Bayesian Estimation in the Presence of Deterministic Nuisance Parameters—Part I: Performance Bounds


How accurately will one estimate a random parameter subject to unknown deterministic nuisance parameters? The hybrid Cramér-Rao bound (HCRB) provides an answer to this question for a restricted category of estimators. The HCRB is the most standard performance certain on the mean-sq.-error (MSE) for random parameter estimation issues that involve deterministic parameters. The HCRB is useful when one is interested in each the random and also the deterministic parameters and in the coupling between their estimation errors. This sure refers to regionally weak-sense unbiased estimators with respect to (w.r.t.) the deterministic parameters. But, if these parameters are nuisance, it is unnecessary to restrict their estimation as unbiased. This paper is the primary of a 2-part study of Bayesian parameter estimation in the presence of deterministic nuisance parameters. It begins with a study on order relations between existing Cramér-Rao (CR)-sort bounds of mean-unbiased Bayesian estimators. Then, a replacement CR-sort certain is developed with no assumption of unbiasedness on the nuisance parameters. Alternatively, Lehmann’s concept of unbiasedness is employed rather than conventional mean-unbiasedness. It's imposed on a risk that measures the space between the estimator and therefore the minimum MSE (MMSE) estimator which assumes good data of the nuisance parameters. Within the succeeding paper, asymptotic performances of some Bayesian estimators with maximum likelihood primarily based estimates for the nuisance parameters are investigated. The proposed risk-unbiased certain (RUB) is proved to be asymptotically achieved by the MMSE estimator with most chance estimates for the nuisance parameters, while the present CR-sort bounds are not necessarily achievable.

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