Misspecification and inference : A review and case studies

Sammanfattning: When conducting inference we usually have to make some assumptions. A common assumption is that the parametric model which describes the behavior of the investigated random phenomena is correctly specied. If not, some of the inferential methods does not provide valid inference, e.g. the method of maximum likelihood. This thesis investigates and presents some of the results regarding misspecied parametric models to illustrate the consequences of misspecication and how the parameter estimates are affected. The main question investigated is wether we still can learn something about the true parameter even though the model is misspecied. An important result is that the quasi-maximum likelihood estimate of a misspecied estimation model converges almost surely towards the parameter minimizing the distance between the true model and the estimation model. Using simulations, it is illustrated how this estimator in certain situations converges almost surely towards the true parameter times a scalar. This result also seems to hold for a situation not covered by any theorems. Furthermore, a general class of estimators called M-estimators is presented for the theoretic framework of misspecied models. An example is given when the theory of M-estimators come to use under model misspecication.