Stefan Sperlich

Nonparametric Estimation And Testing Of Interaction In Additive Models

We consider an additive model with second order interaction terms. It is shown how the components of this model can be estimated using marginal integration, and the asymptotic distribution of the estimators is derived. Moreover, two test statistics for testing the presence of interactions are proposed. Asymptotics for the test functions are obtained, but in this case the asymptotics produce inaccurate results unless the number of observations is very large. For small or moderate sample sizes a bootstrap procedure is suggested and is shown to work well on a simulated example. Finally, our methods are illustrated on a five-dimensional production function for a set of Wisconsin farm data. In particular, the separability hypothesis for the production function is discussed.

BOOTSTRAP INFERENCE IN SEMIPARAMETRIC GENERALIZED ADDITIVE MODELS

Semiparametric generalized additive models are a powerful tool in quantitative econometrics. The main focus is the application of bootstrap methods. It is shown that bootstrap can be used for bias correction, hypothesis testing (e.g. component-wise analysis) and the construction of uniform confidence bands. Various bootstrap tests for model specification and parametrization are given, in particular for testing additivity and link function specification. The practical performance of our methods is illustrated in simulations and in an application to East-West German migration.

Integration and backfitting methods in additive models-finite sample properties and comparison

We examine and compare the finite sample performance of the competing back-fitting and integration methods for estimating additive nonparametric regression using simulated data. Although, the asymptotic properties of the integration estimator, and to some extent the backfitting, method too, are well understood, its small sample properties are not well investigated. Apart from some small experiments in the above cited papers, there is little hard evidence concerning the exact distribution of the estimates. It is our purpose to provide an extensive finite sample comparison between the backfitting procedure and the integration procedure using simulated data.

Estimation of a semiparametric transformation model

This paper proposes consistent estimators for transformation parameters in semiparametric models. The problem is to find the optimal transformation into the space of models with a predetermined regression structure like additive or multiplicative separability. We give results for the estimation of the transformation when the rest of the model is estimated non- or semi-parametrically and fulfills some consistency conditions. We propose two methods for the estimation of the transformation parameter maximizing a profile likelihood function or minimizing the mean squared distance from independence. First the problem of identification of such models is discussed. We then state asymptotic results for a general class of nonparametric estimators. Finally, we give some particular examples of nonparametric estimators of transformed separable models. The small sample performance is studied in several simulations.

Derivative estimation and testing in generalized additive models

Estimation and testing procedures for generalized additive (interaction) model are developed. We present extensions of several existing procedures for additive models when the link is the identity. This set of methods includes estimation of all component functions and their derivatives, testing functional forms and in particular variable selection. Theorems and simulation results are presented for the fundamentally new procedures. These comprise of, in particular, the introduction of local polynomial smoothing for this kind of models and the testing, including variable selection. Our method is straightforward to implement and the simulation studies show good performance in even small data sets.

Estimation of Derivatives for Additive Separable Models

Additive regression models have a long history in nonparametric regression. It is well known that these models can be estimated at the one dimensional rate. Until recently, however, these models have been estimated by a backfitting procedure. Although the procedure converges quickly, its iterative nature makes analyzing its statistical properties difficult. Furthermore it is unclear how to estimate derivatives with this approach since it does not give a closed form for the estimator. Recently, an integration approach has been studied that allows for the derivation of a closed form for the estimator. This paper extends this approach to the simultaneous estimation of both the function and its derivatives by combining the integration procedure with a local polynomial approach. Finally the merits of this procedure with respect to the estimation of a production function subject to separability conditions are discussed. The procedure is applied to livestock production data from Wisconsin. It is shown that there is some evidence of increasing return to scale for larger farms.

Structural tests in additive regression

We consider the component analysis problem for a regression model with an additive structure. The problem is to test whether some of the additive components are of polynomial structure (e.g., linear) without specifying the structure of the remaining components. A particular case is the problem of selecting the significant covariates. The method that we present is based on the wavelet transform using the Haar basis, which allows for applications under mild conditions on the design and smoothness of the regression function. The results demonstrate that each component of the model can be tested with the rate corresponding to the case if all of the remaining components were known. The proposed procedure is also computationally straightforward. Simulation results and a real data example about female labor supply demonstrate the tests good performance.

A comparison of different nonparametric methods for inference on additive models

In this article, we highlight the main differences of available methods for the analysis of regression functions that are probably additive separable. We first discuss definition and interpretation of the most common estimators in practice explaining the different ideas of modeling behind each estimator as well as what the procedures are doing to the data. Computational aspects are mentioned explicitly. The discussion concludes with a simulation study on the mean squared error for different marginal integration approaches. Next, various test statistics for checking additive separability are introduced and accomplished with asymptotic theory. For the statistics, different smoothing and bootstrap methods, we perform a detailed simulation study. A main focus in the reported results is directed on the (non)reliability of the methods when the covariates are strongly correlated among themselves. We found that the most striking differences lie in the different pre-smoothers that are used, but less in the different constructions of test statistics or bootstrap methods. Moreover, although some of the observed differences are strong, they surprisingly cannot be revealed by asymptotic theory.

Do-Validation for Kernel Density Estimation

Bandwidth selection in kernel density estimation is one of the fundamental model selection problems of mathematical statistics. The study of this problem took major steps forward with the articles of Hall and Marron (1987) and Hall and Johnstone (1992). Since then, the focus seems to have been on various versions of implementing the so-called plug-in method aimed at estimating the minimum mean integrated squared error (MISE). The most successful of these efforts still seems to be the plug-in method of Sheather and Jones (1991) or Park and Marron (1990) that we also use as a benchmark in this article. In this article we derive a new theorem deriving the asymptotic theory for linear combinations of bandwidths obtained from different selectors as, for example, direct and indirect cross-validation and plug-in, where we take advantage of recent advances in the study of indirect cross-validation; see Hart and Yi (1998), Hart and Lee (2005), and Savchuk, Hart, and Sheather (2008, 2010). We conclude that the slow convergence of data-driven bandwidths implies that once asymptotic theory is close to that of the plug-in, then it is the practical implementation that counts. This insight led us to a bandwidth selector search with the symmetrized version of one-sided cross-validation as a clear winner.

Continuous Chain Ladder: Reformulating and generalizing a classical insurance problem

The single most important number in the accounts of a non-life insurance company is likely to be the estimate of the outlying liabilities. Since non-life insurance is a major part of our financial industry (amounting to up to 5% of BNP in western countries), it is perhaps surprising that mathematical statisticians and experts of operational research (the natural experts of the underlying problem) have left the intellectual work on estimating this number to actuaries. This paper establishes this important problem in a vocabulary accessible to experts of operations research and mathematical statistics and it can be seen as an open invitation to these two important groups of scholars to join this research. The paper introduces a number of new methodologies and approaches to estimating outstanding liabilities in non-life insurance. In particular it reformulates the classical actuarial technique as a histogram type of approach and improves this classical technique by replacing this histogram by a kernel smoother.