Consider the (k^{text {th }}) central moment functional given by [T(F)=int_{-infty}^{infty}left[t-int_{-infty}^{infty} t d F(t)ight]^{k} d F(t)] where

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Consider the \(k^{\text {th }}\) central moment functional given by

\[T(F)=\int_{-\infty}^{\infty}\left[t-\int_{-\infty}^{\infty} t d F(t)ight]^{k} d F(t)\]

where \(F \in \mathcal{F}\), a collection of distribution functions where \(\mu_{k}

a. Using the method demonstrated in Example 9.8, write this functional in a form suitable for the application of Theorem 9.1.

b. Using Theorem 9.1, find the first two Gâteaux differentials of \(T(F)\).

c. Determine whether Corollary 9.1 applies to this functional and derive the asymptotic normality of \(T\left(\hat{F}_{n}ight)\) if it does.

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