Let \(\mathbf{X}_{i}(1 \leq i \leq n)\) be an i.i.d. sample of random vectors and let \(\mathbf{h}\) be a vector-valued symmetric function \(m\) arguments. Then,

\[\widehat{\boldsymbol{\theta}}=\left(\begin{array}{c}n \\m\end{array}\right)^{-1} \sum_{\left(i_{1}, \ldots, i_{m}\right) \in C_{m}^{n}} \mathbf{h}\left(\mathbf{X}_{i_{1}}, \ldots, \mathbf{X}_{i_{m}}\right)\]

is an unbiased estimate of \(\boldsymbol{\theta}\). This shows that \(\widehat{\boldsymbol{\theta}}\) in (1.22) is an unbiased estimate of \(\boldsymbol{\theta}=E\left[\mathbf{h}\left(\mathbf{X}_{1}, \ldots, \mathbf{X}_{m}\right)\right]\).