Enno Mammen

NONPARAMETRIC ESTIMATION OF AN ADDITIVE MODEL WITH A LINK FUNCTION

This paper describes an estimator of the additive components of a nonparametric additive model with a known link function. When the additive components are twice continuously differentiable, the estimator is asymptotically normally distributed with a rate of convergence in probability of n-2/5. This is true regardless of the (finite) dimension of the explanatory variable. Thus, in contrast to the existing asymptotically normal estimator, the new estimator has no curse of dimensionality. Moreover, the asymptotic distribution of each additive component is the same as it would be if the other components were known with certainty.

Smoothing Splines and Shape Restrictions

Consider a partial linear model, where the expectation of arandom variable Y depends on covariates (x,z) through F( theta_0 x + m_0 (z)), with theta_0 an unknown parameter, and m_0 an unknown function. We apply the theory of empirical processes to derive the asymptotic properties of the penalized quasi-likelihood estimator.

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.

Thresholding algorithms, maxisets and well-concentrated bases

The aim of this paper is to synthetically analyse the performances of thresholding and wavelet estimation methods. In this connection, it is useful to describe the maximal sets where these methods attain a special rate of convergence. We relate these “maxisets” to other problems naturally arising in the context of non parametric estimation, as approximation theory or information reduction. A second part of the paper is devoted to isolate two very special properties especially shared by wavelet bases, which allow them to behave almost as in an Hilbertian context even for Lp risks.

Time Series Modelling with Semiparametric Factor Dynamics

High-dimensional regression problems, which reveal dynamic behavior, are typically analyzed by time propagation of a few number of factors. The inference on the whole system is then based on the low-dimensional time series analysis. Such high-dimensional problems occur frequently in many different fields of science. In this article we address the problem of inference when the factors and factor loadings are estimated by semiparametric methods. This more flexible modeling approach poses an important question: Is it justified, from an inferential point of view, to base statistical inference on the estimated times series factors? We show that the difference of the inference based on the estimated time series and “true” unobserved time series is asymptotically negligible. Our results justify fitting vector autoregressive processes to the estimated factors, which allows one to study the dynamics of the whole high-dimensional system with a low-dimensional representation. We illustrate the theory with a simulation study. Also, we apply the method to a study of the dynamic behavior of implied volatilities and to the analysis of functional magnetic resonance imaging (fMRI) data.

Some asymptotics for multimodality tests based on kernel density estimates

SummaryA test due to B.W. Silverman for modality of a probability density is based on counting modes of a kernel density estimator, and the idea of critical smoothing. An asymptotic formula is given for the expected number of modes. This, together with other methods, establishes the rate of convergence of the critically smoothed bandwidth. These ideas are extended to provide insight concerning the behaviour of the test based on bootstrap critical values.

Universal smoothing factor selection in density estimation: theory and practice

AbstractIn earlier work with Gabor Lugosi, we introduced a method to select a smoothing factor for kernel density estimation such that, forall densities in all dimensions, theL1 error of the corresponding kernel estimate is not larger than 3+∈ times the error of the estimate with the optimal smoothing factor plus a constant times

Bandwidth selection for smooth backfitting in additive models

\sqrt {\log n/n}