83,288 research outputs found
Quantum Theory of Superresolution for Two Incoherent Optical Point Sources
Rayleigh's criterion for resolving two incoherent point sources has been the
most influential measure of optical imaging resolution for over a century. In
the context of statistical image processing, violation of the criterion is
especially detrimental to the estimation of the separation between the sources,
and modern farfield superresolution techniques rely on suppressing the emission
of close sources to enhance the localization precision. Using quantum optics,
quantum metrology, and statistical analysis, here we show that, even if two
close incoherent sources emit simultaneously, measurements with linear optics
and photon counting can estimate their separation from the far field almost as
precisely as conventional methods do for isolated sources, rendering Rayleigh's
criterion irrelevant to the problem. Our results demonstrate that
superresolution can be achieved not only for fluorophores but also for stars.Comment: 18 pages, 11 figures. v1: First draft. v2: Improved the presentation
and added a section on the issues of unknown centroid and misalignment. v3:
published in Physical Review
Identifying the New Keynesian Phillips Curve
Phillips curves are central to discussions of inflation dynamics and monetary policy. New Keynesian Phillips curves describe how past inflation, expected future inflation, and a measure of real marginal cost or an output gap drive the current inflation rate. This paper studies the (potential) weak identification of these curves under GMM and traces this syndrome to a lack of persistence in either exogenous variables or shocks. We employ analytic methods to understand the identification problem in several statistical environments: under strict exogeneity, in a vector autoregression, and in the canonical three-equation, New Keynesian model. Given U.S., U.K., and Canadian data, we revisit the empirical evidence and construct tests and confidence intervals based on exact and pivotal Anderson-Rubin statistics that are robust to weak identification. These tests find little evidence of forward-looking inflation dynamics.Phillips curve, Keynesian, identification, inflation
Identifying the New Keynesian Phillips curve
Phillips curves are central to discussions of inflation dynamics and monetary policy. New Keynesian Phillips curves describe how past inflation, expected future inflation, and a measure of real marginal cost or an output gap drive the current inflation rate. This paper studies the (potential) weak identification of these curves under generalized methods of moments (GMM) and traces this syndrome to a lack of persistence in either exogenous variables or shocks. The authors employ analytic methods to understand the identification problem in several statistical environments: under strict exogeneity, in a vector autoregression, and in the canonical three-equation, New Keynesian model. Given U.S., U.K., and Canadian data, they revisit the empirical evidence and construct tests and confidence intervals based on exact and pivotal Anderson-Rubin statistics that are robust to weak identification. These tests find little evidence of forward-looking inflation dynamics.
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
Optimal waveform estimation for classical and quantum systems via time-symmetric smoothing
Classical and quantum theories of time-symmetric smoothing, which can be used
to optimally estimate waveforms in classical and quantum systems, are derived
using a discrete-time approach, and the similarities between the two theories
are emphasized. Application of the quantum theory to homodyne phase-locked loop
design for phase estimation with narrowband squeezed optical beams is studied.
The relation between the proposed theory and Aharonov et al.'s weak value
theory is also explored.Comment: 13 pages, 5 figures, v2: changed the title to a more descriptive one,
corrected a minor mistake in Sec. IV, accepted by Physical Review
Discussion of: Treelets--An adaptive multi-scale basis for sparse unordered data
We would like to congratulate Lee, Nadler and Wasserman on their contribution
to clustering and data reduction methods for high and low situations. A
composite of clustering and traditional principal components analysis, treelets
is an innovative method for multi-resolution analysis of unordered data. It is
an improvement over traditional PCA and an important contribution to clustering
methodology. Their paper [arXiv:0707.0481] presents theory and supporting
applications addressing the two main goals of the treelet method: (1) Uncover
the underlying structure of the data and (2) Data reduction prior to
statistical learning methods. We will organize our discussion into two main
parts to address their methodology in terms of each of these two goals. We will
present and discuss treelets in terms of a clustering algorithm and an
improvement over traditional PCA. We will also discuss the applicability of
treelets to more general data, in particular, the application of treelets to
microarray data.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS137F the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Learning and Designing Stochastic Processes from Logical Constraints
Stochastic processes offer a flexible mathematical formalism to model and
reason about systems. Most analysis tools, however, start from the premises
that models are fully specified, so that any parameters controlling the
system's dynamics must be known exactly. As this is seldom the case, many
methods have been devised over the last decade to infer (learn) such parameters
from observations of the state of the system. In this paper, we depart from
this approach by assuming that our observations are {\it qualitative}
properties encoded as satisfaction of linear temporal logic formulae, as
opposed to quantitative observations of the state of the system. An important
feature of this approach is that it unifies naturally the system identification
and the system design problems, where the properties, instead of observations,
represent requirements to be satisfied. We develop a principled statistical
estimation procedure based on maximising the likelihood of the system's
parameters, using recent ideas from statistical machine learning. We
demonstrate the efficacy and broad applicability of our method on a range of
simple but non-trivial examples, including rumour spreading in social networks
and hybrid models of gene regulation
Hybrid Shrinkage Estimators Using Penalty Bases For The Ordinal One-Way Layout
This paper constructs improved estimators of the means in the Gaussian
saturated one-way layout with an ordinal factor. The least squares estimator
for the mean vector in this saturated model is usually inadmissible. The hybrid
shrinkage estimators of this paper exploit the possibility of slow variation in
the dependence of the means on the ordered factor levels but do not assume it
and respond well to faster variation if present. To motivate the development,
candidate penalized least squares (PLS) estimators for the mean vector of a
one-way layout are represented as shrinkage estimators relative to the penalty
basis for the regression space. This canonical representation suggests further
classes of candidate estimators for the unknown means: monotone shrinkage (MS)
estimators or soft-thresholding (ST) estimators or, most generally, hybrid
shrinkage (HS) estimators that combine the preceding two strategies. Adaptation
selects the estimator within a candidate class that minimizes estimated risk.
Under the Gaussian saturated one-way layout model, such adaptive estimators
minimize risk asymptotically over the class of candidate estimators as the
number of factor levels tends to infinity. Thereby, adaptive HS estimators
asymptotically dominate adaptive MS and adaptive ST estimators as well as the
least squares estimator. Local annihilators of polynomials, among them
difference operators, generate penalty bases suitable for a range of numerical
examples.Comment: Published at http://dx.doi.org/10.1214/009053604000000652 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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