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Inhaltsverzeichnis

Frontmatter

Chapter I. Introduction

Abstract
A statistic can frequently be considered as a functional on a space of distribution functions. Often such a statistical functional possesses differentiability properties which provide information about its asymptotic behavior. These basic ideas were introduced by R. von Mises (1947), who developed a theory for the analysis of the asymptotic distribution of statistical functionals, using a form of Taylor expansion involving the derivatives of the functionals.
Luisa Turrin Fernholz

Chapter II. Von Mises’ Method

Abstract
In this chapter we present the general structure of von Mises’ approach to the analysis of the asymptotic behavior of statistical functional. The basic technique was introduced by von Mises (1947) and has been extended in various directions by several authors: Filippova (1962), Reeds (1976), Huber (1977, 1981), and Serfling (1980). One result of these extensions is that the field has become divergent, with ad hoc techniques applied in different situations. In the chapters that follow we shall try to establish a unified methodology that can be applied to wide classes of statistics.
Luisa Turrin Fernholz

Chapter III. Hadamard Differentiation

Abstract
As we observed in the last chapter, von Mises’ method is dependent upon the differentiability properties of statistical functionals. In preparation for the study of these properties we present here some basic definitions and results on differentiation of functions defined on topological vector spaces, and prove an implicit function theorem which will later be applied to implicitly defined statistical functionals. More details on the topics covered here can be found in Averbukh and Smolyanov (1968), Yamamuro (1974), and Keller (1974).
Luisa Turrin Fernholz

Chapter IV. Some Probability Theory on C[0,1] and D[0,1]

Abstract
We have seen in Chapter II that to prove asymptotic normality by von Mises’ method it is necessary to show that a statistical functional is differentiable and that the remainder term of its von Mises expansion satisfies the convergence condition (2.7). In this chapter we show that statistical functionals induce functionals on the space D[0,1] of functions on [0,1] with at most discontinuities of the first kind, and that problems of differentiability and convergence can be considered in this setting. Both the differentiability of the functional and the convergence of the remainder depend on the choice of topology on the domain of the functional. A stronger topology will allow more functionals to be differentiable, but will interfere with the convergence of the remainder. We shall use the uniform topology on D[0,1] and we shall show that with this topology the remainder term satisfies the convergence condition (2.7). This result will first be proved on C[0,1], the space of continuous functions on [0,1] with the uniform topology, and then be extended to D[0,1]. In the following chapters we shall show that wide classes of statistical functionals induce Hadamard differentiable functionals on D[0,1] with the uniform topology, and therefore with this choice of topology we are able to construct a broadly applicable von Mises calculus.
Luisa Turrin Fernholz

Chapter V. M-, L-, and R-Estimators

Abstract
In this chapter we shall introduce the three basic types of robust estimators, M-, L-, and R-estimators, and shall study properties of the corresponding statistical functionals. The results established here will later be used to show that the functionals induced on D[0,1] by these estimators are Hadamard differentiable. As we have seen in the previous chapter, the Hadamard differentiability of the induced functionals on D[0,1] is sufficient to imply the asymptotic normality of the estimators.
Luisa Turrin Fernholz

Chapter VI. Calculus on Function Spaces

Abstract
A necessary step in the application of von Mises’ method is to establish that a statistical functional is in some sense differentiable. To accomplish this, we present here the rudiments of a calculus for functionals defined on D[0,1], which will enable us to deal with the statistical functionals that we considered in the last chapter, as well as others that we shall introduce later.
Luisa Turrin Fernholz

Chapter VII. Applications

Abstract
In this chapter we shall show that certain statistical functionals are asymptotically normal by applying the techniques developed in the previous chapters. First, we shall consider M-, L-, and R-estimators. Besides these, we shall treat a somewhat more complicated statistic, a gap-compromise estimator.
Luisa Turrin Fernholz

Chapter VIII. Asymptotic Efficiency

Abstract
In this chapter we show that Hadamard differentiability can be used to prove asymptotic efficiency for statistical functionals. Huber (1977) gave a proof that Fréchet differentiable functionals are asymptotically efficient if and only if the influence curve satisfies certain conditions. However he also noted that “the rather stringent regularity conditions — Fréchet differentiability — will rarely be satisfied”. Here we show that Huber’s result holds under the weaker assumption of Hadamard differentiability. Since we have shown that several classes of statistical functionals are Hadamard differentiable, this approach to asymptotic efficiency through Hadamard differentiability has wide applicability.
Luisa Turrin Fernholz

Backmatter

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