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.
Highlights
This contribution extends the theory of integer equivariant (IE) estimation as introduced in Teunissen (2003)
The best integer equivariant (BIE) estimator has the universal property that its mean squared error (MSE) is never larger than that of any I estimator or any LU estimator. This shows that from the MSE perspective one should always prefer the use of the BIE baseline over that of the integer least-squares (ILS) baseline and best linear unbiased (BLU) baseline
If bis an arbitrary I estimator and ban arbitrary LU estimator, the best integer equivariant (BIE) estimator bBIE, which is optimal in the minimum mean squared error sense, will have a mean squared error (MSE) that satisfies
Summary
This contribution extends the theory of integer equivariant (IE) estimation as introduced in Teunissen (2003). 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. 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. In Brack et al (2014), a sequential approach to best integer equivariant estimation was developed and tested, while Odolinski and Teunissen (2020) analyzed the normal distribution-based
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