The ECD-BIE Estimation Theorem in Spectral Form

Abstract The theory of best integer equivariant (BIE) estimation provides the minimum mean squared error estimator within the class of integer equivariant (IE) estimators and has found increasing application in mixed-integer models such as those arising in carrier-phase GNSS positioning. Recent developments extended BIE estimation to the broader class of elliptically contoured distributions (ECDs), thereby allowing the use of heavier-tailed models. In this contribution, we further develop this theory by deriving a spectral form of the ECD-BIE theorem. Building on the periodic representation of IE-estimators, whereby each estimator can be expressed as a correction to the best linear unbiased estimator (BLUE) that is integer-periodic in the ambiguity domain, we show how this spatial representation can be transformed into a Fourier-based spectral form. This establishes twin representations of ECD-BIE estimators in both spatial and spectral domains. The relative merits of these formulations are discussed and illustrated for the normal, contaminated normal, and multivariate Student distributions.