Cambridge eccentric losing male budgerigar, for example (4,4)
I believe the answer is:
cage bird
'cambridge eccentric losing' is the definition.
I don't know anything about this answer so I can't tell whether it can be defined by this definition.
'male budgerigar for example' is the wordplay.
'male' becomes 'cage' (I can't justify this - if you can you should give a lot more credence to this answer).
'budgerigar for example' becomes 'bird' (budgerigar is a kind of bird).
'cage'+'bird'='CAGE-BIRD'
(Other definitions for cage bird that I've seen before include "One tweeting often shut up" , "Canary, say" .)