On the use of generalized inverse matrices in sampling theory

In theory, it is customary to define general regression estimators in terms of full-rank weighting models (i.e., the design matrix that corresponds to the weighting model is of full rank). For such weighting models, it is well known that the general regression weights reproduce the (known) population totals of the auxiliary variables involved. In practice, however, the weighting model often is not of full rank, especially when the weighting model is for incomplete post-stratification. By means of the theory of generalized inverse matrices, it is shown under which circumstances this consistency property remains valid. In this paper,, we discuss the non-trivial example of consistent weighting between persons and households as proposed by Lemaître and Dufour (1987). We then show how the theory is implemented in Bascula.