This appendix discusses the case where models do not fit data perfectly well and we nevertheless try to get by as well as we can (Gilboa 2009 §7.1). It is the case almost exclusively met in descriptive applications. We will use a simple least-squares criterion to fit data.

Nonparametric measurements and parametric fittings for imperfect models: general discussion

This section presents a general discussion with methodological considerations. It can be skipped by readers who only want to use techniques for fitting data. In most descriptive applications, the model we use does not describe the empirical reality perfectly well and the preference conditions of our model are violated to some extent. One reason is that there usually is randomness and noise in the data. Another reason is that there may be systematic deviations. We then nevertheless continue to use our model if no more realistic and tractable model is available. We then try to determine the parameters of our model that best fit the data, for instance by minimizing a distance, such as a sum of squared differences, between the predictions of the model and the actual data. Alternatively, we may add a probabilistic error theory to the deterministic decision model (called the core model) and determine the parameters that maximize the likelihood of the data.