The usual sample variance $$ \mathrm{Var}(X) = \frac{1}{n-1} \sum_{i = 1}^n (X_i - \bar X)^2 \tag{1} $$ of an iid. sample $X_1, \dots, X_n$ is known to be a U-statistic with kernel $$ h(X_i, X_j) = \frac{1}{2} (X_i - X_j)^2. $$ This means that we can write $$ \mathrm{Var}(X) = \mathrm{Average}_{1 \le i < j \le n} h(X_i, X_j). \tag{2} $$ As a nice consequence of this fact, we immediately know $\mathrm{Var}(X)$ is a minimum variance unbiased estimator of the true variance of $X$.

The usual proof that (2) = (1) is relatively simple but lengthy. So I wanted to know if anyone knows a short, elegant proof of this identity?


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