We propose a coupled bootstrap estimator for the test error of an arbitrary algorithm that estimates the mean in a Poisson sequence, often called the Poisson means problem. The idea behind our method is to generate two carefully-designed data vectors from the original data vector, by using synthetic binomial noise. One such vector acts as the training sample and the second acts as the test sample. To stabilize the test error estimate, we average this over multiple draws of the synthetic noise. A key property of our coupled bootstrap estimator is that it is unbiased for the test error in a problem where the original mean has been shrunken by a small factor, driven by the success probability $p$ in the binomial noise. Further, in the limit as $p \to 0$, we show that the proposed estimator recovers a known unbiased estimator for the test error, under no assumptions on the algorithm at hand (in particular, no smoothness assumptions). Our methodology applies to two central loss functions that can be used to a test error metric: Poisson deviance and squared loss. Through a bias-variance decomposition, for each loss function, we analyze the effects of the binomial success probability and the number of bootstrap samples and on the accuracy of the estimator. We also investigate our method empirically across a variety of settings, using simulated as well as real data.

翻译：我们提出了一种耦合自举估计器（coupled bootstrap estimator），用于估计任意算法在泊松序列中估计均值（通常称为泊松均值问题）时的测试误差。该方法的核心思想是通过引入合成二项噪声，从原始数据向量中精心构造两个数据向量：一个充当训练样本，另一个充当测试样本。为稳定测试误差估计，我们对多次合成的噪声结果取平均。该耦合自举估计器的一个关键性质是：当原始均值被二项噪声的成功概率 $p$ 缩小一个微小因子时，它能够对测试误差进行无偏估计。进一步地，当 $p \to 0$ 时，我们证明所提出的估计器能恢复一个已知的测试误差无偏估计，且无需对算法施加任何假设（特别是无需光滑性假设）。我们的方法适用于两种可用于测试误差度量的核心损失函数：泊松偏差（Poisson deviance）和平方损失（squared loss）。通过偏差-方差分解，我们针对每种损失函数分析了二项成功概率和自举样本数量对估计器精度的影响。此外，我们通过模拟数据和真实数据在多种设定下对方法进行了实证研究。