Consistency of least squares estimates in a system of linear correlation models
We consider a system of linear response models with random explanatory variables in which the global matrix parameter is subject to arbitrary constraints. A generalized least squares estimate (GLSE) of the global parameter is defined by its property of minimizing some norm of the global residual over an affine manifold, called the support, containing the global parameter range. The crucial relation is the one between the true global parameter value and the so-called global mean square (msq) regression parameter value defined by the notion of msq regression following Cramér (1945). We prove that as soon as the support manifold is given, it is contained in another affine manifold, called the region of convergence, so that as the global sample size tends to infinity the GLSE converges or diverges a.s. according as the global msq regression parameter value belongs to this region or not; the a.s. limit of the GLSE is just the orthogonal projection of the global msq regression parameter value on the support manifold. The proof is based on the large sample a.s. uniform boundedness of a random linear operator which appears in the expression of the GLSE error. The uniformity of the convergence and of the consistency when the support manifold varies is also established.