Optimal Strong Convergence Research Articles

Error estimates of a semidiscrete finite element method for fractional stochastic diffusion‐wave equations

In this paper, we consider the Galerkin finite element method for solving the fractional stochastic diffusion‐wave equations driven by multiplicative noise, which can be used to describe the propagation of mechanical waves in viscoelastic media with random effects. The optimal strong convergence error estimates with respect to the semidiscrete finite element approximation in space are established. Finally, a numerical example is presented to verify the reliability of the theoretical study.

Conforming finite element methods for the stochastic Cahn–Hilliard–Cook equation

This paper is concerned with the finite element approximation of the stochastic Cahn–Hilliard–Cook equation driven by an infinite dimensional Wiener type noise. The Argyris finite elements are used to discretize the spatial variables while the infinite dimensional (cylindrical) Wiener process is approximated by truncated stochastic series spanned by the spectral basis of the covariance operator. The optimal strong convergence order in L2 and H˙−2 norms is obtained. Unlike the mixed finite element method studied in the existing literature, our method allows the covariance operator of the Wiener process to have an infinite trace, including the space–time white noise is allowed in our model. Numerical experiments are presented to illustrate the theoretical analysis.

Convergence of the compensated split-step \u03b8-method for nonlinear jump-diffusion systems

In this paper, our aim is to develop a compensated split-step θ (CSSθ) method for nonlinear jump-diffusion systems. First, we prove the convergence of the proposed method under a one-sided Lipschitz condition on the drift coefficient, and global Lipschitz condition on the diffusion and jump coefficients. Then we further show that the optimal strong convergence rate of CSSθ can be recovered, if the drift coefficient satisfies a polynomial growth condition. At last, a nonlinear test equation is simulated to verify the results obtained from theory. The results show that the CSSθ method is efficient for simulating the nonlinear jump-diffusion systems.

EULER–MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS

In this paper we show that stochastic differential equations with jumps and non-Lipschitz coefficients have $(\xi,W,N_{p})$-pathwise unique strong solutions by the Euler--Maruyama approximation. Moreover, the Euler--Maruyama discretisation has an optimal strong convergence rate.

Edge Element Methods for Maxwell's Equations with Strong Convergence for Gauss' Laws

In this paper we propose and investigate some edge element approximations for three Maxwell systems in three dimensions: the stationary Maxwell equations, the time-harmonic Maxwell equations and the time-dependent Maxwell equations. These approximations have three novel features. First, the resulting discrete edge element systems can be solved by some existing preconditioned solvers with optimal convergence rate independent of nite element meshes, including the stationary Maxwell equations. Second, they ensure the optimal strong convergence of the Gauss’ laws in some appropriate norm, in addition to the standard optimal convergence in energy-norm, under the general weak regularity assumptions that hold for both convex and non-convex polyhedral domains and for the discontinuous coecients that may have large jumps across the interfaces between dierent media. Finally, no saddle-point discrete systems are needed to solve for the stationary Maxwell equations, unlike most existing edge element schemes.

Time-Splitting Methods to Solve the Stochastic Incompressible Stokes Equation

For the stochastic incompressible time-dependent Stokes equation, we study different time-splitting methods that decouple the computation of velocity and pressure iterates in every iteration step. Optimal strong convergence is shown for Chorin's time-splitting scheme in the case of solenoidal noise, while computational counterexamples show poor convergence behavior in the case of general stochastic forcing. This suboptimal performance may be traced back to the nonregular pressure process in the case of general noise. A modified version of the deterministic time-splitting method that distinguishes between the deterministic and stochastic pressure removes this deficiency, leading to optimal convergence behavior.

Nonparametric covariate adjustment for receiver operating characteristic curves

AbstractThe accuracy of a diagnostic test is typically characterized using the receiver operating characteristic (ROC) curve. Summarizing indexes such as the area under the ROC curve (AUC) are used to compare different tests as well as to measure the difference between two populations. Often additional information is available on some of the covariates which are known to influence the accuracy of such measures. The authors propose nonparametric methods for covariate adjustment of the AUC. Models with normal errors and possibly non‐normal errors are discussed and analyzed separately. Nonparametric regression is used for estimating mean and variance functions in both scenarios. In the model that relaxes the assumption of normality, the authors propose a covariate‐adjusted Mann–Whitney estimator for AUC estimation which effectively uses available data to construct working samples at any covariate value of interest and is computationally efficient for implementation. This provides a generalization of the Mann–Whitney approach for comparing two populations by taking covariate effects into account. The authors derive asymptotic properties for the AUC estimators in both settings, including asymptotic normality, optimal strong uniform convergence rates and mean squared error (MSE) consistency. The MSE of the AUC estimators was also assessed in smaller samples by simulation. Data from an agricultural study were used to illustrate the methods of analysis. The Canadian Journal of Statistics 38:27–46; 2010 © 2009 Statistical Society of Canada

The approximation of a Crank–Nicolson scheme for the stochastic Navier–Stokes equations

In this paper we prove the convergence of stochastic Navier–Stokes equations driven by white noise. A linearized version of the implicit Crank–Nicolson scheme is considered for the approximation of the solutions to the N–S equations. The noise is defined as the distributional derivative of a Wiener process and approximated by using the generalized L 2 -projection operator. Optimal strong convergence error estimates in the L 2 norm are obtained.

