  We investigate the issue of post-selection inference for a fixed and a mixed
parameter in a linear mixed model using a conditional Akaike information
criterion as a model selection procedure. Within the framework of linear mixed
models we develop complete theory to construct confidence intervals for
regression and mixed parameters under three frameworks: nested and general
model sets as well as misspecified models. Our theoretical analysis is
accompanied by a simulation experiment and a post-selection examination on mean
income across Galicia's counties. Our numerical studies confirm a good
performance of our new procedure. Moreover, they reveal a startling robustness
to the model misspecification of a naive method to construct the confidence
intervals for a mixed parameter which is in contrast to our findings for the
fixed parameters.
