Fast celebrity on holiday, missing California (10)

I believe the answer is:

starvation

'fast' is the definition.
(to fast is to starve oneself)

'celebrity on holiday missing california' is the wordplay.
'celebrity on' becomes 'star' (I can't explain this - if you can you should believe this answer much more).
'holiday' becomes 'vacation' (synonyms).
'missing' is a deletion indicator.
'california' becomes 'ca'.
'vacation' with 'ca' taken away is 'vation'.
'star'+'vation'='STARVATION'

(Another definition for starvation that I've seen is " Desperate need".)

I've seen this clue in The Times.

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