We analyze the problem of discrete distribution estimation under $\ell_1$
loss. We provide non-asymptotic upper and lower bounds on the maximum risk of
the empirical distribution (the maximum likelihood estimator), and the minimax
risk in regimes where the alphabet size $S$ may grow with the number of
observations $n$. We show that among distributions with bounded entropy $H$,
the asymptotic maximum risk for the empirical distribution is $2H/\ln n$, while
the asymptotic minimax risk is $H/\ln n$. Moreover, Moreover, we show that a
hard-thresholding estimator oblivious to the unknown upper bound $H$, is
asymptotically minimax. However, if we constrain the estimates to lie in the
simplex of probability distributions, then the asymptotic minimax risk is again
$2H/\ln n$. We draw connections between our work and the literature on density
estimation, entropy estimation, total variation distance ($\ell_1$ divergence)
estimation, joint distribution estimation in stochastic processes, normal mean
estimation, and adaptive estimation.

该研究分析了离散分布估计问题，并提供了最大风险和最小极小风险的上下界，进而得出在特定条件下最大风险极小风险的渐近性能。通过该研究可得出在经验分布估计中的渐近最大风险和最小极小风险，并且通过对概率分量估计确定了渐近最小极小风险。