Mean quadratic error of an estimate of splines of first order in a model of linear regression

We study the mean quadratic error of an estimate of splines of the first order, which is obtained by the method of least squares under the assumption that the data represents a superposition of proper values of a spline and a white noise. A quantitative formula for the quadratic mean error is found and its asymptotics is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 3, pp. 429–432, March, 1991.     

Multivariate Cramér-Rao inequality for prediction and efficient predictors

We derive and discuss a matricial Cramér-Rao type inequality for the quadratic prediction error matrix. A study of the attainment of the bound follows. Then we introduce an unbiased predictor for a bivariate Poisson process and prove that it is efficient, i.e. its quadratic error attains the Cramér-Rao bound.     

平衡随机模型中方差分量的非负估计

众所周知, 对于平衡随机模型, 方差分量的方差分析估计为一致最小方差无偏估计. 本文基于方差分量的方差分析估计, 构造了一个二次不变估计类, 它包含了一些常用重要估计. 证明了该估计类在一定条件下在均方误差意义下一致优于方差分析估计, 并在此估计类基础上, 给出了方差分量的两种非负估计, 它们在均方误差意义下分别一致优于方差分析估计和限制极大似然估计, 且有显式解、容易计算.     

Discrimination with jointly equicorrelated multi-level multivariate data

In this article we study a linear as well as a quadratic discriminant function for multi-level multivariate repeated measurement data under the assumption of multivariate normality. We assume that the m-variate observations have jointly equicorrelated covariance structure in addition to a Kronecker product structure on the mean vector. The new discriminant functions are very effective in discriminating individuals when the number of observations is very small. The proposed classification rules are demonstrated on a real data set. The error rates of the proposed classification rules are found to be much less than the error rates of the traditional classification rules, when in fact the traditional classification rules fail most of the time owing to the small sample sizes.     

Mathematical analysis of Zienkiewicz—Zhu's derivative patch recovery technique

Zienkiewicz-Zhu's derivative patch recovery technique is analyzed for general quadrilateral finite elements. Under certain regular conditions on the meshes, the arithmetic mean of the absolute error of the recovered gradient at the nodal points is superconvergent for the second-order elliptic operators. For rectangular meshes and the Laplacian, the recovered gradient is superconvergent in the maximum norm at the nodal points. Furthermore, it is proved for a model two-point boundary-value problem that the recovery technique results in an “ultra-convergent” derivative recovery at the nodal points for quadratic finite elements when uniform meshes are used. © 1996 John Wiley & Sons, Inc.     

Double k-Class Estimators in Regression Models with Non-spherical Disturbances

In this paper, we consider a family of feasible generalised double k-class estimators in a linear regression model with non-spherical disturbances. We derive the large sample asymptotic distribution of the proposed family of estimators and compare its performance with the feasible generalized least squares and Stein-rule estimators using the mean squared error matrix and risk under quadratic loss criteria. A Monte-Carlo experiment investigates the finite sample behaviour of the proposed family of estimators.     

A lower bound on the performance of the quadratic discriminant function

The quadratic discriminant function is often used to separate two classes of points in a multidimensional space. When the two classes are normally distributed, this results in the optimum separation. In some cases however, the assumption of normality is a poor one and the classification error is increased. The current paper derives an upper bound for the classification error due to a quadratic decision surface. The bound is strict when the class means and covariances and the quadratic discriminant surface satisfy certain specified symmetry conditions.     

Hidden conic quadratic representation of some nonconvex quadratic optimization problems

The problem of minimizing a quadratic objective function subject to one or two quadratic constraints is known to have a hidden convexity property, even when the quadratic forms are indefinite. The equivalent convex problem is a semidefinite one, and the equivalence is based on the celebrated S-lemma. In this paper, we show that when the quadratic forms are simultaneously diagonalizable (SD), it is possible to derive an equivalent convex problem, which is a conic quadratic (CQ) one, and as such is significantly more tractable than a semidefinite problem. The SD condition holds for free for many problems arising in applications, in particular, when deriving robust counterparts of quadratic, or conic quadratic, constraints affected by implementation error. The proof of the hidden CQ property is constructive and does not rely on the S-lemma. This fact may be significant in discovering hidden convexity in some nonquadratic problems.     

