Showing papers on "Gâteaux derivative published in 1985"

Asymptotic distribution theory of statistical functionals: The compact derivative approach for robust estimators

Abstract: Derivatives of statistical functionals have been used to derive the asymptotic distributions ofL-,M- andR-estimators. This approach is often heuristic because the types of derivatives chosen have serious limitations. The Gâteaux derivative is too weak and the Frechet derivative is too strong. In between lies the compact derivative. This paper obtains strong results in a rigorous manner using the compact derivative onC0(R). This choice of space allows results for a broader class of functionals than previous choices, and the fact that\(\left\{ {\sqrt n \left( {\tilde F_n - F} \right)} \right\}\) is often tight provides the compact set required. A major result is the derivation of the compact derivative of the inverse c.d.f. when the range space is endowed with the uniform norm. It has applications to the asymptotic theory ofL-,M- andR-estimators. We illustrate the power of this result by applications toL-estimators in settings including the one sample problem, data grouped by quantiles, and censored survival time data.

Gâteaux differentiable points with special representation

Abstract: If (x,7) is a bounded sequence in a Banach space, is there an element x = a, x,7 such that E7= IjXa,7x,jj < o and the directional derivative of the norm at x, D(x, x,), exists for every n? In fact, there are such x's dense in the closed span of {x,7 }. An application of this fact is made to a proof of Rybakov's theorem on vector measures. A real-valued function f defined on a linear topological space X is said to be Gateaux differentiable at x E X if, for every y E X, Df (x; y) = lim f (x + ty) -f (x) t 0 t exists and converges to x*(y) for a unique x* E X*, where X* denotes the linear space of all continuous linear functionals on X. If Df(x; y) exists for a particular direction y E X, then we call this limit the Gateaux derivative off at x in the direction of y. The notation for the directional derivative of the norm is D(x; y) = lim lix + ty|| lxii t 0 t A real-valued function f on a subset A of a linear space X is said to be subdifferentiable at x E A if there exists x * E X* such that x*(y x) < f (y) -f (x) for ally in A. We say that x* is a subgradient of f at x. We denote by af(x) the set of all subgradients of f at x and call this set the subdifferential off at x. THEOREM 1. Let f be a continuous convex function defined on a Banach space X, and (xn) a bounded sequence in X; then there is an element x = Zn1 anxn such that n= 1InIXnaII < X and Df(x; Xn) exists for every n. Further, such x's are dense in the closed span of xn's. PROOF. Suppose (xn) is a bounded sequence in X and f is a continuous convex function on X. Define T: 11 X by T(a)= X where a = (an)E 11 n = 1 Received by the editors March 2, 1984 and, in revised form, June 5, 1984. 1980 Mathematics Subject Classification. Primary 46A55; Secondary 46A32, 52A07.