随机设计非线性混合模型的统计分析

本文研究了个体观察次数为随机的非线性 混合效应模型中参数的点估计以及区间估计. 在仅给出适当的矩条件下, 给出了固定效应、随机效应的方差阵以及误差方差的矩估计, 并证明了估计量的相合性及渐近正态性. 为给出误差方差以及随机效应方差分量的置信区间, 本文也给出了误差及随机效应的四阶矩估计. 随机模拟说明了方法的有效性.     

Adaptive sparse grid algorithms with applications to electromagnetic scattering under uncertainty

We discuss adaptive sparse grid algorithms for stochastic differential equations with a particular focus on applications to electromagnetic scattering by structures with holes of uncertain size, location, and quantity. Stochastic collocation (SC) methods are used in combination with an adaptive sparse grid approach based on nested Gauss-Patterson grids. As an error estimator we demonstrate how the nested structure allows an effective error estimation through Richardson extrapolation. This is shown to allow excellent error estimation and it also provides an efficient means by which to estimate the solution at the next level of the refinement. We introduce an adaptive approach for the computation of problems with discrete random variables and demonstrate its efficiency for scattering problems with a random number of holes. The results are compared with results based on Monte Carlo methods and with Stroud based integration, confirming the accuracy and efficiency of the proposed techniques.     

Normal distribution based pseudo ML for missing data: With applications to mean and covariance structure analysis

When missing data are either missing completely at random (MCAR) or missing at random (MAR), the maximum likelihood (ML) estimation procedure preserves many of its properties. However, in any statistical modeling, the distribution specification for the likelihood function is at best only an approximation to the real world. In particular, since the normal-distribution-based ML is typically applied to data with heterogeneous marginal skewness and kurtosis, it is necessary to know whether such a practice still generates consistent parameter estimates. When the manifest variables are linear combinations of independent random components and missing data are MAR, this paper shows that the normal-distribution-based MLE is consistent regardless of the distribution of the sample. Examples also show that the consistency of the MLE is not guaranteed for all nonnormally distributed samples. When the population follows a confirmatory factor model, and data are missing due to the magnitude of the factors, the MLE may not be consistent even when data are normally distributed. When data are missing due to the magnitude of measurement errors/uniqueness, MLEs for many of the covariance parameters related to the missing variables are still consistent. This paper also identifies and discusses the factors that affect the asymptotic biases of the MLE when data are not missing at random. In addition, the paper also shows that, under certain data models and MAR mechanism, the MLE is asymptotically normally distributed and the asymptotic covariance matrix is consistently estimated by the commonly used sandwich-type covariance matrix. The results indicate that certain formulas and/or conclusions in the existing literature may not be entirely correct.     

A novel stochastic locally transversal linearization (LTL) technique for engineering dynamical systems: Strong solutions

Most available integration techniques for stochastically driven engineering dynamical systems are based on stochastic Taylor expansions of the response variables and thus require numerical modelling of multiple stochastic integrals (MSI-s). Since the latter is an extremely involved numerical task and becomes inaccurate for higher level MSI-s, these methods fail to achieve an accuracy beyond a limited order. Recently, the first author has proposed a locally transversal linearization (LTL) technique that completely avoids the use of Taylor-like expansions in the construction of the integration map [Proc. Roy. Soc. Ser. A, 457 (2001) 539; Int. J. Numer. Methods Eng. 61 (2004) 764]. A crucial step in the implementation of the LTL method is to arrive at a conditionally linearized solution which is then made to transversally intersect the non-linear solution manifold in the associated phase space. The present paper is the first part of an investigation consisting of two parts to considerably simplify and numerically expedite the generation of the conditionally linearized solution without affecting the local and global error orders of the original LTL method. In particular, the derivation of the conditionally linear form of the stochastic differential equations is done in such a way that the corresponding fundamental solution matrix (and consequently its inverse) remains unchanged during the entire integration process. In this part of the work, only strong (path wise) stochastic solutions are constructed. Through formal error estimates, it is verified that the present version of the LTL method has the same error orders as its older counterpart. Further, a host of numerical examples on stochastically driven non-linear oscillators are presented to illustrate its superior computational speed and ease of implementation. In a companion paper related to this work, a weak form of the alternative LTL approach will be derived.     

