论文标题
圆形卷积的最小值测试和二次功能估计
Minimax testing and quadratic functional estimation for circular convolution
论文作者
论文摘要
在圆形卷积模型中，我们的目标是使用被加性测量误差污染的观测值推断圆形随机变量的密度。我们强调了这两个问题的相互作用：最佳测试和二次功能估计。在一般规律性的假设下，我们确定了二次功能估计的最小值风险的上限。上边界由两个术语组成，一个术语模拟经典的偏见变化权衡，第二个术语在二次功能估计中引起典型的肘部效应。使用二次函数的最小值估计器作为测试统计量，我们为非参数替代方案的非扰动最小值半径提供了上限。有趣的是，在估计情况下导致肘部效应的一词在测试半径中消失。我们为测试问题提供了匹配的下限。通过证明测试问题的任何下限也为二次功能估计问题产生下限，我们获得了估计风险的下限。最后，我们证明了该术语的匹配下限，从而导致估计问题中的肘部效应。考虑到Sobolev空间以及普通或超级平滑误差密度，结果说明了结果。
In a circular convolution model, we aim to infer on the density of a circular random variable using observations contaminated by an additive measurement error. We highlight the interplay of the two problems: optimal testing and quadratic functional estimation. Under general regularity assumptions, we determine an upper bound for the minimax risk of estimation for the quadratic functional. The upper bound consists of two terms, one that mimics a classical bias-variance trade-off and a second that causes the typical elbow effect in quadratic functional estimation. Using a minimax optimal estimator of the quadratic functional as a test statistic, we derive an upper bound for the nonasymptotic minimax radius of testing for nonparametric alternatives. Interestingly, the term causing the elbow effect in the estimation case vanishes in the radius of testing. We provide a matching lower bound for the testing problem. By showing that any lower bound for the testing problem also yields a lower bound for the quadratic functional estimation problem, we obtain a lower bound for the risk of estimation. Lastly, we prove a matching lower bound for the term causing the elbow effect in the estimation problem. The results are illustrated considering Sobolev spaces and ordinary or super smooth error densities.
