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

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

2010-01-09

arxiv.org/abs/1001.1396v1

Mathematics

Probability

Scientific paper

Let $(\mathbf{W,W'})$ be an exchangeable pair of vectors in $\mathbb{R}^k$. Suppose this pair satisfies \beas E(\mathbf{W}'|\mathbf{W})=(I_k-\Lambda)\mathbf{W}+\mathbf{R(W)}. \enas If $||\mathbf{W-W'}||_2\le K$ and $\mathbf{R(W)}=0$, then concentration of measure results of following form is proved for all $\mathbf{w}\succeq 0$ 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^2\nu_1}), \enas for an explicit constant $\nu_1$, where $\succeq$ 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 $\mathbf{R(W)}\neq 0$ 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.

Affiliated with

Also associated with

No associations

Rating

Multivariate concentration of measure type results using exchangeable pairs and size biasing does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Multivariate concentration of measure type results using exchangeable pairs and size biasing, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Multivariate concentration of measure type results using exchangeable pairs and size biasing will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-580364