Estimation problems with constrained parameter spaces arise in various settings. In many of these problems, the observations available to the statistician can be modelled as arising from the noisy realization of the image of a random linear operator; an important special case is random design regression. We derive sharp rates of estimation for arbitrary compact elliptical parameter sets and demonstrate how they depend on the distribution of the random linear operator. Our main result is a functional that characterizes the minimax rate of estimation in terms of the noise level, the law of the random operator, and elliptical norms that define the error metric and the parameter space. This nonasymptotic result is sharp up to an explicit universal constant, and it becomes asymptotically exact as the radius of the parameter space is allowed to grow. We demonstrate the generality of the result by applying it to both parametric and nonparametric regression problems, including those involving distribution shift or dependent covariates.

翻译：具有约束参数空间的估计问题出现在各种场景中。在许多此类问题中，统计学家可获得的观测数据可建模为由随机线性算子像的含噪实现产生；一个重要的特例是随机设计回归。我们为任意紧椭球参数集推导了尖锐的估计速率，并展示了这些速率如何依赖于随机线性算子的分布。我们的主要结果是一个泛函，它根据噪声水平、随机算子的律以及定义误差度量和参数空间的椭球范数来刻画极小极大估计速率。这一非渐近结果精确至显式通用常数范围内，并随着参数空间半径允许增长而渐近精确。我们通过将该结果应用于参数和非参数回归问题（包括涉及分布偏移或相依协变量的问题）来展示其普适性。