Wolfgang Karl Härdle

Applied Multivariate Statistical Analysis

I Descriptive Techniques: Comparison of Batches.- II Multivariate Random Variables: A Short Excursion into Matrix Algebra.- Moving to Higher Dimensions.- Multivariate Distributions.- Theory of the Multinormal.- Theory of Estimation.- Hypothesis Testing.- III Multivariate Techniques: Regression Models.- Variable Selection.- Decomposition of Data Matrices by Factors.- Principal Components Analysis.- Factor Analysis.- Cluster Analysis.- Discriminant Analysis.- Correspondence Analysis.- Canonical Correlation Analysis.- Multidimensional Scaling.- Conjoint Measurement Analysis.- Applications in Finance.- Computationally Intensive Techniques.- IV Appendix: Symbols and Notations.- Data.

Investigating Smooth Multiple Regression by the Method of Average Derivatives

Abstract Let (x 1, …, xk, y) be a random vector where y denotes a response on the vector x of predictor variables. In this article we propose a technique [termed average derivative estimation (ADE)] for studying the mean response m(x) = E(y | x) through the estimation of the k vector of average derivatives δ = E(m′). The ADE procedure involves two stages: first estimate δ using an estimator , and then approximate m(x) by ), where ĝ is an estimator of the univariate regression of y on . We argue that the ADE procedure exhibits several attractive characteristics: data summarization through interpretable coefficients, graphical depiction of the possible nonlinearity between y and , and theoretical properties consistent with dimension reduction. We motivate the ADE procedure using examples of models that take the form . In this framework, δ is shown to be proportional to β and [mcirc](x) infers m(x) exactly. The focus of the procedure is on the estimator , which is based on a simple average of kernel smoother...

Partially Linear Models

In the last ten years, there has been increasing interest and activity in the general area of partially linear regression smoothing in statistics. Many methods and techniques have been proposed and studied. This monograph hopes to bring an up-to-date presentation of the state of the art of partially linear regression techniques. The emphasis of this monograph is on methodologies rather than on the theory, with a particular focus on applications of partially linear regression techniques to various statistical problems. These problems include least squares regression, asymptotically efficient estimation, bootstrap resampling, censored data analysis, linear measurement error models, nonlinear measurement models, nonlinear and nonparametric time series models. We hope that this monograph will serve as a useful reference for theoretical and applied statisticians and to graduate students and others who are interested in the area of partially linear regression. While advanced mathematical ideas have been valuable in some of the theoretical development, the methodological power of partially linear regression can be demonstrated and discussed without advanced mathematics. This monograph can be divided into three parts: part one–Chapter 1 through Chapter 4; part two–Chapter 5; and part three–Chapter 6. In the first part, we discuss various estimators for partially linear regression models, establish theo- retical results for the estimators, propose estimation procedures, and implement the proposed estimation procedures through real and simulated examples. The second part is of more theoretical interest. In this part, we construct several adaptive and efficient estimates for the parametric component. We show that the LS estimator of the parametric component can be modified to have both Bahadur asymptotic efficiency and second order asymptotic efficiency. In the third part, we consider partially linear time series models. First, we propose a test procedure to determine whether a partially linear model can be used to fit a given set of data. Asymptotic test criteria and power investigations are presented. Second, we propose a Cross-Validation (CV) based criterion to select the optimum linear subset from a partially linear regression and estab- lish a CV selection criterion for the bandwidth involved in the nonparametric kernel estimation. The CV selection criterion can be applied to the case where the observations fitted by the partially linear model (1.1.1) are independent and identically distributed (i.i.d.). Due to this reason, we have not provided a sepa- rate chapter to discuss the selection problem for the i.i.d. case. Third, we provide recent developments in nonparametric and semiparametric time series regression. This work of the authors was supported partially by the Sonderforschungs- bereich373“QuantifikationundSimulationO konomischerProzesse”.Thesecond author was also supported by the National Natural Science Foundation of China and an Alexander von Humboldt Fellowship at the Humboldt University, while the third author was also supported by the Australian Research Council. The second and third authors would like to thank their teachers: Professors Raymond Car- roll, Guijing Chen, Xiru Chen, Ping Cheng and Lincheng Zhao for their valuable inspiration on the two authors’ research efforts. We would like to express our sin- cere thanks to our colleagues and collaborators for many helpful discussions and stimulating collaborations, in particular, Vo Anh, Shengyan Hong, Enno Mam- men, Howell Tong, Axel Werwatz and Rodney Wolff. For various ways in which they helped us, we would like to thank Adrian Baddeley, Rong Chen, Anthony Pettitt, Maxwell King, Michael Schimek, George Seber, Alastair Scott, Naisyin Wang, Qiwei Yao, Lijian Yang and Lixing Zhu. The authors are grateful to everyone who has encouraged and supported us to finish this undertaking. Any remaining errors are ours.

Applied Nonparametric Methods

We review different approaches to nonparametric density and regression estimation. Kernel estimators are motivated from local averaging and solving ill-posed problems. Kernel estimators are compared to k-NN estimators, orthogonal series and splines. Pointwise and uniform confidence bands are described, and the choice of smoothing parameter is discussed. Finally, the method is applied to nonparametric prediction of time series and to semiparametric estimation.

