PhD project
The performance of statistical inference after model checking

Most standard statistical inference procedures rely on model assumptions 
such as normality, i.i.d. and the like. Often, such assumptions are formally 
tested before applying the inference. An example is to apply a normality 
test (such as Kolmogorov-Smirnov) before carrying out a two-sample t-test, 
which then is only applied if normality is not rejected (one may apply an 
alternative test such as the Wilcoxon test if normality is rejected). 

Unfortunately, such a combined procedure doesn't ensure that the model
assumptions are really fulfilled when applying the t-test, because the 
standard theory for the t-test does not take into account that data have 
been selected by a previous model check, which technically can be shown to 
violate the model assumption that it was meant to enforce ("goodness of fit
paradox", Hennig 2007). It may happen with a certain probability that the
normality assumption is rejected for proper normal data, and also that the
normality assumption is not rejected if it is in fact wrong.

This project is about investigating, theoretically or by simulations, what the
performance of such a combined procedure actually is. Apart from the
normality test/t-test combination, other such combinations can be investigated,
too, such as the use of the runs test of independence before applying inference
that assumes independence.

C. Hennig: Falsification of propensity models by statistical tests and the goodness-of-fit paradox. Philosophia Mathematica 15 (2007), 166-192 .