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)
Question:
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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a To show that operatornameVarleftwidetildemathbfh1leftmathbfZ1 ight ightEleftmathbfh1leftmathbfZ1 ight mathbfh1topleftmathbfZ1 ight ightboldsymbolthe...View the full answer
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Related Book For 
Applied Categorical And Count Data Analysis
ISBN: 9780367568276
2nd Edition
Authors: Wan Tang, Hua He, Xin M. Tu
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