24,524 research outputs found
Asymptotic properties of robust complex covariance matrix estimates
In many statistical signal processing applications, the estimation of
nuisance parameters and parameters of interest is strongly linked to the
resulting performance. Generally, these applications deal with complex data.
This paper focuses on covariance matrix estimation problems in non-Gaussian
environments and particularly, the M-estimators in the context of elliptical
distributions. Firstly, this paper extends to the complex case the results of
Tyler in [1]. More precisely, the asymptotic distribution of these estimators
as well as the asymptotic distribution of any homogeneous function of degree 0
of the M-estimates are derived. On the other hand, we show the improvement of
such results on two applications: DOA (directions of arrival) estimation using
the MUSIC (MUltiple SIgnal Classification) algorithm and adaptive radar
detection based on the ANMF (Adaptive Normalized Matched Filter) test
Spatial Sign Correlation
A new robust correlation estimator based on the spatial sign covariance
matrix (SSCM) is proposed. We derive its asymptotic distribution and influence
function at elliptical distributions. Finite sample and robustness properties
are studied and compared to other robust correlation estimators by means of
numerical simulations.Comment: 20 pages, 7 figures, 2 table
Performance analysis and optimal selection of large mean-variance portfolios under estimation risk
We study the consistency of sample mean-variance portfolios of arbitrarily
high dimension that are based on Bayesian or shrinkage estimation of the input
parameters as well as weighted sampling. In an asymptotic setting where the
number of assets remains comparable in magnitude to the sample size, we provide
a characterization of the estimation risk by providing deterministic
equivalents of the portfolio out-of-sample performance in terms of the
underlying investment scenario. The previous estimates represent a means of
quantifying the amount of risk underestimation and return overestimation of
improved portfolio constructions beyond standard ones. Well-known for the
latter, if not corrected, these deviations lead to inaccurate and overly
optimistic Sharpe-based investment decisions. Our results are based on recent
contributions in the field of random matrix theory. Along with the asymptotic
analysis, the analytical framework allows us to find bias corrections improving
on the achieved out-of-sample performance of typical portfolio constructions.
Some numerical simulations validate our theoretical findings
Performance Bounds for Parameter Estimation under Misspecified Models: Fundamental findings and applications
Inferring information from a set of acquired data is the main objective of
any signal processing (SP) method. In particular, the common problem of
estimating the value of a vector of parameters from a set of noisy measurements
is at the core of a plethora of scientific and technological advances in the
last decades; for example, wireless communications, radar and sonar,
biomedicine, image processing, and seismology, just to name a few. Developing
an estimation algorithm often begins by assuming a statistical model for the
measured data, i.e. a probability density function (pdf) which if correct,
fully characterizes the behaviour of the collected data/measurements.
Experience with real data, however, often exposes the limitations of any
assumed data model since modelling errors at some level are always present.
Consequently, the true data model and the model assumed to derive the
estimation algorithm could differ. When this happens, the model is said to be
mismatched or misspecified. Therefore, understanding the possible performance
loss or regret that an estimation algorithm could experience under model
misspecification is of crucial importance for any SP practitioner. Further,
understanding the limits on the performance of any estimator subject to model
misspecification is of practical interest. Motivated by the widespread and
practical need to assess the performance of a mismatched estimator, the goal of
this paper is to help to bring attention to the main theoretical findings on
estimation theory, and in particular on lower bounds under model
misspecification, that have been published in the statistical and econometrical
literature in the last fifty years. Secondly, some applications are discussed
to illustrate the broad range of areas and problems to which this framework
extends, and consequently the numerous opportunities available for SP
researchers.Comment: To appear in the IEEE Signal Processing Magazin
Asymptotic robustness of Kelly's GLRT and Adaptive Matched Filter detector under model misspecification
A fundamental assumption underling any Hypothesis Testing (HT) problem is
that the available data follow the parametric model assumed to derive the test
statistic. Nevertheless, a perfect match between the true and the assumed data
models cannot be achieved in many practical applications. In all these cases,
it is advisable to use a robust decision test, i.e. a test whose statistic
preserves (at least asymptotically) the same probability density function (pdf)
for a suitable set of possible input data models under the null hypothesis.
Building upon the seminal work of Kent (1982), in this paper we investigate the
impact of the model mismatch in a recurring HT problem in radar signal
processing applications: testing the mean of a set of Complex Elliptically
Symmetric (CES) distributed random vectors under a possible misspecified,
Gaussian data model. In particular, by using this general misspecified
framework, a new look to two popular detectors, the Kelly's Generalized
Likelihood Ration Test (GLRT) and the Adaptive Matched Filter (AMF), is
provided and their robustness properties investigated.Comment: ISI World Statistics Congress 2017 (ISI2017), Marrakech, Morocco,
16-21 July 201
The spatial sign covariance matrix and its application for robust correlation estimation
8 pages, 2 figures, to be published in the conference proceedings of 11th international conference "Computer Data Analysis & Modeling 2016" http://www.ajs.or.at/index.php/ajs/about/editorialPolicies#openAccessPolicyPeer reviewedPublisher PD
Convergence and Fluctuations of Regularized Tyler Estimators
This article studies the behavior of regularized Tyler estimators (RTEs) of
scatter matrices. The key advantages of these estimators are twofold. First,
they guarantee by construction a good conditioning of the estimate and second,
being a derivative of robust Tyler estimators, they inherit their robustness
properties, notably their resilience to the presence of outliers. Nevertheless,
one major problem that poses the use of RTEs in practice is represented by the
question of setting the regularization parameter . While a high value of
is likely to push all the eigenvalues away from zero, it comes at the
cost of a larger bias with respect to the population covariance matrix. A deep
understanding of the statistics of RTEs is essential to come up with
appropriate choices for the regularization parameter. This is not an easy task
and might be out of reach, unless one considers asymptotic regimes wherein the
number of observations and/or their size increase together. First
asymptotic results have recently been obtained under the assumption that
and are large and commensurable. Interestingly, no results concerning the
regime of going to infinity with fixed exist, even though the
investigation of this assumption has usually predated the analysis of the most
difficult and large case. This motivates our work. In particular, we
prove in the present paper that the RTEs converge to a deterministic matrix
when with fixed, which is expressed as a function of the
theoretical covariance matrix. We also derive the fluctuations of the RTEs
around this deterministic matrix and establish that these fluctuations converge
in distribution to a multivariate Gaussian distribution with zero mean and a
covariance depending on the population covariance and the parameter
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