Bayesian Variable Selection for Logistic Models Using Auxiliary Mixture Sampling

This article presents a Markov chain Monte Carlo algorithm for both variable and covariance selection in the context of logistic mixed effects models. This algorithm allows us to sample solely from standard densities with no additional tuning. We apply a stochastic search variable approach to select explanatory variables as well as to determine the structure of the random effects covariance matrix.

Prior determination of explanatory variables and random effects is not a prerequisite because the definite structure is chosen in a data-driven manner in the course of the modeling procedure. To illustrate the method, we give two bank data examples.     

Discrete-time reduced-order estimator design for bilinear stochastic systems with error covariance assignment

This paper is concerned with the design of reduced-order stateestimators for bilinear stochastic discrete-time systems subjectedto estimation error covariance assignment. The purpose of theproblem addressed is to design the reduced-order state estimators for the bilinear stochastic discrete-time systems such thatthe steady-state estimation error covariances achieve the prespecified values. A simple, effective matrix inequality approach is developed to solve this problem. Specifically, (1) the parameterisationof estimation error covariances that certain bilinear errordynamic processes may possess is presented, (2) the characterisationof all reduced-order state estimators that assign such errorcovariances is explicitly derived, and (3) the solvabilityof the assignability conditions is discussed. Furthermore,an illustrative example is used to demonstrate the effectivenessof the proposed design procedure.     

A conservative bound on the estimation error covariance matrix in the presence of correlated driving noise and correlated discrete measurement noise

In an earlier paper, the conservative and minimal bound to the crosscorrelation terms between estimation error and a random forcing function was presented. That bound was found to be a particular linear combination of the estimation error covariance and the forcing function covariance involving a free scalar parameter. The bound was then substituted for the cross-correlation terms in the differential equation for the estimation error covariance matrix in order to approximate its behavior between discrete measurement times. The time history of the free parameter which minimized a linear combination of the elements of the estimated covariance matrix at the next measurement time was found as the noniterative solution to an optimal control problem with a matrix state.In this paper, necessary and sufficient conditions are presented for the problem of minimizing a linear combination of the elements of the approximated estimation error covariance at the end of an interval in which are linearly incorporated a finite number of discrete vector measurements corrupted by white and/or correlated measurement noise. Although the determination of the optimal trajectory in general requires iteration, a particularly simple algorithm is presented. Numerical results are presented for the case of a satellite in a highly elliptic orbit about a model Earth.     

Study and modelling of two apple quality attributes: the soluble solids content and the firmness

Recently, the basic dynamics of fruit characteristics have been modelled using a stochastic approach. The time evolution of apple quality attributes was represented by means of a system of differential equations in which the initial conditions and model parameters are both random. In this work, a complete study of two apple quality attributes, the soluble solids content and the firmness, is carried out. For each of these characteristics, the system of differential equations is linear and the state variables and the parameters are represented as random variables with their statistical properties (mean values, variances, covariances, joint probability density function) known at the initial time. The dynamic behaviour of these statistical properties is analysed. The variance propagation algorithm is used to obtain an analytical expression of the dynamic behaviour of the mean value, the variance, the covariance and the probability density function. A Monte Carlo method and the Latin hypercube method were developed to obtain a numerical expression of the dynamic behaviour of these statistical quantities and particularly to follow the time evolution of joint probability density function which represents one but the best mean to characterize random phenomena linked with fruit quality attributes.     

An Adaptive Hybrid Spectral Method for Stochastic Helmholtz Problems

The implementation of an adaptive hybrid spectral method for Helmholtz equations with random parameters is addressed in this work. New error indicators for generalized polynomial chaos for stochastic approximations and spectral element methods for physical approximations are developed, and systematic adaptive strategies are proposed associated with these error indicators. Numerical results show that these error indicators provide effective estimates for the approximation errors, and the overall adaptive procedure results in efficient approximation method for the stochastic Helmholtz equations.     

