest is a
shape matrix, that is, a normalized version of the corresponding scatter or
dispersion matrix. In this paper, we propose a depth concept for shape matrices
that involves data points only through their directions from the center of the
distribution. We use the terminology Tyler shape depth since the resulting
estimator of shape, namely the deepest shape matrix, is the median-based
counterpart of the M-estimator of shape of Tyler (1987). Beyond estimation,
shape depth, like its Tyler antecedent, also allows hypothesis testing on
shape. Its main benefit, however, lies in the ranking of shape matrices it
provides, whose practical relevance is illustrated in principal component
analysis and in shape-based outlier detection. We study the invariance,
quasi-concavity and continuity properties of Tyler shape depth, the topological
and boundedness properties of the corresponding depth regions, existence of a
deepest shape matrix and prove Fisher consistency in the elliptical case.
Finally, we derive a Glivenko-Cantelli-type result and establish almost sure
consistency of the deepest shape matrix estimator.Comment: 28 pages, 5 figure
On high-dimensional sign tests
Sign tests are among the most successful procedures in multivariate
nonparametric statistics. In this paper, we consider several testing problems
in multivariate analysis, directional statistics and multivariate time series
analysis, and we show that, under appropriate symmetry assumptions, the
fixed- multivariate sign tests remain valid in the high-dimensional case.
Remarkably, our asymptotic results are universal, in the sense that, unlike in
most previous works in high-dimensional statistics, may go to infinity in
an arbitrary way as does. We conduct simulations that (i) confirm our
asymptotic results, (ii) reveal that, even for relatively large , chi-square
critical values are to be favoured over the (asymptotically equivalent)
Gaussian ones and (iii) show that, for testing i.i.d.-ness against serial
dependence in the high-dimensional case, Portmanteau sign tests outperform
their competitors in terms of validity-robustness.Comment: Published at http://dx.doi.org/10.3150/15-BEJ710 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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