Best integer equivariant estimation for elliptically contoured distributions
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ORIGINAL ARTICLE
Best integer equivariant estimation for elliptically contoured distributions P. J. G. Teunissen1,2,3 Received: 18 March 2020 / Accepted: 27 July 2020 © The Author(s) 2020
Abstract This contribution extends the theory of integer equivariant estimation (Teunissen in J Geodesy 77:402–410, 2003) by developing the principle of best integer equivariant (BIE) estimation for the class of elliptically contoured distributions. The presented theory provides new minimum mean squared error solutions to the problem of GNSS carrier-phase ambiguity resolution for a wide range of distributions. The associated BIE estimators are universally optimal in the sense that they have an accuracy which is never poorer than that of any integer estimator and any linear unbiased estimator. Next to the BIE estimator for the multivariate normal distribution, special attention is given to the BIE estimators for the contaminated normal and the multivariate t-distribution, both of which have heavier tails than the normal. Their computational formulae are presented and discussed in relation to that of the normal distribution. Keywords Global navigation satellite system (GNSS ) · Integer equivariant (IE) estimation · Best integer equivariant (BIE) · Best linear unbiased estimation (BLUE) · Elliptically contoured distribution (ECD) · Multivariate normal · Contaminated normal · Multivariate t-distribution · LAMBDA method
1 Introduction This contribution extends the theory of integer equivariant (IE) estimation as introduced in Teunissen (2003). The theory of IE estimation provides a solution to the problem of carrier-phase ambiguity resolution, which is key to highprecision GNSS positioning and navigation. It is well known that the so-called fixed GNSS baseline estimator is superior to its ‘float’ counterpart if the integer ambiguity success rate is sufficiently close to its maximum value of one. Although this is a strong result, the necessary condition on the success rate does not make it hold for all measurement scenarios. This restriction was the motivation to search for a class of estimators that could provide a universally optimal estimator while still benefiting from the integerness of the carrier-phase
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P. J. G. Teunissen [email protected]
1
Department of Geoscience and Remote Sensing, Delft University of Technology, Delft, The Netherlands
2
GNSS Research Centre, Curtin University of Technology, Perth, Australia
3
Department of Infrastructure Engineering, The University of Melbourne, Melbourne, Australia
ambiguities. The solution was found in the class of IE estimators, with its best integer equivariant (BIE) estimator being best in this class in the mean squared error sense (Teunissen 2003). The class of IE estimators obeys the integer remove– restore principle and was shown to be larger than the class of integer (I) estimators as well as larger than the class of linear unbiased (LU) estimators. As a consequence, the BIE estimator has the universal property that its mean squared error (MSE) is never larger
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