The Constrained CramÉr–Rao Bound From the Perspective of Fitting a Model

Moore, Terrence J.; Kozick, R.J.; Sadler, Brian M.

doi:10.1109/lsp.2006.891316

Cited by 32 publications

(33 citation statements)

References 12 publications

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“…From Lemma 9 under the assumption in (11),θ CCRB satisfies (15). Thus,θ CCRB from (110) is a locally X -unbiased estimator…”

Section: (120)mentioning

confidence: 89%

“…Thus,θ CML can be considered as a C-unbiased estimator. In addition, it can be observed thatθ CML is not an X -unbiased estimator, since it does not satisfy (14) and (15). The LU-CCRB and the CCRB from (80) and (79), respectively, are evaluated in the lower plot of Fig.…”

Section: B Complex Amplitude Estimation With Amplitude Constraint Anmentioning

confidence: 99%

“…A popular estimator for constrained parameter estimation is the constrained maximum likelihood (CML) estimator [11], [15], [30]- [36]. This estimator is obtained by maximizing the likelihood function subject to parametric constraints.…”

Section: Introductionmentioning

confidence: 99%

“…only estimators with zero bias are considered. In early works on constrained parameter estimation [8], [11], [15], [36], the CCRB was assumed to be a lower bound for estimators that satisfy the constraints and have zero bias in the constrained set. It was shown in [37] that zero-bias requirement may be too strict.…”

Section: Introductionmentioning

confidence: 99%

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Cram$\acute{\text{e}}$r–Rao Bound for Constrained Parameter Estimation Using Lehmann-Unbiasedness

Nitzan

Routtenberg

Tabrikian

2019

IEEE Trans. Signal Process.

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The constrained Cramér-Rao bound (CCRB) is a lower bound on the mean-squared-error (MSE) of estimators that satisfy some unbiasedness conditions. Although the CCRB unbiasedness conditions are satisfied asymptotically by the constrained maximum likelihood (CML) estimator, in the nonasymptotic region these conditions are usually too strict and the commonly-used estimators, such as the CML estimator, do not satisfy them. Therefore, the CCRB may not be a lower bound on the MSE matrix of such estimators. In this paper, we propose a new definition for unbiasedness under constraints, denoted by C-unbiasedness, which is based on using Lehmann-unbiasedness with a weighted MSE (WMSE) risk and taking into account the parametric constraints. In addition to C-unbiasedness, a Cramér-Rao-type bound on the WMSE of C-unbiased estimators, denoted as Lehmann-unbiased CCRB (LU-CCRB), is derived. This bound is a scalar bound that depends on the chosen weighted combination of estimation errors. It is shown that C-unbiasedness is less restrictive than the CCRB unbiasedness conditions. Thus, the set of estimators that satisfy the CCRB unbiasedness conditions is a subset of the set of C-unbiased estimators and the proposed LU-CCRB may be an informative lower bound in cases where the corresponding CCRB is not. In the simulations, we examine linear and nonlinear estimation problems under nonlinear constraints in which the CML estimator is shown to be C-unbiased and the LU-CCRB is an informative lower bound on the WMSE, while the corresponding CCRB on the WMSE is not a lower bound and is not informative in the non-asymptotic region.

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“…From Lemma 9 under the assumption in (11),θ CCRB satisfies (15). Thus,θ CCRB from (110) is a locally X -unbiased estimator…”

Section: (120)mentioning

confidence: 89%

Section: B Complex Amplitude Estimation With Amplitude Constraint Anmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Cram$\acute{\text{e}}$r–Rao Bound for Constrained Parameter Estimation Using Lehmann-Unbiasedness

Nitzan

Routtenberg

Tabrikian

2019

IEEE Trans. Signal Process.

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“…To simplify the derivation, we follow the re-parameterization approach [32] and establish CRLB(u o ) via relating it to the CRLB of θ o = [φ, ϕ] T , where φ and ϕ are the source geodetic latitude and longitude (see (1)). Specifically, we have [20] …”

Section: Crlbmentioning

confidence: 99%

Dual-satellite source geolocation with time and frequency offsets and satellite location errors

Liu

Yang

Mihaylova

2017

2017 20th International Conference on Information Fusion (Fusion)

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Abstract-This paper considers locating a static source on Earth using the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements obtained by a dualsatellite geolocation system. The TDOA and FDOA from the source are subject to unknown time and frequency offsets because the two satellites are imperfectly time-synchronized or frequencylocked. The satellite locations are not known accurately as well. To make the source position identifiable and mitigate the effect of satellite location errors, calibration stations at known positions are used. Achieving the maximum likelihood (ML) geolocation performance usually requires jointly estimating the source position and extra variables (i.e., time and frequency offsets as well as satellite locations), which is computationally intensive. In this paper, a novel closed-form geolocation algorithm is proposed. It first fuses the TDOA and FDOA measurements from the source and calibration stations to produce a single pair of TDOA and FDOA for source geolocation. This measurement fusion step eliminates the time and frequency offsets while taking into account the presence of satellite location errors. The source position is then found via standard TDOA-FDOA geolocation. The developed algorithm has low complexity and performance analysis shows that it attains the Cramér-Rao lower bound (CRLB) under Gaussian noises and mild conditions. Simulations using a challenging scenario with a short-baseline dual-satellite system verify the theoretical developments and demonstrate the good performance of the proposed algorithm.

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2013

Wireless Communications

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The Constrained CramÉr–Rao Bound From the Perspective of Fitting a Model

Cited by 32 publications

References 12 publications

Cram$\acute{\text{e}}$r–Rao Bound for Constrained Parameter Estimation Using Lehmann-Unbiasedness

Cram$\acute{\text{e}}$r–Rao Bound for Constrained Parameter Estimation Using Lehmann-Unbiasedness

Dual-satellite source geolocation with time and frequency offsets and satellite location errors

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