448 research outputs found
Lower Bounds on Exponential Moments of the Quadratic Error in Parameter Estimation
Considering the problem of risk-sensitive parameter estimation, we propose a
fairly wide family of lower bounds on the exponential moments of the quadratic
error, both in the Bayesian and the non--Bayesian regime. This family of
bounds, which is based on a change of measures, offers considerable freedom in
the choice of the reference measure, and our efforts are devoted to explore
this freedom to a certain extent. Our focus is mostly on signal models that are
relevant to communication problems, namely, models of a parameter-dependent
signal (modulated signal) corrupted by additive white Gaussian noise, but the
methodology proposed is also applicable to other types of parametric families,
such as models of linear systems driven by random input signals (white noise,
in most cases), and others. In addition to the well known motivations of the
risk-sensitive cost function (i.e., the exponential quadratic cost function),
which is most notably, the robustness to model uncertainty, we also view this
cost function as a tool for studying fundamental limits concerning the tail
behavior of the estimation error. Another interesting aspect, that we
demonstrate in a certain parametric model, is that the risk-sensitive cost
function may be subjected to phase transitions, owing to some analogies with
statistical mechanics.Comment: 28 pages; 4 figures; submitted for publicatio
Analysis of the Bayesian Cramer-Rao lower bound in astrometry: Studying the impact of prior information in the location of an object
Context. The best precision that can be achieved to estimate the location of
a stellar-like object is a topic of permanent interest in the astrometric
community.
Aims. We analyse bounds for the best position estimation of a stellar-like
object on a CCD detector array in a Bayesian setting where the position is
unknown, but where we have access to a prior distribution. In contrast to a
parametric setting where we estimate a parameter from observations, the
Bayesian approach estimates a random object (i.e., the position is a random
variable) from observations that are statistically dependent on the position.
Methods. We characterize the Bayesian Cramer-Rao (CR) that bounds the minimum
mean square error (MMSE) of the best estimator of the position of a point
source on a linear CCD-like detector, as a function of the properties of
detector, the source, and the background.
Results. We quantify and analyse the increase in astrometric performance from
the use of a prior distribution of the object position, which is not available
in the classical parametric setting. This gain is shown to be significant for
various observational regimes, in particular in the case of faint objects or
when the observations are taken under poor conditions. Furthermore, we present
numerical evidence that the MMSE estimator of this problem tightly achieves the
Bayesian CR bound. This is a remarkable result, demonstrating that all the
performance gains presented in our analysis can be achieved with the MMSE
estimator.
Conclusions The Bayesian CR bound can be used as a benchmark indicator of the
expected maximum positional precision of a set of astrometric measurements in
which prior information can be incorporated. This bound can be achieved through
the conditional mean estimator, in contrast to the parametric case where no
unbiased estimator precisely reaches the CR bound.Comment: 17 pages, 12 figures. Accepted for publication on Astronomy &
Astrophysic
Use and Abuse of the Fisher Information Matrix in the Assessment of Gravitational-Wave Parameter-Estimation Prospects
The Fisher-matrix formalism is used routinely in the literature on
gravitational-wave detection to characterize the parameter-estimation
performance of gravitational-wave measurements, given parametrized models of
the waveforms, and assuming detector noise of known colored Gaussian
distribution. Unfortunately, the Fisher matrix can be a poor predictor of the
amount of information obtained from typical observations, especially for
waveforms with several parameters and relatively low expected signal-to-noise
ratios (SNR), or for waveforms depending weakly on one or more parameters, when
their priors are not taken into proper consideration. In this paper I discuss
these pitfalls; show how they occur, even for relatively strong signals, with a
commonly used template family for binary-inspiral waveforms; and describe
practical recipes to recognize them and cope with them.
Specifically, I answer the following questions: (i) What is the significance
of (quasi-)singular Fisher matrices, and how must we deal with them? (ii) When
is it necessary to take into account prior probability distributions for the
source parameters? (iii) When is the signal-to-noise ratio high enough to
believe the Fisher-matrix result? In addition, I provide general expressions
for the higher-order, beyond--Fisher-matrix terms in the 1/SNR expansions for
the expected parameter accuracies.Comment: 24 pages, 3 figures, previously known as "A User Manual for the
Fisher Information Matrix"; final, corrected PRD versio
A Search for Tensor, Vector, and Scalar Polarizations in the Stochastic Gravitational-Wave Background
The detection of gravitational waves with Advanced LIGO and Advanced Virgo has enabled novel tests of general relativity, including direct study of the polarization of gravitational waves. While general relativity allows for only two tensor gravitational-wave polarizations, general metric theories can additionally predict two vector and two scalar polarizations. The polarization of gravitational waves is encoded in the spectral shape of the stochastic gravitational-wave background, formed by the superposition of cosmological and individually-unresolved astrophysical sources. Using data recorded by Advanced LIGO during its first observing run, we search for a stochastic background of generically-polarized gravitational waves. We find no evidence for a background of any polarization, and place the first direct bounds on the contributions of vector and scalar polarizations to the stochastic background. Under log-uniform priors for the energy in each polarization, we limit the energy-densities of tensor, vector, and scalar modes at 95% credibility to ΩT
Search for Tensor, Vector, and Scalar Polarizations In the Stochastic Gravitational-Wave Background
The detection of gravitational waves with Advanced LIGO and Advanced Virgo has enabled novel tests of general relativity, including direct study of the polarization of gravitational waves. While general relativity allows for only two tensor gravitational-wave polarizations, general metric theories can additionally predict two vector and two scalar polarizations. The polarization of gravitational waves is encoded in the spectral shape of the stochastic gravitational-wave background, formed by the superposition of cosmological and individually unresolved astrophysical sources. Using data recorded by Advanced LIGO during its first observing run, we search for a stochastic background of generically polarized gravitational waves. We find no evidence for a background of any polarization, and place the first direct bounds on the contributions of vector and scalar polarizations to the stochastic background. Under log-uniform priors for the energy in each polarization, we limit the energy densities of tensor, vector, and scalar modes at 95% credibility to Ω0
Setting upper limits on the strength of periodic gravitational waves from PSR J1939+2134 using the first science data from the GEO 600 and LIGO detectors
Data collected by the GEO 600 and LIGO interferometric gravitational wave detectors during their first observational science run were searched for continuous gravitational waves from the pulsar J1939+2134 at twice its rotation frequency. Two independent analysis methods were used and are demonstrated in this paper: a frequency domain method and a time domain method. Both achieve consistent null results, placing new upper limits on the strength of the pulsar's gravitational wave emission. A model emission mechanism is used to interpret the limits as a constraint on the pulsar's equatorial ellipticity
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