Consider the skewness functional given by

\[T(F)=\int_{-\infty}^{\infty}\left[t-\int_{-\infty}^{\infty} t d F(t)ight]^{3} d F(t)\]

where \(F \in \mathcal{F}\), a collection of distribution functions with mean \(\mu\) and variance \(\sigma^{2}\) and third central moment \(\gamma\).

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.

Transcribed Image Text:

Theorem 9.1. Consider a functional of the form T(F) - L-L = r h(x1,...,xr) II dF (x;), i=1 where F = F, and F is a collection of distribution functions. Then if k r, k |AT(F; G F) = |||(r=i+1)]; L L .. h (x1,..., 2 X r-k dF (ya d[G(xi) F(xi)] - F(x;)]}. Tk, Y1,..., Yr-k) II If kr then AT(F; G-F) = 0. i=1