The most common and most likely answer to this crossword clue is the 18 letter word TOOÂ CLEVERÂ BYÂ HALF
I've seen this crossword clue in The Guardian
T
O
O
Â
C
L
E
V
E
R
Â
B
Y
Â
H
A
L
F
Buddy explains!
The straight part
We know that "Arrogant" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case arrogant = too clever by half.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.