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## Über dieses Buch

The role of the weak convergence technique via weighted empirical processes has proved to be very useful in advancing the development of the asymptotic theory of the so called robust inference procedures corresponding to non-smooth score functions from linear models to nonlinear dynamic models in the 1990's. This monograph is an ex­ panded version of the monograph Weighted Empiricals and Linear Models, IMS Lecture Notes-Monograph, 21 published in 1992, that includes some aspects of this development. The new inclusions are as follows. Theorems 2. 2. 4 and 2. 2. 5 give an extension of the Theorem 2. 2. 3 (old Theorem 2. 2b. 1) to the unbounded random weights case. These results are found useful in Chapters 7 and 8 when dealing with ho­ moscedastic and conditionally heteroscedastic autoregressive models, actively researched family of dynamic models in time series analysis in the 1990's. The weak convergence results pertaining to the partial sum process given in Theorems 2. 2. 6 . and 2. 2. 7 are found useful in fitting a parametric autoregressive model as is expounded in Section 7. 7 in some detail. Section 6. 6 discusses the related problem of fit­ ting a regression model, using a certain partial sum process. Inboth sections a certain transform of the underlying process is shown to provide asymptotically distribution free tests. Other important changes are as follows. Theorem 7. 3.

## Inhaltsverzeichnis

### 1. Introduction

Abstract
A weighted empirical process (W.E.P.) corresponding to the random variables (r.v.’s) Xn1, ..., X nn and the non-random real weights dn1, ..., d nn is defined to be
$$U_d (x): = \sum\limits_{i = 1}^n {d_{ni} I(X_{ni} \leqslant x),x \in \mathbb{R} \geqslant 1}$$
The weights {d ni } need not be nonnegative.
Hira L. Koul

### 2. Asymptotic Properties of W.E.P.’s

Abstract
Let, for each n ≥ 1, ηn1, …, η nn be independent r.v.’s taking values in [0,1] with respective d.f.’s Gn1, …, G nn and dn1, …, d nn be real numbers. Define {fy(2.1.1)|15-1} Observe that W d belongs to D[0,1] for each n and any triangular array {d ni , 1 ≤ in}, while V h of (1.4.1) belongs to D(ℝ) for each n and any triangular array {h ni , 1 ≤ in}.
Hira L. Koul

### 3. Linear Rank and Signed Rank Statistics

Abstract
Let {X ni , F ni } be as in (2.2.23) and {c ni } be p × 1 real vectors. The rank and the absolute rank of the i th residual for 1 ≤ in, u ∈ ℝ p , are defined, respectively, as
$$\begin{array}{l} R_{iu} = \sum\limits_{j = 1}^n {I(X_{nj} - u'c_{nj} \le X_{ni} - u'c_{ni} )} , \\ R_{iu}^ + = \sum\limits_{j = 1}^n {I(|X_{nj} - u'c_{nj} | \le |X_{ni} - u'c_{ni} |)} \\ \end{array}$$
(3.1.1)
% MathType!End!2!1! Let ϕ be a nondecreasing real valued function on [0,1] and define
$$\begin{gathered} Td(\phi ,u) = \sum\limits_{i = 1}^n {d_{ni} \phi \left( {\frac{{R_{iu} }} {{n + 1}}} \right),} \hfill \\ T_d^ + (\phi ,u) = \sum\limits_{i = 1}^n {d_{ni} \phi ^ + \left( {\frac{{R_{iu}^ + }} {{n + 1}}} \right)s(X_{ni} - u'c_{ni} ),} \hfill \\ \end{gathered}$$
(3.1.2)
for u ∈ ℝ p , where
$$\phi ^ + (s) = \phi ((s + 1)/2),0 \le s \le 1;s(x) = I(x > 0) - I(x < 0).$$
Hira L. Koul

### 4. M, R and Some Scale Estimators

Abstract
In the last four decades statistics has seen the emergence and consolidation of many competitors of the Least Square estimator of β of (1.1.1). The most prominent are the so-called M- and R- estimators. The class of M-estimators was introduced by Huber (1973) and its computational aspects and some robustness properties are available in Huber (1981). The class of R-estimators is based on the ideas of Hodges and Lehmann (1963) and has been developed by Adichie (1967), Jurečková (1971) and Jaeckel (1972).
Hira L. Koul

### 5. Minimum Distance Estimators

Abstract
The practice of obtaining estimators of parameters by minimizing a certain distance between some functions of observations and parameters has long been present in statistics. The classical examples of this method are the Least Square and the minimum Chi Square estimators.
Hira L. Koul

### 6. Goodness-of-fit Tests in Regression

Abstract
In this chapter we shall discuss the two problems of the goodness-of-fit. The first one pertains to the error d.f. of the linear model (1.1.1) and the second one pertains to fitting a parametric regression model to a regression function. The proposed tests will be based on certain residual weighted empiricals for the first problem and a partial sum process of the residuals for the second problem. The first five sections of this chapter deal with the first problem and Section 6.6, with several subsections, discusses the second problem. To begin with we shall focus on the first problem.
Hira L. Koul

### 7. Autoregression

Abstract
The purpose of the Chapters 7 and 8 is to offer a unified functional approach to some aspects of robust estimation and goodness-of-fit testing problems in autoregressive (AR) and conditionally heteroseedastic autoregressive (ARCH) models. We shall first focus on the well celebrated p-th order linear AR models. For these models, the similarity of the functional approach developed in the previous chapters in connection with linear regression models is transparent. This chapter thus extends the domain of applications of the statistical methodology of the previous chapters to the one of the most applied models with dependent observations. Chapter 8 discusses the development of similar approach in some general non-linear AR and ARCH models.
Hira L. Koul

### 8. Nonlinear Autoregression

Abstract
The decade of 1990’s has seen an exponential growth in the applications of nonlinear autoregressive models (AR) to economics, finance and other sciences. Tong (1990) illustrates the usefulness of homoscedastic AR models in a large class of applied examples from physical sciences while Gouriéroux (1997) contains several examples from economics and finance where the ARCH (autoregressive conditional heteroscedastic) models of Engle (1982) and its various generalizations are found useful. Most of the existing literature has focused on developing classical inference procedures in these models. The theoretical development of the analogues of the estimators discussed in the previous sections that are known to be robust against outliers in the innovations in linear AR models has relatively lagged behind.
Hira L. Koul

### 9. Appendix

Abstract
We include here some results relevant to the weak convergence of processes in D[0,1] and C[0,1] for the sake of easy reference and without proofs. Our source is the book by Billingsley (1968) (B) on Convergence of Probability Measures.
Hira L. Koul

Without Abstract
Hira L. Koul

### Backmatter

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