Nonlinear Coefficient Functions Research Articles

Statistical inference for partially time-varying coefficient errors-in-variables models

This paper studies the partially time-varying coefficient models where some covariates are measured with additive errors. In order to overcome the bias of the usual profile least squares estimation when measurement errors are ignored, we propose a modified profile least squares estimator of the regression parameter and construct estimators of the nonlinear coefficient function and error variance. The proposed three estimators are proved to be asymptotically normal under mild conditions. In addition, we introduce the profile likelihood ratio test and then demonstrate that it follows an asymptotically χ2 distribution under the null hypothesis. Finite sample behavior of the estimators is investigated via simulations too.

Small Time Asymptotics for Stochastic Evolution Equations

We obtain a large deviation principle describing the small time asymptotics of the solution of a stochastic evolution equation with multiplicative noise. Our assumptions are a condition on the linear drift operator that is satisfied by generators of analytic semigroups and Lipschitz continuity of the nonlinear coefficient functions. Methods originally used by Peszat (Probab. Theory Relat. Fields 98:113–136, 1994) for the small noise asymptotics problem are adapted to solve the small time asymptotics problem. The results obtained in this way improve on some results of Zhang (Ann. Probab. 28(2):537–557, 2000).

Statistical inference in partially time-varying coefficient models

A partially time-varying coefficient time series model is introduced to characterize the nonlinearity and trending phenomenon. To estimate the regression parameter and the nonlinear coefficient function, the profile least squares approach is applied with the help of local linear approximation. The asymptotic distributions of the proposed estimators are established under mild conditions. Meanwhile, the generalized likelihood ratio test is studied and the test statistics are demonstrated to follow asymptotic χ 2 -distribution under the null hypothesis. Furthermore, some extensions of the proposed model are discussed and several numerical examples are provided to illustrate the finite sample behavior of the proposed methods.

Global solution for a quasi-linear plate system with boundary memory damping

In this work, we consider a quasi-linear plate model with boundary memory damping. We prove that this system has a unique global solution when the initial data are small enough and the non-linear coefficient function, the memory damping as well as the geometry of the domain satisfy suitable assumptions. We also prove the exponential decay of the energy of the system.

Numerical Methods for the Determination of Material Properties in Soil Science*

This article deals with the determination of nonlinear coefficient functions in partial differential equations in the field of soil science. We consider two examples to illustrate the numerical determination of nonlinear coefficient functions. In detail, these are the determination of the sorption characteristic of a chemical and the determination of the unsaturated hydraulic properties of a porous medium from measurements obtained from suitable column experiments. This inverse problem is treated by minimizing a least square functional. To cope with the ill-posedness, we apply a parametrization of the unknown nonlinear coefficient function, which is defined by an appropriate interpolation. The parametrization does not use a priori assumptions, which are not justified by physical properties. This kind of parametrization permits a hierarchical approach in the number of the degrees of freedom used. According to the hierarchical structure, we integrate the determination of the coefficient functions into a multi-level procedure. The investigation of the stability of the parametrization is based on the singular values of the sensitivity matrix.

Identification of hydraulic parameters in layered soils based on a quasi-Newton method

In models for one-dimensional unsaturated flow in layered soils based on the general flow equation, non-linear coefficient functions appear. The inverse problem consists of the identification of parameters in models for these coefficients. In this study, parameters are identified for layered soils. The general flow equation is solved iteratively, adjusting the values of the parameters until the solution matches a set of in situ water content data. The optimization method is based on a quasi-Newton method with Richardson extrapolation. Data sets of different size and with various degrees of perturbation are used to investigate existence, uniqueness and accuracy of parameter estimates. To stabilize parameter estimates, a weighting procedure is proposed based on the values of the water content data. The correspondence of parameter estimates, based on relatively small sets of unperturbed data, to the true parameter values indicates a high degree of numerical accuracy of the method. Parameter estimates based on perturbed data sets suggest that ill-posedness of the estimation problem is a matter of data insufficiency and modeling errors, rather than of data error alone.