4,645 research outputs found
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
Bounds on Portfolio Quality
The signal-noise ratio of a portfolio of p assets, its expected return
divided by its risk, is couched as an estimation problem on the sphere. When
the portfolio is built using noisy data, the expected value of the signal-noise
ratio is bounded from above via a Cramer-Rao bound, for the case of Gaussian
returns. The bound holds for `biased' estimators, thus there appears to be no
bias-variance tradeoff for the problem of maximizing the signal-noise ratio. An
approximate distribution of the signal-noise ratio for the Markowitz portfolio
is given, and shown to be fairly accurate via Monte Carlo simulations, for
Gaussian returns as well as more exotic returns distributions. These findings
imply that if the maximal population signal-noise ratio grows slower than the
universe size to the 1/4 power, there may be no diversification benefit, rather
expected signal-noise ratio can decrease with additional assets. As a practical
matter, this may explain why the Markowitz portfolio is typically applied to
small asset universes. Finally, the theorem is expanded to cover more general
models of returns and trading schemes, including the conditional expectation
case where mean returns are linear in some observable features, subspace
constraints (i.e., dimensionality reduction), and hedging constraints
Frequentist coverage of adaptive nonparametric Bayesian credible sets
We investigate the frequentist coverage of Bayesian credible sets in a
nonparametric setting. We consider a scale of priors of varying regularity and
choose the regularity by an empirical Bayes method. Next we consider a central
set of prescribed posterior probability in the posterior distribution of the
chosen regularity. We show that such an adaptive Bayes credible set gives
correct uncertainty quantification of "polished tail" parameters, in the sense
of high probability of coverage of such parameters. On the negative side, we
show by theory and example that adaptation of the prior necessarily leads to
gross and haphazard uncertainty quantification for some true parameters that
are still within the hyperrectangle regularity scale.Comment: Published at http://dx.doi.org/10.1214/14-AOS1270 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Distributed multi-agent Gaussian regression via finite-dimensional approximations
We consider the problem of distributedly estimating Gaussian processes in
multi-agent frameworks. Each agent collects few measurements and aims to
collaboratively reconstruct a common estimate based on all data. Agents are
assumed with limited computational and communication capabilities and to gather
noisy measurements in total on input locations independently drawn from a
known common probability density. The optimal solution would require agents to
exchange all the input locations and measurements and then invert an matrix, a non-scalable task. Differently, we propose two suboptimal
approaches using the first orthonormal eigenfunctions obtained from the
\ac{KL} expansion of the chosen kernel, where typically . The benefits
are that the computation and communication complexities scale with and not
with , and computing the required statistics can be performed via standard
average consensus algorithms. We obtain probabilistic non-asymptotic bounds
that determine a priori the desired level of estimation accuracy, and new
distributed strategies relying on Stein's unbiased risk estimate (SURE)
paradigms for tuning the regularization parameters and applicable to generic
basis functions (thus not necessarily kernel eigenfunctions) and that can again
be implemented via average consensus. The proposed estimators and bounds are
finally tested on both synthetic and real field data
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