Let (left{W_{n}ight}_{n=1}^{infty}) be a sequence of independent and identically distributed random variables from a distribution (F) with

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Let \(\left\{W_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables from a distribution \(F\) with mean \(\eta\) and variance \(\theta\). Prove that the parameter \(\theta\) can be represented in the smooth function model with \(d=4, \mathbf{X}_{n}^{\prime}=\left(W_{n}, W_{n}^{2}, W_{n}^{3}, W_{n}^{4}ight)\) for all \(n \in \mathbb{N}, g(\mathbf{x})=x_{2}-\left(x_{1}ight)^{2}\), and

\[h(\mathbf{x})=x_{4}-4 x_{1} x_{3}+6 x_{1}^{2} x_{2}-3 x_{1}^{4}-\left(x_{2}-x_{1}^{2}ight)^{2},\]

where \(\mathbf{x}^{\prime}=\left(x_{1}, x_{2}, x_{3}, x_{4}ight)\). This will verify the results of Example 7.7.

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