Nonparametric curve estimation from time series

Because of the sheer size and scope of the plastics industry, the title Developments in Plastics Technology now covers an incredibly wide range of subjects or topics. No single volume can survey the whole field in any depth and what follows is, therefore, a series of chapters on selected topics. The topics were selected by us, the editors, because of their immediate relevance to the plastics industry. When one considers the advancements of the plastics processing machinery (in terms of its speed of operation and conciseness of control), it was felt that several chapters should be included which related to the types of control systems used and the correct usage of hydraulics. The importance of using cellular, rubber-modified and engineering-type plastics has had a major impact on the plastics industry and therefore a chapter on each of these subjects has been included. The two remaining chapters are on the characterisation and behaviour of polymer structures, both subjects again being of current academic or industrial interest. Each of the contributions was written by a specialist in that field and to them all, we, the editors, extend our heartfelt thanks, as writing a contribution for a book such as this, while doing a full-time job, is no easy task.

How Far are Automatically Chosen Regression Smoothing Parameters from their Optimum

Abstract We address the problem of smoothing parameter selection for nonparametric curve estimators in the specific context of kernel regression estimation. Call the “optimal bandwidth” the minimizer of the average squared error. We consider several automatically selected bandwidths that approximate the optimum. How far are the automatically selected bandwidths from the optimum? The answer is studied theoretically and through simulations. The theoretical results include a central limit theorem that quantifies the convergence rate and gives the differences asymptotic distribution. The convergence rate turns out to be excruciatingly slow. This is not too disappointing, because this rate is of the same order as the convergence rate of the difference between the minimizers of the average squared error and the mean average squared error. In some simulations by John Rice, the selectors considered here performed quite differently from each other. We anticipated that these differences would be reflected in differ...

Copula Theory and Its Applications

In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applied in various fields, and several methods of constructing multivariate copulas. 1.1 Historical Introduction The history of copulas may be said to begin with Frechet [70]. He studied the following problem, which is stated here in dimension 2: given the distribution functions F1 and F2 of two random variables X1 and X2 defined on the same probability space (Ω ,F ,P), what can be said about the set Γ (F1,F2) of the bivariate d.f.’s whose marginals are F1 and F2? It is immediate to note that the set Γ (F1,F2), now called the Frechet class of F1 and F2, is not empty since, if X1 and X2 are independent, then the distribution function (x1,x2) → F(x1,x2) = F1(x1)F2(x2) always belongs to Γ (F1,F2). But, it was not clear which the other elements of Γ (F1,F2) were. Preliminary studies about this problem were conducted in [65, 71, 90] (see also [31, 182] for a historical overview). But, in 1959, Sklar obtained the deepest result in this respect, by introducing the notion, and the name, of a copula, and proving the theorem that now bears his name [192]. In his own words [194]: Fabrizio Durante Department of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Linz Austria e-mail: fabrizio.durante@jku.at Carlo Sempi Dipartimento di Matematica “Ennio De Giorgi”, Universita del Salento, Lecce, Italy e-mail: carlo.sempi@unisalento.it P. Jaworski et al. (eds.), Copula Theory and Its Applications, Lecture Notes in Statistics 198, DOI 10.1007/978-3-642-12465-5_1, c

Local Polynomial Estimators of the Volatility Function in Nonparametric Autoregression

Abstract In this paper we consider a class of dynamic models in which both the conditional mean and the conditional variance (volatility) are unknown functions of the past. We first derive probabilistic conditions under which nonparametric estimation of these functions is possible. We then construct an estimator based on local polynomial fitting. We examine the rates of convergence of these estimators and give a result on their asymptotic normality. The local polynomial fitting of the volatility function is applied to different foreign exchange rate series. We find an asymmetric U-shaped ‘smiling face’ form of the volatility function.

Statistical Tools for Finance and Insurance

Statistical Tools for Finance and Insurance presents ready-to-use solutions, theoretical developments and method construction for many practical problems in quantitative finance and insurance. Written by practitioners and leading academics in the field this book offers a unique combination of topics from which every market analyst and risk manager will benefit. Features of the book: Offers insight into new methods and the applicability of the stochastic technology; Provides the tools, instruments and (online) algorithms for recent techniques in quantitative finance and modern treatments in insurance calculations; Covers topics such as heavy tailed distributions, implied trinomial trees, support vector machines, valuation of mortgage-backed securities, pricing of CAT bonds, simulation of risk processes, and ruin probability approximation; Presents extensive examples; The downloadable electronic edition of the book offers interactive tools.

Direct Semiparametric Estimation of Single-Index Models with Discrete Covariates

Abstract Others have developed average derivative estimators of the parameter β in the model E(Y|X = x) = G(xβ), where G is an unknown function and X is a random vector. These estimators are noniterative and easy to compute but require that X be continuously distributed. This article develops a noniterative, easily computed estimator of β for models in which some components of X are discrete. The estimator is n ½ consistent and asymptotically normal. An application to data on product innovation by German manufacturers illustrates the estimators usefulness.