Rates of convergence for partial mass problems
Given two probability measures \(p\) and \(R\) on \(\mathbb{R}^k\) and \(\alpha\in [0,1]\), \(R\) is said to be a trimming of \(P\) if it is absolutely continuous with respect to \(P\) and the Radon-Nikodym derivative satisfies \(dR/dP\leq 1/(1-\alpha)\). The set of \(\alpha\)-trimmings of \(P\) is denoted by \(\mathcal{R}_{\alpha}(P)\). The authors consider the minimisation problem \[ \inf_{R\in\mathcal{R}_{\alpha}(P)} d(R,P_0), \] where \(P_0\) is a reference probability and \(d\) is either the Kolmogorov distance \(d_K\) or the \(L^p\)-Wasserstein distance \(W_p\). If they exist, the following probabilities are considered \[ P_{n,\alpha}:=\arg\,\min_{\mathcal{R}_{\alpha}(P_n)} d(R,P_0)\quad\text{and}\quad P_{\alpha}:=\arg\,\min_{\mathcal{R}_{\alpha}(P)} d(R,P_0), \] where \(P_n\) is the empirical measure on a sample. The authors obtain rates of convergence for \(W_p\): \[ W_p(\mathcal{R}_{\alpha}(P_n),P)=o_P\left(\frac{(\log n)^{\nu}}{n^{1/k}}\right), \quad \nu>1/k, \] and, for \(d_K\): \[ d_k(\mathcal{R}_{\alpha}(P_n),P)=o_P\left(\frac{(\log n)^{\nu}}{n}\right)^{\frac{k+1}{2k}}. \]
scientific article; zbMATH DE number 6162387
scientific article