Question: Consider the function (mathbf{h}left(mathbf{Z}_{1}, mathbf{Z}_{2} ight)) for the U-statistic in (1.22). (a) Show [operatorname{Var}left[widetilde{mathbf{h}}_{1}left(mathbf{Z}_{1} ight) ight]=Eleft(mathbf{h}_{1}left(mathbf{Z}_{1} ight) mathbf{h}_{1}^{top}left(mathbf{Z}_{1} ight) ight)-boldsymbol{theta} boldsymbol{theta}^{top}] (b) Use the iterated

Consider the function $\mathbf{h}\left(\mathbf{Z}_{1}, \mathbf{Z}_{2}\right)$ for the U-statistic in (1.22).

(a) Show

\[\operatorname{Var}\left[\widetilde{\mathbf{h}}_{1}\left(\mathbf{Z}_{1}\right)\right]=E\left(\mathbf{h}_{1}\left(\mathbf{Z}_{1}\right) \mathbf{h}_{1}^{\top}\left(\mathbf{Z}_{1}\right)\right)-\boldsymbol{\theta} \boldsymbol{\theta}^{\top}\]

(b) Use the iterated conditional expectation to show

\[E\left(\mathbf{h}_{1}\left(\mathbf{Z}_{1}\right) \mathbf{h}_{1}^{\top}\left(\mathbf{Z}_{1}\right)\right)=E\left(\mathbf{h}\left(\mathbf{Z}_{1}, \mathbf{Z}_{2}\right) \mathbf{h}^{\top}\left(\mathbf{Z}_{1}, \mathbf{Z}_{3}\right)\right) .\]

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