Attainability of two-point testing rates for finite-sample location estimation
研究了Le Cam两点检验下界在有限样本位置估计中能否达到。设计了一个运行时间接近线性且无需调参的算法，对对称对数凹混合分布实现该下界，并证明对称单峰分布下不可实现。适合理论统计学者判断是否精读。
Le Cam’s two-point testing method yields perhaps the simplest lower bound for estimating the mean of a distribution: roughly, if it is impossible to well distinguish a distribution centered at μ from the same distribution centered at μ+Δ, then it is impossible to estimate the mean by better than Δ/2. It is setting-dependent, whether or not a nearly matching upper bound is attainable. We study the conditions under which the two-point testing lower bound can be attained for univariate mean estimation; both in the setting of location estimation (where the distribution is known up to translation) and adaptive location estimation (unknown distribution). Roughly, we will say an estimate nearly attains the two-point testing lower bound if it incurs error that is at most polylogarithmically larger than the Hellinger modulus of continuity for Ω˜(n) samples. Adaptive location estimation is particularly interesting, as some distributions admit much better guarantees than sub-Gaussian rates (e.g., Unif(μ−1,μ+1) permit error Θ(1n), while the sub-Gaussian rate is Θ(1n)), yet it is not obvious whether these rates may be adaptively attained by one unified approach. Our main result designs an algorithm that nearly attains the two-point testing rate for mixtures of symmetric, log-concave distributions with a common mean. Moreover, this algorithm runs in near-linear time and is parameter-free. In contrast, we show the two-point testing rate is not nearly attainable, even for symmetric, unimodal distributions. We complement this with results for location estimation, showing the two-point testing rate is nearly attainable for unimodal distributions but unattainable for symmetric distributions.