v-statistics

Suppose we are estimating the symmetric statistic $S=S(X_1,\dots ,X_k)$ where $\mathbb{E}S=\theta$. Given data $X_1,\dots ,X_n$ the plug-in estimator

$$V_n(S)=\frac{1}{n^k}\sum _{\beta }S(X_{\beta }),$$

is the v-statistic for $S$, where $\beta$ is a permutation $X_{i_1},\dots ,X_{i_k}$. V-statistics are named after von-Mises, having been introduced by him in 1947. They are not unbiased. Indeed even for $k=2$ we have

$$V_n(S)=\frac{1}{n^2}\sum _{i\ne j}S(X_i,X_j)+\frac{1}{n^2}\sum _{i}S(X_i,X_i),$$

so $\mathbb{E}{V}_n(S)=\frac{n-1}{n}\theta +\frac{1}{n}\mathbb{E}S(X,X)$. As $n\to \infty$ this approaches $\theta$ but is biased for any finite $n$, unlike u-statistics.

Under some conditions (eg if the $X_i$ take values in a compact set), v-statistics are reverse submartingales. Under these conditions, like u-statistics, we can form CSs for them (confidence sequences for convex functionals).