The Performance of Statistical Inference after Model Checking

Most standard statistical inference procedures rely on model assumptions such as normality, independent and identically distributed and the like. Often in practice, such assumptions are formally tested before applying the inference. Such a procedure does not ensure that the model assumptions are really fulfilled because the standard theory for popular inference tests does not take into account that the data has been selected by a previous model check. Applying a misspecification test violates the very model assumption it was meant to enforce. (``misspecification paradox''). In practice it is useful to have an alternative test in the case that the misspecification test rejects the model assumption. However, this does not completely address the misspecification paradox because there is still a certain probability that the model assumption is rejected when it is in fact true, and vice versa. This thesis is about investigating, theoretically and by simulations, the performance of such a combined procedure. A novel simulation process is proposed where samples can be randomly chosen from a situation where the model assumption is fulfilled or violated. A few combinations of distributions and statistical tests are considered and both level and power are presented and discussed. Although the levels show no strong evidence of choosing the combined procedure over the tests run without model checking, the power plots show that in certain conditions, it can be more powerful. A theory is presented where it is shown that in a particular situation and with reasonable assumptions, the combined procedure does have a higher power compared to unconditional tests. The assumptions were relaxed a little and the same conclusions could be made. Finally, a three stage testing procedure in two different scenarios, distributional shape and linear regression significance, are presented and discussed. The same conclusions can be made from the levels and powers