Anticipative calculus for Lévy processes and stochastic differential equations*

We develop an anticipative calculus for Lévy processes with finite second moment for analysing anticipating stochastic differential equations. The calculus is based on the chaos expansion of square-integrable random variables in terms of iterated integrals with respect to the compensated Poisson random measure. We define a space of smooth and generalized random variables in terms of such chaos expansions, and present anticipative stochastic integration, the Wick product and the so-called 𝒮-transform. These concepts serve as tools for studying general Wick type stochastic differential equations with anticipative initial conditions. We apply the 𝒮-transform to find the unique solutions to a class of linear stochastic differential equations. The solutions can be expressed in terms of the Wick product.     

Local error estimates for moderately smooth problems: Part II—SDEs and SDAEs with small noise

The paper consists of two parts. In the first part of the paper, we proposed a procedure to estimate local errors of low order methods applied to solve initial value problems in ordinary differential equations (ODEs) and index-1 differential-algebraic equations (DAEs). Based on the idea of Defect Correction we developed local error estimates for the case when the problem data is only moderately smooth, which is typically the case in stochastic differential equations. In this second part, we will consider the estimation of local errors in context of mean-square convergent methods for stochastic differential equations (SDEs) with small noise and index-1 stochastic differential-algebraic equations (SDAEs). Numerical experiments illustrate the performance of the mesh adaptation based on the local error estimation developed in this paper. The first author acknowledges support by the BMBF-project 03RONAVN and the second author support by the Austrian Science Fund Project P17253.     

On the linearization of the equations of motion of a rotating disk

In this paper, we present a consistent approach to reduce the fully nonlinear equations of a rotating disk to the classical linear equation derived by Lamb and Southwell and the nonlinear equations derived by Nowinski. The approach recognizes the fact that the out-of-plane deflection and the in-plane deflections are of different orders of magnitude. By using the ratio between the plate thickness and the outer radius as a measurement and carefully examining the reasonable magnitudes of all the variables involved, the fully nonlinear equations can be non-dimensionalized with all the terms being sorted according to their orders of magnitude. It is found that the classical linear equation derived by Lamb and Southwell can be recovered if all the terms of the lowest order of magnitude in the fully nonlinear equations are retained. If all the terms of the lowest two orders of magnitude are retained, Nowinski’s equations can then be recovered. Furthermore, the terms arising from in-plane deformation and rotary inertia are of the highest order and can be ignored in most of the applications.     

A new formulation of some discrete-time stochastic-parameter state estimation problems

Linear unbiased full-order state estimation problem for discrete-time models with stochastic parameters and additive finite energy type disturbance signals is reformulated in terms of linear matrix inequalities. Two estimation problems that are considered are the design for mean-square bounded estimation error and the design for the mean-square stochastic version of the suboptimal H_∞ estimator. These two designs are shown to apply to both the estimation with random sensor delay and estimation under observation uncertainty.     

A posteriori error estimation and error control for finite element approximations of the time-dependent Navier–Stokes equations

We present an approach to estimate numerical errors in finite element approximations of the time-dependent Navier–Stokes equations along with a strategy to control these errors. The error estimators and the error control procedure are based on the residuals of the Navier–Stokes equations, which are shown to be comparable to error components in the velocity variable. The present methodology applies to the estimation of numerical errors due to the spatial discretization only. Its performance is demonstrated for two-dimensional channel flows past a cylinder in the periodic regime.     

Sequences of bias-adjusted covariance matrix estimators under heteroskedasticity of unknown form

The linear regression model is commonly used by practitioners to model the relationship between the variable of interest and a set of explanatory variables. The assumption that all error variances are the same, known as homoskedasticity, is oftentimes violated when cross sectional data are used. Consistent standard errors for the ordinary least squares estimators of the regression parameters can be computed following the approach proposed by White (Econometrica 48:817–838, 1980). Such standard errors, however, are considerably biased in samples of typical sizes. An improved covariance matrix estimator was proposed by Qian and Wang (J Stat Comput Simul 70:161–174, 2001). In this paper, we improve upon the Qian–Wang estimator by defining a sequence of bias-adjusted estimators with increasing accuracy. The numerical results show that the Qian–Wang estimator is typically much less biased than the estimator proposed by Halbert White and that our correction to the former can be quite effective in small samples. Finally, we show that the Qian–Wang estimator can be generalized into a broad class of heteroskedasticity-consistent covariance matrix estimators, and our results can be easily extended to such a class of estimators.     

