论文标题
具有方向依赖精度的多元平均估计
Multivariate mean estimation with direction-dependent accuracy
论文作者
论文摘要
我们考虑基于$ n $独立的，相同分布的观察值估算随机向量的平均值的问题。我们证明了一个估计器的存在，该估计器在各个方向上都有近乎最佳的错误，其中随机向量的一个维度边际的差异并不小：概率$ 1-δ$，该过程返回$ \whμ_n$，可以满足每个方向的$ \ u \ in s^{d-1} $ u \ in} $ u \ u \ n} $ u \ n- r \ [ \ frac {c} {\ sqrt {n}} \ left（σ（U） \ var（\ inr {x，u}）$和$ c $是常数。为了实现这一目标，我们只需要以一定的力矩等效性假设的形式比协方差矩阵的存在略多。 证明依赖于新颖的界限，因为经验和真实概率的比率在某些类别的随机变量上均匀地保持。
We consider the problem of estimating the mean of a random vector based on $N$ independent, identically distributed observations. We prove the existence of an estimator that has a near-optimal error in all directions in which the variance of the one dimensional marginal of the random vector is not too small: with probability $1-δ$, the procedure returns $\whμ_N$ which satisfies that for every direction $u \in S^{d-1}$, \[ \inr{\whμ_N - μ, u}\le \frac{C}{\sqrt{N}} \left( σ(u)\sqrt{\log(1/δ)} + \left(\E\|X-\EXP X\|_2^2\right)^{1/2} \right)~, \] where $σ^2(u) = \var(\inr{X,u})$ and $C$ is a constant. To achieve this, we require only slightly more than the existence of the covariance matrix, in the form of a certain moment-equivalence assumption. The proof relies on novel bounds for the ratio of empirical and true probabilities that hold uniformly over certain classes of random variables.
