This paper deals with the general possibly singular linear model. It is assumed that in addition to the sample information we have some nonstochastic prior information concerning the unknown regression coefficients that can be expressed in form of linear independent inequality constraints. Since these constraints are part and parcel of the model, an inequality constrained generalized least squares (ICGLS) problem arises with some hitherto unknown aspects. Based on a projector theoretical approach, we show how the set of ICGLS selections under the constrained model is related to the set of GLS selections under the associated unconstrained model. As a by-product we obtain an interesting method for determining an ICGLS selection from a GLS selection. The insights gained from our considerations might also be useful in a future study of the statistical properties of ICGLS estimators. Certain special model cases are also considered. Some results discussed by Werner and by Firoozi are reobtained.
Werner, H. J., & Yapar, C. (1996). On inequality constrained generalized least squares selections in the general possibly singular Gauss-Markov model: A projector theoretical approach. Linear Algebra and Its Applications, 237–238, 359–393. https://doi.org/10.1016/0024-3795(94)00357-2