Regression estimation for a bounded response over a bounded region

The usual multiple linear regression model of a response variable on p explanatory variables is considered. It is pointed out that two additional assumptions are often appropriate: That the model is valid only when the explanatory variables lie in some bounded region of Euclidean p-space; and that the expected response is bounded (possibly just from one side). The maximum likelihood estimator (MLE) of the vector of regression coefficients is derived under the assumption that the region of X values is a centered ellipse; this includes both extrapolation and interpolation problems. The MLE is a type of ridge estimator when the region corresponds to extrapolation in a multicollinear problem. The implications of this result for ridge estimation are discussed.