纠缠单样本分布中位置参数的估计
Estimating location parameters in entangled single-sample distributions
摘要：
我们考虑独立采样数据的共同均值的估计问题，其中样本是从具有共同均值的对称单峰分布中以可能不相同的方式抽取的。这概括了高斯混合建模的设置，因为不同混合分量的数量可能会随着观测的数量而不同。我们提出了一个估计，适应数据的异质性水平，实现近最优的i.i.d.设置和一些异构设置，其中"低噪声“点的分数小到$\frac{\log n}{n}$。我们的估计是一个混合的模态区间，短，和中位数估计从经典统计，然而，关键的技术贡献依赖于新的经验过程理论的结果，我们得出独立的，但非独立同分布。数据在多元设置中，我们推广我们的理论，径向对称分布的混合物的平均估计，并推导出极小极大下界的预期误差的任何估计是不可知的个别数据点的规模。最后，我们描述了我们的估计适用于线性回归的扩展。在多元均值估计和回归设置中，我们提出了计算上可行的版本，我们的估计运行在时间多项式的数据点的数量。
英文摘要：
We consider the problem of estimating the common mean of independently sampled data, where samples are drawn in a possibly non-identical manner from symmetric, unimodal distributions with a common mean. This generalizes the setting of Gaussian mixture modeling, since the number of distinct mixture components may diverge with the number of observations. We propose an estimator that adapts to the level of heterogeneity in the data, achieving near-optimality in both the i.i.d. setting and some heterogeneous settings, where the fraction of ``low-noise'' points is as small as $\frac{\log n}{n}$. Our estimator is a hybrid of the modal interval, shorth, and median estimators from classical statistics; however, the key technical contributions rely on novel empirical process theory results that we derive for independent but non-i.i.d. data. In the multivariate setting, we generalize our theory to mean estimation for mixtures of radially symmetric distributions, and derive minimax lower bounds on the expected error of any estimator that is agnostic to the scales of individual data points. Finally, we describe an extension of our estimators applicable to linear regression. In the multivariate mean estimation and regression settings, we present computationally feasible versions of our estimators that run in time polynomial in the number of data points.
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