Strong convergence in nonparametric regression with truncated dependent data

In this paper we derive rates of uniform strong convergence for the kernel estimator of the regression function in a left-truncation model. It is assumed that the lifetime observations with multivariate covariates form a stationary α -mixing sequence. The estimation of the covariate’s density is considered as well. Under the assumption that the lifetime observations are bounded, we show that, by an appropriate choice of the bandwidth, both estimators of the covariate’s density and regression function attain the optimal strong convergence rate known from independent complete samples.

Testing serial correlation in semiparametric varying-coefficient partially linear EV models

This paper studies estimation and serial correlation test of a semiparametric varying-coefficient partially linear EV model of the form Y = Xτβ + Zτα(T) + e, ξ = X + η with the identifying condition E[(e, ητ)τ] = 0, Cov[(e, ητ)τ] = σ2Ip+1. The estimators of interested regression parameters β, and the model error variance σ2, as well as the nonparametric components α(T), are constructed. Under some regular conditions, we show that the estimators of the unknown vector β and the unknown parameter σ2 are strongly consistent and asymptotically normal and that the estimator of α(T) achieves the optimal strong convergence rate of the usual nonparametric regression. Based on these estimators and asymptotic properties, we propose the VN,p test statistic and empirical log-likelihood ratio statistic for testing serial correlation in the model. The proposed statistics are shown to have asymptotic normal or chi-square distributions under the null hypothesis of no serial correlation. Some simulation studies are conducted to illustrate the finite sample performance of the proposed tests.

Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems

We generalise the current theory of optimal strong convergence rates for implicit Euler-based methods by allowing for Poisson-driven jumps in a stochastic differential equation (SDE). More precisely, we show that under one-sided Lipschitz and polynomial growth conditions on the drift coefficient and global Lipschitz conditions on the diffusion and jump coefficients, three variants of backward Euler converge with strong order of one half. The analysis exploits a relation between the backward and explicit Euler methods.

The law of iterated logarithm of estimators for partially linear panel data models

This paper studies the asymptotic theory for a semiparametric partially linear panel data model with a one-way error component structure, which has a wide range of applications in many important areas. We establish the law of iterated logarithm of the feasible semiparametric generalized least squares estimator of the linear regression coefficients as well as the estimators of the one-way error variances. In addition, the optimal strong convergence rate of the estimator of the function for the nonlinear component is derived as well.

Estimation of a semiparametric varying-coefficient partially linear errors-in-variables model

This paper studies the estimation of a varying-coefficient partially linear regression model which is a generalization of the partially linear regression model and varying-coefficient regression model [Fan and Huang, Manuscript, University of North Carolina, Chapel Hill, USA, 2002]. We focus on the case where some covariates are measured with additive errors. The usual profile least squares and local polynomial estimations lead to biased estimators of the parametric and nonparametric components, respectively, when measurement errors are ignored. By correcting the attenuation we propose a modified profile least squares estimator for the parametric component and a local polynomial estimator for the nonparametric component. We show that the former is consistent, asymptotically normal and achieves the rate in the law of the iterated logarithm, and the latter achieves the optimal strong convergence rate of the usual nonparametric regression. In addition, a consistent estimator is also developed for the error variance. These results can be used to make asymptotically valid statistical inferences. Some simulation studies are conducted to illustrate the finite sample performance of the proposed estimators.

Galerkin Finite Element Methods for Stochastic Parabolic Partial Differential Equations

We study the finite element method for stochastic parabolic partial differential equations driven by nuclear or space-time white noise in the multidimensional case. The discretization with respect to space is done by piecewise linear finite elements, and in time we apply the backward Euler method. The noise is approximated by using the generalized L2-projection operator. Optimal strong convergence error estimates in the L2 and $\dot{H}^{-1}$ norms with respect to the spatial variable are obtained. The proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem. The computational analysis and numerical example are given.

Semidiscrete Galerkin Approximation for a Linear Stochastic Parabolic Partial Differential Equation Driven by an Additive Noise

We study the semidiscrete Galerkin approximation of a stochastic parabolic partial differential equation forced by an additive space-time noise. The discretization in space is done by a piecewise linear finite element method. The space-time noise is approximated by using the generalized L2 projection operator. Optimal strong convergence error estimates in the L2 and \(\dot{H}^{-1}\) norms with respect to the spatial variable are obtained. The proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem. The error estimates are applicable in the multi-dimensional case.

Kernel Density and Hazard Rate Estimation for Censored Data under α-Mixing Condition

We derive rates of uniform strong convergence for kernel density estimators and hazard rate estimators in the presence of right censoring. It is assumed that the failure times (survival times) form a stationary α-mixing sequence. Moreover, we show that, by an appropriate choice of the bandwidth, both estimators attain the optimal strong convergence rate known from independent complete samples. The results represent an improvement over that of Cai's paper (cf. Cai (1998b, J. Multivariate Anal., 67, 23–34)).