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arXiv · 1411.1467

Minimax Estimation of Discrete Distributions under $\ell_1$ Loss

Yanjun Han·Jiantao Jiao·Tsachy Weissman
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Abstract

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.

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Yanjun Han, Jiantao Jiao, Tsachy Weissman. 2014-11-06. Minimax Estimation of Discrete Distributions under $\ell_1$ Loss. https://arxiv.org/abs/1411.1467

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