Variance as a U-statistic
= Variance as a U-statistic
{title2=$U_n=\frac1{n-1}\sum_i(X_i-\overline X)^2$}
The symmetric kernel $h(x,y)=(x-y)^2/2$ has expectation equal to the population variance for independent identically distributed observations with finite second moment. Its <U-statistic> reduces to the displayed unbiased <sample variance>, using $\sum_{i<j}(X_i-X_j)^2=n\sum_i(X_i-\overline X)^2$.