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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Corollary 9.1. Let {X} be a sequence of independent and identically dis- tributed random variables from a distribution F. Let 0 = T(F) be a functional parameter of the form r T(F)= = [ h(x1,...,x) [[ dF (x;), II i=1 (9.7) with estimator n = T(Fn), where Fn is the empirical distribution function of X1, X. Define = E[t(X;|F)], o = V[t(X;|F)], where n AT(FF)=n t(X|F), i=1. for some function (X|F), assume that 0 < < , and that AT (F, F-F) is not functionally equal to zero, then n/2 (n - 0-) Z as n where Z has a N(0,2) distribution.