Chapter 44 Testing non-nested hypotheses

The comparison of different hypotheses, i.e. of competing models, is the basis of model specification. It may be performed along two main lines. The first one consists in associating with each model a loss function and in retaining the specification implying the smallest (estimated) loss. In practice, the loss function is defined either by updating some a priori knowledge on the models given the available observations (the Bayesian point of view), or by introducing some criterion taking into account the trade-off between the goodness of fit and the complexity of the model. The second approach is hypothesis testing theory. However, the determination of the decision rule is not done on the same basis as model choice. The basis of hypothesis testing theory is to introduce the probability of errors. This chapter focuses on the case where none of the hypotheses is a particular case of another one.