105 research outputs found
Statistics of the MLE and Approximate Upper and Lower Bounds - Part 1: Application to TOA Estimation
In nonlinear deterministic parameter estimation, the maximum likelihood
estimator (MLE) is unable to attain the Cramer-Rao lower bound at low and
medium signal-to-noise ratios (SNR) due the threshold and ambiguity phenomena.
In order to evaluate the achieved mean-squared-error (MSE) at those SNR levels,
we propose new MSE approximations (MSEA) and an approximate upper bound by
using the method of interval estimation (MIE). The mean and the distribution of
the MLE are approximated as well. The MIE consists in splitting the a priori
domain of the unknown parameter into intervals and computing the statistics of
the estimator in each interval. Also, we derive an approximate lower bound
(ALB) based on the Taylor series expansion of noise and an ALB family by
employing the binary detection principle. The accurateness of the proposed
MSEAs and the tightness of the derived approximate bounds are validated by
considering the example of time-of-arrival estimation
A Useful Form of the Abel Bound and Its Application to Estimator Threshold Prediction
International audienceThis correspondence investigates the Abel bound in order to predict the estimators mean square error (mse) threshold effect. A tractable and computationally efficient form of this bound is derived. This form combines the Chapman–Robbins and the Cramér–Rao bounds. This bound is applied to a data-aided carrier frequency estimation problem for which a closed-form expression is provided. An indicator of the signal-to-noise ratio threshold is proposed. A comparison with recent results on the Barankin bound (Chapman–Robbins version) shows the superiority of the Abel-bound version to predict the mse threshold without increasing the computational complexity
On a Theoretical Background for Computing Reliable Approximations of the Barankin Bound
The Barankin bound is locally the greatest possible lower bound for the variance of any unbiased estimator of a deterministic pa- rameter, under certain relatively mild conditions. Much more essential, Barankin's work determines the su cient and necessary conditions un- der which an unbiased estimator with nite variance exists. Nevertheless, the computing of this bound, along with the proof of existence or non- existence of the estimator, has shown to be extremely challenging in most cases. Thereby, many approaches have been made to attain easily com- putable approximations of the bound, given it exists. Focusing on the rather central matter of existence, we provide a simple theoretical frame within which our approximations of the bound give a clear insight on whether an unbiased estimator does exist.Sociedad Argentina de Informática e Investigación Operativa (SADIO
On a Theoretical Background for Computing Reliable Approximations of the Barankin Bound
The Barankin bound is locally the greatest possible lower bound for the variance of any unbiased estimator of a deterministic pa- rameter, under certain relatively mild conditions. Much more essential, Barankin's work determines the su cient and necessary conditions un- der which an unbiased estimator with nite variance exists. Nevertheless, the computing of this bound, along with the proof of existence or non- existence of the estimator, has shown to be extremely challenging in most cases. Thereby, many approaches have been made to attain easily com- putable approximations of the bound, given it exists. Focusing on the rather central matter of existence, we provide a simple theoretical frame within which our approximations of the bound give a clear insight on whether an unbiased estimator does exist.Sociedad Argentina de Informática e Investigación Operativa (SADIO
Hierarchies of Frequentist Bounds for Quantum Metrology: From Cram\'er-Rao to Barankin
We derive lower bounds on the variance of estimators in quantum metrology by
choosing test observables that define constraints on the unbiasedness of the
estimator. The quantum bounds are obtained by analytical optimization over all
possible quantum measurements and estimators that satisfy the given
constraints. We obtain hierarchies of increasingly tight bounds that include
the quantum Cram\'er-Rao bound at the lowest order. In the opposite limit, the
quantum Barankin bound is the variance of the locally best unbiased estimator
in quantum metrology. Our results reveal generalizations of the quantum Fisher
information that are able to avoid regularity conditions and identify threshold
behavior in quantum measurements with mixed states, caused by finite data.Comment: 6+7 pages, 1+1 figure
Weiss–Weinstein Bound for Data-Aided Carrier Estimation
