An Exponential Inequality for <i>U</i>-Statistics with Applications to Testing

A new exponential inequality for degenerate U-statistics is presented and it is shown that the bound on the tail probability is quadratic for small to medium values of the deviation and linear for larger values.

Abstract

We present a new exponential inequality for degenerate U -statistics. The bound on the tail probability is quadratic for small to medium values of the deviation and linear for larger values. We apply this bound to a family of test statistics and provide the key step in an optimality result for adaptive tests [1].