An Efron-Stein Inequality for Nonsymmetric Statistics

Abstract

If $S(x_1, x_2,\cdots, x_n)$ is any function of $n$ variables and if $X_i, \hat{X}_i, 1 \leq i \leq n$ are $2n$ i.i.d. random variables then $\operatorname{var} S \leq \frac{1}{2} E \sum^n_{i=1} (S - S_i)^2$ where $S = S(X_1, X_2,\cdots, X_n)$ and $S_i$ is given by replacing the $i$th observation with $\hat{X}_i$, so $S_i = S(X_1, X_2,\cdots, \hat{X}_i,\cdots, X_n)$. This is applied to sharpen known variance bounds in the long common subsequence problem.