Cramér-Rao Bound Under Norm Constraint

Nitzan, Eyal; Routtenberg, Tirza; Tabrikian, Joseph

doi:10.1109/lsp.2019.2929411

Cited by 4 publications

(4 citation statements)

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“…This theory is crucial for the system design, error analysis, and quality assessments of existing estimation methods and for the development of new non-Bayesian estimators. In particular, the CCRB [26][27][28][29], which is associated with the CML estimator, is unsuited as a bound on the performance of Good-Turing estimators outside the asymptotic region, while it provides a lower bound on the MSE of any χ-unbiased estimator [28][29][30]. Our recent works on non-Bayesian estimation after selection [31][32][33] suggest that conditional schemes, in which the performance criterion depends on the observed data, require different CRB-type bounds.…”

Section: B Related Workmentioning

confidence: 99%

“…In this subsection we develop the conventional CCRB and the unbiasedness condition for estimating θ under the considered model. The CCRB [26,27] provides a lower bound on the MSE of any locally χ-unbiased estimator [28][29][30], which is a weaker requirement than ordinary mean unbiasedness, and is defined as follows.…”

Section: B Ccrb and Constrained Unbiasednessmentioning

confidence: 99%

“…The CCRB on the MSE of any unbiased estimator in the sense of Definition 1 at θ ∈ Ω θ is given by [27][28][29][30]…”

Section: B Ccrb and Constrained Unbiasednessmentioning

confidence: 99%

“…The measure of closeness is determined by the considered cost function, C( θ, θ). Examples for Lehmann unbiasedness with different cost functions and under parametric constraints can be found in [28,29,[31][32][33][34]43].…”

Section: A Lehmann Unbiasednessmentioning

confidence: 99%

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Non-Bayesian Parametric Missing-Mass Estimation

Cohen,

Routtenberg,

Tong

2021

Preprint

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We consider the classical problem of missing-mass estimation, which deals with estimating the total probability of unseen elements in a sample. The missing-mass estimation problem has various applications in machine learning, statistics, language processing, ecology, sensor networks, and others. The naive, constrained maximum likelihood (CML) estimator is inappropriate for this problem since it tends to overestimate the probability of the observed elements. Similarly, the conventional constrained Cramér-Rao bound (CCRB), which is a lower bound on the mean-squared-error (MSE) of unbiased estimators, does not provide a relevant bound on the performance for this problem. In this paper, we introduce a frequentist, non-Bayesian parametric model of the problem of missing-mass estimation. We introduce the concept of missing-mass unbiasedness by using the Lehmann unbiasedness definition. We derive a non-Bayesian CCRB-type lower bound on the missing-mass MSE (mmMSE), named the missing-mass CCRB (mmCCRB), based on the missing-mass unbiasedness. The missing-mass unbiasedness and the proposed mmCCRB can be used to evaluate the performance of existing estimators. Based on the new mmCCRB, we propose a new method to improve existing estimators by an iterative missing-mass Fisher scoring method. Finally, we demonstrate via numerical simulations that the proposed mmCCRB is a valid and informative lower bound on the mmMSE of state-of-the-art estimators for this problem: the CML, the Good-Turing, and Laplace estimators. We also show that the performance of the Laplace estimator is improved by using the new Fisher-scoring method.

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