Strong consistency of least squares estimates in polygonal regression with random explanatory variables (Q1376673)

Regression models with random explanatory variables arise in some important practical situations involving bipartite observations \((X(t), Y(t))\) on \(n\) items. Of particular interest are polygonal regression models \[ Y'(t)= \sum_{i=1}^k b_i'(X(t)) q_iI_{S(i)} (X(t))+ e'(t),\tag{1} \] where a prime denotes transpose, \(Y(t)\) are \(r\times 1\) response observation vectors, \(e(t)\) residuals, \(b_i(.)\)'s are known \(l(i)\times 1\) vector-valued functions on the common range space \(H\) of \(X(t)\), \(q_i\) unknown \(l(i)\times r\) matrix parameters, and \(S(i)\) specify disjoint sets in \(H\), with \(I_{S(i)}\) the indicator of \(S(i)\). Constraints may be imposed on the parameters \(q_i\) in order to ensure the smoothness to some order of the regression. Models like these arise when the unknown regression function is approximated by linear parametric functions in every domain \(S(i)\) and also when the response variable follows a mixture of conditional distributions with linear conditional mean structure given the explanatory variable in several domains \(S(i)\). The purpose of this paper is to establish the strong consistency of generalized least squares estimators (GLSE). The moment restriction needed is necessary and sufficient for the GLSE consistency uniform with respect to the projection support and especially for the GLSE consistency in the extreme case where the projection support has the maximal dimension.