Voronoi cells, probabilistic bounds, and hypothesis testing in mixed integer linear models

Although real-valued linear models, whether or not of full rank, have been thoroughly investigated and are well documented, very little is known about statistical and probabilistic aspects of a mixed integer linear model, which arose from space geodesy and serves as the standard starting model for precise positioning using the global positioning system (GPS). Voronoi cells play a fundamental role in the least squares estimation of the integer unknowns of the model. In this paper, we first develop a method to construct Voronoi cells and study how to fit figures of simple shape to a Voronoi cell, both from inside and outside. We then derive a number of new lower and upper bounds on the probability that the integers of the model are correctly estimated. Finally, we discuss the tests of two hypotheses on the integer mean.