  This work addresses the problem of conducting valid inference for additive
and linear mixed models after model selection. One possible solution to
overcome overconfident inference results after model selection is selective
inference, which constitutes a post-selection inference framework, yielding
valid inference statements by conditioning on the selection event. We extend
recent work on selective inference to the class of additive and linear mixed
models for any type of model selection mechanism that can be expressed as a
function of the outcome variable (and potentially on covariates on which it
conditions). We investigate the properties of our proposal in simulation
studies and apply the framework to a data set in monetary economics. Due to the
generality of our proposed approach, the presented approach also works for
non-standard selection procedures, which we demonstrate in our application.
Here, the final additive mixed model is selected using a hierarchical selection
procedure, which is based on the conditional Akaike information criterion and
involves varying data set sizes.
