After transfer of websites from one Plesk server to another, Webalizer statistics stopped working.
This problem may occur If source server's version of Webalizer is newer than the one on destination server. In such case, older version may fail to read Webalizer data files, which contain traffic information for current month.
webalizer -V on both source and destination servers and compare versions. If destination server has older version of Webalizer, then this article's resolution applies.
Upgrade Webalizer on destination server to the same version as on source server or newer.