Based on two independent samples X_1,...,X_m and X_{m+1},...,X_n drawn from multivariate distributions with unknown Lebesgue densities p and q respectively, we propose an exact multiple test in order to identify simultaneously regions of significant deviations between p and q. The construction is built from randomized nearest-neighbor statistics. It does not require any preliminary information about the multivariate densities such as compact support, strict positivity or smoothness and shape properties. The properly adjusted multiple testing procedure is shown to be sharp-optimal for typical arrangements of the observation values which appear with probability close to one. The proof relies on a new coupling Bernstein type exponential inequality, reflecting the non-subgaussian tail behavior of a combinatorial process. For power investigation of the proposed method a reparametrized minimax set-up is introduced, reducing the composite hypothesis "p=q" to a simple one with the multivariate mixed density (m/n)p+(1-m/n)q as infinite dimensional nuisance parameter. Within this framework, the test is shown to be spatially and sharply asymptotically adaptive with respect to uniform loss on isotropic Hölder classes. The exact minimax risk asymptotics are obtained in terms of solutions of the optimal recovery.
A Three-Way Testing Framework for Quantifying Epistemic Calibration Uncertainty in SBI
DEs, ICMC（数学与统计系，计算与应用数学研究所）；Federal Univ. of São Carlos（圣卡洛斯联邦大学）；Univ. of São Paulo（圣保罗大学）；Brazil（巴西）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Inria（法国国家数字科学研究所）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立理工学院）；LJK（拉格朗日计算与分析实验室）
Booth School of Business, University of Chicago（芝加哥大学布斯商学院）；Volgenau School of Engineering, George Mason University（乔治梅森大学沃尔格瑙工程学院）；School of Information Systems, Muma College of Business, University of South Florida（南佛罗里达大学穆马商学院信息系统学院）
