On the Gauss–Helmert model with a singular dispersion matrix where BQBQ is of smaller rank than BB

The case of a singular dispersion matrix within the Gauss–Helmert Model has been considered before, usually assuming a sufficiently small rank deficiency in order to guarantee a unique solution for both the residual vector as well as the estimated parameter vector of type Best Linear Uniformly Unbiased Estimate (BLUUE). In this contribution the emphasis is shifted towards establishing necessary and sufficient conditions for a unique residual vector, along with a unique estimate of type Best Linear Uniformly Minimum Bias Estimate (BLUMBE) for the parameter vector. Should uniformly unbiased estimates exist, the BLUMBE obviously becomes the BLUUE.