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We propose a coupled bootstrap (CB) method 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 bootstrap B of the synthetic noise. A key property of the CB estimator is that it is unbiased for the test error in a Poisson 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 $B \to \infty$ and $p \to 0$, we show that the CB estimator recovers a known unbiased estimator for test error based on Hudson's lemma, under no assumptions on the given algorithm for estimating the mean (in particular, no smoothness assumptions). Our methodology applies to two central loss functions that can be used to define test error: Poisson deviance and squared loss. Via 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., Comment: 26 pages, 9 figures

The adoption of Uncrewed Aerial Vehicles (UAVs) for last-mile deliveries will affect the energy productivity of package delivery and require new methods to understand the associated energy consumption and greenhouse gas (GHG) emissions. Here we combine empirical testing of 187 quadcopter flights with first principles analysis to develop a usable energy model for drone package delivery. We develop a machine-learning algorithm to assess energy use across three different flight regimes: takeoff, cruise, and landing. Our model shows that, in the US, a small electric quadcopter drone with a payload of 1 kg would consume approximately 0.05 MJ/km and result in 41 g of CO$_{2}$e per package. The energy per package delivered by drones (0.19 MJ/package) can be up to 96\% lower than conventional transportation modes. Our open model and generalizable coefficients can assist stakeholders in understanding and improving the energy use of drone package delivery.

We develop a new approach for estimating the risk of an arbitrary estimator of the mean vector in the classical normal means problem. The key idea is to generate two auxiliary data vectors, by adding carefully constructed normal noise vectors to the original data. We then train the estimator of interest on the first auxiliary vector and test it on the second. In order to stabilize the risk estimate, we average this procedure over multiple draws of the synthetic noise vector. A key aspect of this coupled bootstrap (CB) approach is that it delivers an unbiased estimate of risk under no assumptions on the estimator of the mean vector, albeit for a modified and slightly "harder" version of the original problem, where the noise variance is elevated. We prove that, under the assumptions required for the validity of Stein's unbiased risk estimator (SURE), a limiting version of the CB estimator recovers SURE exactly. We then analyze a bias-variance decomposition of the error of the CB estimator, which elucidates the effects of the variance of the auxiliary noise and the number of bootstrap samples on the accuracy of the estimator. Lastly, we demonstrate that the CB estimator performs favorably in various simulated experiments., Comment: 32 pages, 9 figures

Trail running is an endurance sport in which athletes face severe physical challenges. Due to the growing number of participants, the organization of limited staff, equipment, and medical support in these races now plays a key role. Monitoring runner's performance is a difficult task that requires knowledge of the terrain and of the runner's ability. In the past, choices were solely based on the organizers' experience without reliance on data. However, this approach is neither scalable nor transferable. Instead, we propose a firm statistical methodology to perform this task, both before and during the race. Our proposed framework, Trail Running Assessment of Performance (TRAP), studies (1) the the assessment of the runner's ability to reach the next checkpoint, (2) the prediction of the runner's expected passage time at the next checkpoint, and (3) corresponding prediction intervals for the passage time. To obtain data on the ability of runners, we introduce a Python package, ScrapITRA, to access the race history of runners from the International Trail Running Association (ITRA). We apply our methodology, using the ITRA data along with checkpoint and terrain-level information, to the "holy grail" of ultra-trail running, the Ultra-Trail du Mont-Blanc (UTMB) race, demonstrating the predictive power of our methodology.

Hypothesis testing in contingency tables is usually based on asymptotic results, thereby restricting its proper use to large samples. To study these tests in small samples, we consider the likelihood ratio test and define an accurate index, the P-value, for the celebrated hypotheses of homogeneity, independence, and Hardy-Weinberg equilibrium. The aim is to understand the use of the asymptotic results of the frequentist Likelihood Ratio Test and the Bayesian FBST -- Full Bayesian Significance Test -- under small-sample scenarios. The proposed exact P-value is used as a benchmark to understand the other indices. We perform analysis in different scenarios, considering different sample sizes and different table dimensions. The exact Fisher test for $2 \times 2$ tables that drastically reduces the sample space is also discussed. The main message of this paper is that all indices have very similar behavior, so the tests based on asymptotic results are very good to be used in any circumstance, even with small sample sizes.