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Minimum mean squared error (MSE) adjustment and the optimal Tykhonov–Phillips regularization parameter via reproducing best invariant quadratic uniformly unbiased estimates (repro-BIQUUE)

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Abstract

In a linear Gauss–Markov model, the parameter estimates from BLUUE (Best Linear Uniformly Unbiased Estimate) are not robust against possible outliers in the observations. Moreover, by giving up the unbiasedness constraint, the mean squared error (MSE) risk may be further reduced, in particular when the problem is ill-posed. In this paper, the α-weighted S-homBLE (Best homogeneously Linear Estimate) is derived via formulas originally used for variance component estimation on the basis of the repro-BIQUUE (reproducing Best Invariant Quadratic Uniformly Unbiased Estimate) principle in a model with stochastic prior information. In the present model, however, such prior information is not included, which allows the comparison of the stochastic approach (α-weighted S-homBLE) with the well-established algebraic approach of Tykhonov–Phillips regularization, also known as R-HAPS (Hybrid APproximation Solution), whenever the inverse of the “substitute matrix” S exists and is chosen as the R matrix that defines the relative impact of the regularizing term on the final result.

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Authors and Affiliations

  1. School of Earth Sciences, The Ohio State University, Columbus, OH, 43210, USA

    Burkhard Schaffrin

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  1. Burkhard Schaffrin
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Correspondence to Burkhard Schaffrin.

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The delay in publishing this paper is due to a number of unfortunate complications. It was first submitted as a multi-author paper in two parts. Due to some miscommunication among the original authors, it was reassigned to one of the J Geod special issues, but later reassigned at this author’s request to a standard issue of J Geod. This compounded with a difficulty to find willing reviewers to slow the process. We apologize to the author.

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Schaffrin, B. Minimum mean squared error (MSE) adjustment and the optimal Tykhonov–Phillips regularization parameter via reproducing best invariant quadratic uniformly unbiased estimates (repro-BIQUUE). J Geod 82, 113–121 (2008). https://doi.org/10.1007/s00190-007-0162-0

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