Confidence Region for a Linear Relation

Abstract

The analysis of a linear relation is considered, when there are replications and all the variables involved are subject to errors or fluctuations. A test based on the $F$ distribution is derived for testing the hypothesis that the unknown relation is a given linear relation. From this test a joint confidence region for the coefficients of the linear relation is derived. A confidence region for the linear relation is then defined as the set of all points which belong to hyperplanes not rejected by the test. The corresponding confidence coefficient is not known exactly, but it is known to be greater than a previously chosen $P$. In the non-degenerate case, the confidence region is a hyperboloid centered at the centroid of the given points, and it has the property that a hyperplane is not rejected by the test if and only if it is entirely contained in it. This confidence region estimation procedure is compatible with the maximum likelihood estimation of a linear relation, in the sense that the maximum likelihood hyperplane is contained in the confidence region for the linear relation, if this region is not empty.