The multinomial minimax risk for uniformity testing against $l_p$ alternatives converges exactly to $2Phi(-u^*/2)$ in the intermediate regime, proven via a conditional central limit theorem.
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Sharp Lower Bound on the Minimax Risk for Multinomial Uniformity Testing via a Conditional Central Limit Theorem
The multinomial minimax risk for uniformity testing against $l_p$ alternatives converges exactly to $2Phi(-u^*/2)$ in the intermediate regime, proven via a conditional central limit theorem.