Notes on the tightness of the hybrid Cramér-Rao lower bound

Yair Noam; Hagit Messer

doi:10.1109/tsp.2009.2015113

Notes on the tightness of the hybrid Cramér-Rao lower bound

Yair Noam, Hagit Messer

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

85 Scopus citations

Abstract

In this paper, we study the properties of the hybrid Cramér-Rao bound (HCRB). We first address the problem of estimating unknown deterministic parameters in the presence of nuisance random parameters. We specify a necessary and sufficient condition under which the HCRB of the nonrandom parameters is equal to the Cramér-Rao bound (CRB). In this case, the HCRB is asymptotically tight [in high signal-to-noise ratio (SNR) or in large sample scenarios], and, therefore, useful. This condition can be evaluated even when the CRB cannot be evaluated analytically. If this condition is not satisfied, we show that the HCRB on the nonrandom parameters is always looser than the CRB. We then address the problem in which the random parameters are not nuisance. In this case, both random and nonrandom parameters need to be estimated. We provide a necessary and sufficient condition for the HCRB to be tight. Furthermore, we show that if the HCRB is tight, it is obtained by the maximum likelihood /maximum a posteriori probability (ML/MAP) estimator, which is shown to be an unbiased estimator which estimates both random and nonrandom parameters simultaneously optimally (in the minimum mean-square-error sense).

Original language	English
Pages (from-to)	2074-2084
Number of pages	11
Journal	IEEE Transactions on Signal Processing
Volume	57
Issue number	6
DOIs	https://doi.org/10.1109/tsp.2009.2015113
State	Published - 2009
Externally published	Yes

Bibliographical note

Funding Information:
Manuscript received June 09, 2008; accepted January 06, 2009. First published February 10, 2009; current version published May 15, 2009. The associate editor coordinating the review of this paper and approving it for publication was Dr. Peter J. Schreier. This work was supported by a fellowship from The Yitzhak and Chaya Weinstein Research Institute for Signal Processing at Tel-Aviv University.

Funding

Manuscript received June 09, 2008; accepted January 06, 2009. First published February 10, 2009; current version published May 15, 2009. The associate editor coordinating the review of this paper and approving it for publication was Dr. Peter J. Schreier. This work was supported by a fellowship from The Yitzhak and Chaya Weinstein Research Institute for Signal Processing at Tel-Aviv University.

Funders	Funder number
Yitzhak and Chaya Weinstein Research Institute for Signal Processing at Tel-Aviv University

Keywords

Asymptomatic tightness
Cramér-Rao bound (CRB)
Hybrid Cramér-Rao bound (HCRB)
Maximum likelihood/maximum a posteriori probability (ML/MAP)
Parameter estimation
Tightness

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abstract = "In this paper, we study the properties of the hybrid Cram{\'e}r-Rao bound (HCRB). We first address the problem of estimating unknown deterministic parameters in the presence of nuisance random parameters. We specify a necessary and sufficient condition under which the HCRB of the nonrandom parameters is equal to the Cram{\'e}r-Rao bound (CRB). In this case, the HCRB is asymptotically tight [in high signal-to-noise ratio (SNR) or in large sample scenarios], and, therefore, useful. This condition can be evaluated even when the CRB cannot be evaluated analytically. If this condition is not satisfied, we show that the HCRB on the nonrandom parameters is always looser than the CRB. We then address the problem in which the random parameters are not nuisance. In this case, both random and nonrandom parameters need to be estimated. We provide a necessary and sufficient condition for the HCRB to be tight. Furthermore, we show that if the HCRB is tight, it is obtained by the maximum likelihood /maximum a posteriori probability (ML/MAP) estimator, which is shown to be an unbiased estimator which estimates both random and nonrandom parameters simultaneously optimally (in the minimum mean-square-error sense).",

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Notes on the tightness of the hybrid Cramér-Rao lower bound

Abstract

Bibliographical note

Funding

Keywords

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