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

Yair Noam, Hagit Messer

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)2074-2084
Number of pages11
JournalIEEE Transactions on Signal Processing
Issue number6
StatePublished - 2009
Externally publishedYes

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.


  • 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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