误差服从多元t分布的一类线性模型下参数估计的若干注记

研究一类线性模型下参数估计的若干问题．这类模型包含了多个因变量线性模型、增长曲线模型、扩充的增长曲线模型、似乎不相关回归方程组、方差分量模型等常用模型．在这类线性模型下,证明了当误差服从多元t分布时与误差服从多元正态分布时,具有相同的完全统计量和无偏估计,且在后一种情况下的充分统计量必为前一种情况下的充分统计量．对于带有多种协方差结构的前述几种模型,把在误差服从多元正态分布下,相应的协方差阵及有关参数的一致最小风险无偏(UMRU)估计存在性的结论推广到了相应的误差服从多元t分布情形．此外,对于误差服从多元t分布的这类统一的线性模型,给出了回归系数的线性可估函数的无偏估计的协方差阵的C-R下界．     

Local error estimates for moderately smooth problems: Part I – ODEs and DAEs

The paper consists of two parts. In the first part, we propose a procedure to estimate local errors of low order methods applied to solve initial value problems in ordinary differential equations (ODEs) and index 1 differential-algebraic equations (DAEs). Based on the idea of defect correction we develop local error estimates for the case when the problem data is only moderately smooth. Numerical experiments illustrate the performance of the mesh adaptation based on the error estimation developed in this paper. In the second part of the paper, we will consider the estimation of local errors in context of stochastic differential equations with small noise. AMS subject classification (2000)  65L06, 65L80, 65L50, 65L05     

变系数混合效应模型

本文研究了下列变系数混合效应模型: $y_{ij}=z_{ij}^{\tau}b_i+x_{ij}^{\tau}\beta(w_{ij}) +\xe_{ij},\;i=1,\cdots,m;\;j=1,\cdots,n_i$, 其中$b_i$为i.i.d.期望为$\xt$, 协方差阵为$\xs^2_bI_q$的随机效应向量, $\xe_{ij}$是i.i.d.期望为零, 具有有限方差的随机误差. 文中我们不仅给出了函数系数向量$\xb(\cdot)$的局部多项式估计, 同时给出了随机效应期望、方差和随机误差方差的估计, 并给出了这些估计量的渐进正态性和相合性, 研究结果表明了这些估计量的可靠性.     

Rounded data analysis based on multi-layer ranked set sampling

Observations of sampling are often subject to rounding, but are modeled as though they were unrounded. This paper examines the impact of rounding errors on parameter estimation with multi-layer ranked set sampling. It shows that the rounding errors seriously distort the behavior of covariance matrix estimate, and lead to inconsistent estimation. Taking this into account, we present a new approach to implement the estimation for this model, and further establish the strong consistency and asymptotic normality of the proposed estimators. Simulation experiments show that our estimates based on rounded multi-layer ranked set sampling are always more efficient than those based on rounded simple random sampling.     

Efficient Bayesian Inference for Multivariate Factor Stochastic Volatility Models

We discuss efficient Bayesian estimation of dynamic covariance matrices in multivariate time series through a factor stochastic volatility model. In particular, we propose two interweaving strategies to substantially accelerate convergence and mixing of standard MCMC approaches. Similar to marginal data augmentation techniques, the proposed acceleration procedures exploit nonidentifiability issues which frequently arise in factor models. Our new interweaving strategies are easy to implement and come at almost no extra computational cost; nevertheless, they can boost estimation efficiency by several orders of magnitude as is shown in extensive simulation studies. To conclude, the application of our algorithm to a 26-dimensional exchange rate dataset illustrates the superior performance of the new approach for real-world data. Supplementary materials for this article are available online.