The “Maximum Likelihood Solution” of the Problem of Estimating a Linear Functional Relationship

Summary The problem considered is that of estimating a functional relationship (assumed linear) between two unobservable mathematical quantities ξ and η from pairs of observations on these quantities when both sets of observations are subject to normal independent errors. The problem has attracted much interest over a long period and one reads repeatedly that the method of maximum likelihood yields unsatisfactory estimators in this case. In this note we show that in fact no maximum-likelihood solution exists for this problem, which ceases then to provide an example of unsatisfactory maximum-likelihood estimators and merely joins the queue of problems in which the likelihood approach fails to produce estimators. The distinction, although academic, is worth making.