Foundations

Abstract

In every experimental situation the outcome of an experiment depends on a number of factors of influence lie temperature, pressure, different treatments or varieties. This dependence can be described by a functional relationship, the response function μ, which quantifies the effect of the particular experimental condition t = (t₁,…, t _K ) The observation X of an experiment is subject to a random error Z. Hence, an experimental situation will be formalized by the relationship X(t) = μ(t) + Z(t). As the response function μ describes the mean outcome of the experiment the observation has to be centered and it is natural to require that E(X(t)) = μ(t) or, equivalently, that the error Z has zero expectation, E(Z(t)) = 0. The distribution of the random error may depend on the experimental conditions. In case of complete ignorance on the structure of the response function it will be impossible to make any inference. Therefore, we will consider the rather general situation in which the response is described by a linear model in which the response function μ can be finitely parametrized in a linear way as introduced in Subsection 1.1. The performance of the statistical inference depends on the experimental conditions for the different observations, and it is one of the challenging tasks to design an experiment in such a way that the outcome is most reliable. This concept will be explained in Subsection 1.2.