Optimum Allocation in Linear Regression Theory

Abstract

If for the estimation of $\beta_1, \beta_2$ different observations (``sources'') of form (1.1) are potentially available, each of them being repeatable as many times as we please, the question arises which of them the experimenter should utilize, and in what proportions. With appropriate optimality conventions the answer is the following. For the estimation of a single quantity of form $\theta = \alpha_1\beta_1 + \alpha_2\beta_2$ the optimum allocation comprises two sources only; for the estimation of both parameters, the corresponding number is two or three; the best proportions are indicated in Sections 2 and 4 below. Generalizations to more than two parameters and to observations at different costs are briefly discussed. The problem is related to Hotelling's weighing problem [2] and to the topics treated by David and Neyman in [1].