Multivariate concentration of measure type results using exchangeable pairs and size biasing

Abstract

Let $(\mathbf{W},{\mathbf{W}}^{\prime })$ be an exchangeable pair of vectors in ${\mathbb{R}}^k$. Suppose this pair satisfies \beas E(\mathbf{W}'|\mathbf{W})=(I_k-Λ)\mathbf{W}+\mathbf{R(W)}. \enas If $||\mathbf{W}-{\mathbf{W}}^{\prime }|{|}_2\le K$ and $\mathbf{R}(\mathbf{W})=0$, then concentration of measure results of following form is proved for all $\mathbf{w}⪰0$ when the moment generating function of $\mathbf{W}$ is finite. \beas P(\mathbf{W}\succeq\mathbf{w}),P(\mathbf{W}\preceq -\mathbf{w})\le \exp(-\frac{||\mathbf{w}||_2^2}{2K^2ν_1}), \enas for an explicit constant ${\nu }_1$, where $⪰$ stands for coordinate wise $\ge$ ordering. This result is applied to examples like complete non degenerate U-statistics. Also, we deal with the example of doubly indexed permutation statistics where $\mathbf{R}(\mathbf{W})\ne 0$ and obtain similar concentration of measure inequalities. Practical examples from doubly indexed permutation statistics include Mann-Whitney-Wilcoxon statistic and random intersection of two graphs. Both these two examples are used in nonparametric statistical testing. We conclude the paper with a multivariate generalization of a recent concentration result due to Ghosh and Goldstein \cite{cnm} involving bounded size bias couplings.

References in corpus (3)

• Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity condition
• Berry Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing
• Moment inequalities for U-statistics