International audienceThis letter investigates Bayesian bounds on the mean-square error (MSE) applied to a data-aided carrier estimation problem. The presented bounds are derived from a covariance inequality principle: the so-calledWeiss andWeinstein family. These bounds are of utmost interest to find the fundamental MSE limits of an estimator, even for critical scenarios (low signal-to-noise ratio and/or low number of observations). In a data-aided carrier estimation problem, a closed-form expression of the Weiss–Weinstein bound (WWB) that is known to be the tightest bound of the Weiss and Weinstein family is given. A comparison with the maximum likelihood estimator and the other bounds of the Weiss and Weinstein family is given. The WWB is shown to be an efficient tool to approximate this estimator's MSE and to predict the well-known threshold effect
A Fresh Look at the Bayesian Bounds of the Weiss-Weinstein Family
International audienceMinimal bounds on the mean square error (MSE) are generally used in order to predict the best achievable performance of an estimator for a given observation model. In this paper, we are interested in the Bayesian bound of the Weiss–Weinstein family. Among this family, we have Bayesian Cramér-Rao bound, the Bobrovsky–MayerWolf–Zakaï bound, the Bayesian Bhattacharyya bound, the Bobrovsky–Zakaï bound, the Reuven–Messer bound, and the Weiss–Weinstein bound. We present a unification of all these minimal bounds based on a rewriting of the minimum mean square error estimator (MMSEE) and on a constrained optimization problem. With this approach, we obtain a useful theoretical framework to derive new Bayesian bounds. For that purpose, we propose two bounds. First, we propose a generalization of the Bayesian Bhattacharyya bound extending the works of Bobrovsky, Mayer–Wolf, and Zakaï. Second, we propose a bound based on the Bayesian Bhattacharyya bound and on the Reuven–Messer bound, representing a generalization of these bounds. The proposed bound is the Bayesian extension of the deterministic Abel bound and is found to be tighter than the Bayesian Bhattacharyya bound, the Reuven–Messer bound, the Bobrovsky–Zakaï bound, and the Bayesian Cramér–Rao bound. We propose some closed-form expressions of these bounds for a general Gaussian observation model with parameterized mean. In order to illustrate our results, we present simulation results in the context of a spectral analysis problem
Weiss-Weinstein bound of frequency estimation error for very weak GNSS signals
Tightness remains the center quest in all modern estimation bounds. For very
weak signals, this is made possible with judicial choices of prior probability
distribution and bound family. While current bounds in GNSS assess performance
of carrier frequency estimators under Gaussian or uniform assumptions, the
circular nature of frequency is overlooked. In addition, of all bounds in
Bayesian framework, Weiss-Weinstein bound (WWB) stands out since it is free
from regularity conditions or requirements on the prior distribution.
Therefore, WWB is extended for the current frequency estimation problem. A
divide-and-conquer type of hyperparameter tuning method is developed to level
off the curse of computational complexity for the WWB family while enhancing
tightness. Synthetic results show that with von Mises as prior probability
distribution, WWB provides a bound up to 22.5% tighter than Ziv-Zaka\"i bound
(ZZB) when SNR varies between -3.5 dB and -20 dB, where GNSS signal is deemed
extremely weak.Comment: 35 pages, 13 figures, submitted to NAVIGATION, Journal of the
Institute of Navigatio
Statistical Methods for Image Registration and Denoising
This dissertation describes research into image processing techniques that enhance military operational and support activities. The research extends existing work on image registration by introducing a novel method that exploits local correlations to improve the performance of projection-based image registration algorithms. The dissertation also extends the bounds on image registration performance for both projection-based and full-frame image registration algorithms and extends the Barankin bound from the one-dimensional case to the problem of two-dimensional image registration. It is demonstrated that in some instances, the Cramer-Rao lower bound is an overly-optimistic predictor of image registration performance and that under some conditions, the Barankin bound is a better predictor of shift estimator performance. The research also looks at the related problem of single-frame image denoising using block-based methods. The research introduces three algorithms that operate by identifying regions of interest within a noise-corrupted image and then generating noise free estimates of the regions as averages of similar regions in the image
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