90 research outputs found
Computable lower bounds for deterministic parameter estimation
This paper is primarily tutorial in nature and presents a simple approach(norm minimization under linear constraints) for deriving computable lower bounds on the MSE of deterministic parameter estimators with a clear interpretation of the bounds. We also address the issue of lower bounds tightness in comparison with the MSE of ML estimators and their ability to predict the SNR threshold region. Last, as many practical estimation problems must be regarded as joint detection-estimation problems, we remind that the estimation performance must be conditional on detection performance, leading to the open problem of the fundamental limits of the joint detectionestimation performance
On Lower Bounds for Non Standard Deterministic Estimation
We consider deterministic parameter estimation and the situation where the probability density function (p.d.f.) parameterized by unknown deterministic parameters results from the marginalization of a joint p.d.f. depending on random variables as well. Unfortunately, in the general case, this marginalization is mathematically intractable, which prevents from using the known standard deterministic lower bounds (LBs) on the mean squared error (MSE). Actually the general case can be tackled by embedding the initial observation space in a hybrid one where any standard LB can be transformed into a modified one fitted to nonstandard deterministic estimation, at the expense of tightness however. Furthermore, these modified LBs (MLBs) appears
to include the submatrix of hybrid LBs which is an LB for the deterministic parameters. Moreover, since in the nonstandard estimation, maximum likelihood estimators (MLEs) can be no longer derived, suboptimal nonstandard MLEs (NSMLEs) are proposed as being a substitute. We show that any standard LB on the MSE of MLEs has a nonstandard version lower bounding the MSE of NSMLEs. We provide an analysis of the relative performance of the
NSMLEs, as well as a comparison with the MLBs for a large class of estimation problems. Last, the general approach introduced is exemplified, among other things, with a new look at the well-known Gaussian complex observation models
Hybrid Lower Bound On The MSE Based On The Barankin And Weiss-Weinstein Bounds
International audienceThis article investigates hybrid lower bounds in order to predict the estimators mean square error threshold effect. A tractable and computationally efficient form is derived. This form combines the Barankin and the Weiss-Weinstein bounds. This bound is applied to a frequency estimation problem for which a closed-form expression is provided. A comparison with results on the hybrid Barankin bound shows the superiority of this new bound to predict the mean square error threshold
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
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
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 ChapmanâRobbins and the CramĂ©râRao 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 (ChapmanâRobbins version) shows the superiority of the Abel-bound version to predict the mse threshold without increasing the computational complexity
A Constrained Hybrid Cramér-Rao Bound for Parameter Estimation
In statistical signal processing, hybrid parameter estimation refers to the case where the parameters vector to estimate contains both non-random and random parameters. Numerous works have shown the versatility of deterministic constrained CrameÌr-Rao bound for estimation performance analysis and design of a system of measurement. However in many systems both random and non-random parameters may occur simultaneously. In this communication, we propose a constrained hybrid lower bound which take into account of equality constraint on deterministic parameters. The usefulness of the proposed bound is illustrated with an application to radar Doppler estimation
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 WeissâWeinstein family. Among this family, we have Bayesian CramĂ©r-Rao bound, the BobrovskyâMayerWolfâZakaĂŻ bound, the Bayesian Bhattacharyya bound, the BobrovskyâZakaĂŻ bound, the ReuvenâMesser bound, and the WeissâWeinstein 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, MayerâWolf, and ZakaĂŻ. Second, we propose a bound based on the Bayesian Bhattacharyya bound and on the ReuvenâMesser 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 ReuvenâMesser bound, the BobrovskyâZakaĂŻ bound, and the Bayesian CramĂ©râRao 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
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
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