Let \(\mathbf{X}_{i}(1 \leq i \leq n)\) and \(\mathbf{Y}_{i}(1 \leq i \leq n)\) be two independent i.i.d. samples of random vectors from two different population. Let \(\mathbf{h}\) be a vector-valued symmetric function with \(s+t\) arguments. Show that

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

is an unbiased estimate of \(\boldsymbol{\theta}=E\left[\mathbf{h}\left(\mathbf{X}_{1}, \ldots, \mathbf{X}_{s}, \mathbf{Y}_{1}, \ldots, \mathbf{Y}_{t}\right)\right]\).