Cramér-Rao bound for a mixture of real- and integer-valued parameter vectors and its application to the linear regression model

Abstract

Performance lower bounds are known to be a fundamental design tool in parametric estimation theory. A plethora of deterministic bounds exist in the literature, ranging from the general Barankin bound to the well-known Cramér-Rao bound (CRB), the latter providing the optimal mean square error performance of locally unbiased estimators. In this contribution, we are interested in the estimation of mixed realand integer-valued parameter vectors. We propose a closed-form lower bound expression leveraging on the general CRB formulation, being the limiting form of the McAulay-Seidman bound. Such formulation is the key point to take into account integer-valued parameters. As a particular case of the general form, we provide closed-form expressions for the Gaussian observation model. One noteworthy point is the assessment of the asymptotic efficiency of the maximum likelihood estimator for a linear regression model with mixed parameter vectors and known noise covariance matrix, thus complementing the rather rich literature on that topic. A representative carrier-phase based precise positioning example is provided to support the discussion and show the usefulness of the proposed lower bound.

Compact CRB for delay, Doppler, and phase estimation – application to GNSS SPP and RTK performance characterisation

“…23], is a relevant lower bound for the vector of estimates , where b can be regarded as the parameter vector of interest and a a so‐called nuisance parameter vector. Since it is well known that adding unknown parameters leads to an equal or higher CRB, then [48] where denotes the covariance matrix of . In addition, from [38, Ch.…”

“…Indeed the general problem is encountered in a multitude of applications, therefore, a tractable CRB for this problem constitutes a key tool of broad interest. The new CRB is obtained for the standard narrowband signal model, where the Doppler effect on the band‐limited baseband signal is not considered and amounts to a frequency shift. The CRB is expressed in terms of the signal samples, making it especially easy to use irrespective of the considered baseband signal such that the actual sample values are used. Leveraging recent results on the CRB for a mixture of real‐ and integer‐valued parameter vectors [48], summarised in Section 6 for completeness, we exploit both CRBs to properly characterize the ultimate GNSS single point positioning (SPP) and RTK performance. To the best of our knowledge, this is the first time, these positioning techniques are theoretically characterised from the baseband signal model in terms of the CRB and CMLE.…”

Robust Filtering Techniques for RTK Positioning in Harsh Propagation Environments

“…Thus, real-to-integer mapping is the process that assigns the float ambiguity vector to an integer one , such that . While different estimators have been proposed, the integer least squares (ILS) provides the optimal performance [ 34 , 35 , 36 ]. Integer estimators that include a validation step belong to the Integer Aperture (IA) framework [ 37 ].…”

Doppler-aided positioning in GNSS receivers - A performance analysis

“…Observing (8), it is noteworthy that the Doppler (or delay drift) depends on the velocity, but also on the position through this angular velocity vector. Hence, Dopplers bring a direct piece of information on the user position.…”

“…In this formulation, we do not take into account the carrier phase information mentioned right above, mainly because this leads to very specific solutions which do not apply for standard standalone receivers. Nevertheless, the Cramér-Rao Bound (CRB) for a mixture of real and integer-valued parameters, and its use for carrier phase-based positioning techniques performance characterization, has been derived in [8].…”