Application of a Measure of Information to the Design and Comparison of Regression Experiments

A normal regression experiment can be represented by {equation*}{1.1} Y_i = ∑ⱼ₌₁^k Xᵢⱼ θ_j + η_i (i = 1, ⋯, n){equation*} where \ηᵢ/i = 1, ⋯, n\ is a set of normally distributed random variables with zero means and non-singular dispersion matrix C, θ = (θ₁, ⋯, θₖ) is the parameter-vector of interest and X = (Xᵢⱼ) is a known n × k matrix which will be called the allocation matrix. The rows of X will be called the allocation vectors. We denote the experiment by ε(X, C). We assume that C is known; generally it will be a function of $X, C(X)$. The particular realisation of Y will be denoted y. The matrix F = X'C⁻¹X is the Fisher-information-matrix of ε(X, C). When F is non-singular, one answer to the question "What information does y give about θ?" is to quote F⁻¹, the dispersion matrix of the maximum-likelihood-estimates of θ. A strong argument in favour of this is that F⁻¹ is independent of both θ and y. The fact that it is independent of θ means that the answer is not "local"; the fact that it is independent of y leads to simplicity. This approach is taken by Box and Hunter [1] in their work on rotatable designs. However, we must accept the fact that many experimenters wish to have a one-dimensional answer to the question i.e. we must associate with ε(X, C) a single number which we call the "information". For instance Elfving [5] has developed the use of trace F⁻¹. In this paper we adopt the measure of information introduced by Lindley [7]. In Section 2 we generalise Lindley's treatment of the regression situation to include the singular case, explain the uses of the measure and compare it with that of Elfving. Section 3 deals with the analogue of Elfving's main theorem. Theorems 4.1 and 4.2 of Section 4 provide links with the traditional variance approach. In Section 5 we derive the asymptotic form of the measure as the n of (1.1) increases and show that this form can be derived also from Neyman-Pearsonian theory. In Section 6 the influence of nuisance parameters is discussed and an analogue of a theorem of Chernoff [2] is established.