Axel Munk

Box-Type Approximations in Nonparametric Factorial Designs

Abstract Linear rank statistics in nonparametric factorial designs are asymptotically normal and, in general, heteroscedastic. In a comprehensive simulation study, the asymptotic chi-squared law of the corresponding quadratic forms is shown to be a rather poor approximation of the finite-sample distribution. Motivated by this problem, we propose simple finite-sample size approximations for the distribution of quadratic forms in factorial designs under a normal heteroscedastic error structure. These approximations are based on an F distribution with estimated degrees of freedom that generalizes ideas of Patnaik and Box. Simulation studies show that the nominal level is maintained with high accuracy and in most cases the power is comparable to the asymptotic maximin Wald test. Data-driven guidelines are given to select the most appropriate test procedure. These ideas are finally transferred to nonparametric factorial designs where the same quadratic forms as in the parametric case are applied to the vector ...

Convergence Rates of General Regularization Methods for Statistical Inverse Problems and Applications

Previously, the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut-off and Tikhonov-type estimators. Spectral cut-off estimators achieve minimax rates for a broad range of smoothness classes and operators, but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator. Tikhonov estimators are simpler to compute but still involve the inversion of an operator and achieve minimax rates only in restricted smoothness classes. In this paper we introduce a unifying technique to study the mean square error of a large class of regularization methods (spectral methods) including the aforementioned estimators as well as many iterative methods, such as

Estimating the variance in nonparametric regression—what is a reasonable choice?

\nu

Testing heteroscedasticity in nonparametric regression

-methods and the Landweber iteration. The latter estimators converge at the same rate as spectral cut-off but require only matrix-vector products. Our results are applied to various problems; in particular we obtain precise convergence rates for satellite gradiometry,

Self-consistent residual dipolar coupling based model-free analysis for the robust determination of nanosecond to microsecond protein dynamics

L^2

Hidden Markov models for circular and linear-circular time series

-boosting, and errors in variable problems.

Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise

The exact mean-squared error (MSE) of estimators of the variance in nonparametric regression based on quadratic forms is investigated. In particular, two classes of estimators are compared: Hall, Kay and Titteringtons optimal difference-based estimators and a class of ordinary difference-based estimators which generalize methods proposed by Rice and Gasser, Sroka and Jennen-Steinmetz. For small sample sizes the MSE of the first estimator is essentially increased by the magnitude of the integrated first two squared derivatives of the regression function. It is shown that in many situations ordinary difference-based estimators are more appropriate for estimating the variance, because they control the bias much better and hence have a much better overall performance. It is also demonstrated that Rices estimator does not always behave well. Data-driven guidelines are given to select the estimator with the smallest MSE.

Contributions of empirical and quantile processes to the asymptotic theory of goodness-of-fit tests

The importance of being able to detect heteroscedasticity in regression is widely recognized because efficient inference for the regression function requires that heteroscedasticity is taken into account. In this paper a simple consistent test for heteroscedasticity is proposed in a nonparametric regression set‐up. The test is based on an estimator for the best L2‐approximation of the variance function by a constant. Under mild assumptions asymptotic normality of the corresponding test statistic is established even under arbitrary fixed alternatives. Confidence intervals are obtained for a corresponding measure of heteroscedasticity. The finite sample performance and robustness of these procedures are investigated in a simulation study and Box‐type corrections are suggested for small sample sizes.

Convergence Analysis of Generalized Iteratively Reweighted Least Squares Algorithms on Convex Function Spaces

Residual dipolar couplings (RDCs) provide information about the dynamic average orientation of inter-nuclear vectors and amplitudes of motion up to milliseconds. They complement relaxation methods, especially on a time-scale window that we have called supra-