The size dependence of exchange bias field HE and coercivity Hc was studied by measuring exchange biased Fe-FeF2 dot arrays in comparison with an unstructured exchange biased Fe-FeF2 bilayer. The domain sizes in the ferromagnet (FM) and the antiferromagnet (AFM) play an important role for exchange bias (EB), and thus interesting phenomena may be expected when the size of an EB system becomes comparable to these sizes. We observe drastic changes of HE and Hc in nanostructured Fe-FeF2, which are unexpected because they appear even at a structure size which is too large for matching with AFM or FM domain size to play a role. We propose that under certain conditions the hysteresis loop is affected differently in the two branches of the reversal by shape anisotropy due to patterning. This is possible because the EB induces a reversal asymmetry already in the unpatterned bilayer system.