Exact order of Hoffman’s error bounds for elliptic quadratic inequalities derived from vector-valued Chebyshev approximation

In this paper, we introduce the exact order of Hoffman’s error bounds for approximate solutions of elliptic quadratic inequalities. Elliptic quadratic inequalities are closely related to Chebyshev approximation of vector-valued functions (including complex-valued functions). The set of Chebyshev approximations of a vector-valued function defined on a finite set is shown to be Hausdorff strongly unique of order exactly 2 ^s for some nonnegative integer s. As a consequence, the exact order of Hoffman’s error bounds for approximate solutions of elliptic quadratic inequalities is exactly 2 ^-s for some nonnegative integer s. The integer s, called the order of deficiency (which is computable), quantifies how much the Abadie constraint qualification is violated by the elliptic quadratic inequalities. Received: April 15, 1999 / Accepted: February 21, 2000?Published online July 20, 2000     

生长曲线模型中参数的Bayes线性无偏估计

本文在平方损失下导出了生长曲线模型中参数的Bayes线性无偏估计(LUE), 并在均方误差矩阵(MSEM)准则下研究了Bayes LUE相对于广义最小二乘估计(GLSE)的优良性. 对于非满秩情形,获得了可估函数的Bayes LUE并讨论了其优良性问题.     

热传导型半导体器件的二次元特征有限体积元方法及分析:H1模估计

该文构造了热传导型半导体器件的全离散特征有限体积元格式,将特征线方法与有限体积元方法相结合, 采用Lagrange型分片二次多项式空间和分片常数函数空间分别作为试探函数和检验函数空间,并进行误差分析,得到了最优阶H¹模误差估计结果.     

Stochastic moment problem and hedging of generalized Black-Scholes options

In mathematical finance one is interested in the quadratic error which occurs while replacing a continuously adjusted portfolio by a discretely adjusted one. We first study higher order approximations of stochastic integrals. Then we apply the results to quantify quadratic error which occurs in estimating the discretely adjusted hedging risk in pricing European options in a generalized Black-Scholes market.     

The Durand-Kerner polynomials roots-finding method in case of multiple roots

The convergence of the Durand-Kerner algorithm is quadratic in case of simple roots but only linear in case of multiple roots. This paper shows that, at each step, the mean of the components converging to the same root approaches it with an error proportional to the square of the error at the previous step. Since it is also shown that it is possible to estimate the multiplicity order of the roots during the algorithm, a modification of the Durand-Kerner iteration is proposed to preserve a quadratic-like convergence even in case of multiple zeros.This work is supported in part by the Research Program C3 of the French CNRS and MEN, and by the Direction des Recherches et Etudes Techniques (DGA).     

The Pearson system of utility functions

This paper describes a parametric family of utility functions for decision analysis. The parameterization embeds the HARA class in a four-parameter representation for the risk aversion function. The resulting utility functions can have only four shapes: concave, convex, S-shaped, and reverse S-shaped. This makes the family suited for both expected utility and prospect theory. The paper also describes an alternative technique to estimate the four parameters from elicited utilities, which is simpler than standard fitting by minimization of the mean quadratic error.     

Traffic queue estimation

This paper deals with the urban traffic queue modelling and estimation. A general formulation of delays on a road section is used to link measured speeds at its entrance to the current queue state. This measurement model is applied to correct a posteriori the queue state distribution whose a priori evolution is ruled by a time varying Markov chain. The resulting predictor-corrector is tested with a microscopic simulation tool. Results in fixed plan control show that for long road sections (200 meters) the estimation error is unacceptable whereas for short road sections (50–100 meters) the average quadratic error is about 1 vehicle². In real time control conditions, in spite of a significant increase of the estimation quadratic error, results show a delay benefit of 15% with respect to the fixed plan control.     

Continuous piecewise linear finite elements for the Kirchhoff–Love plate equation

A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff–Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the basic plate triangle (BPT). Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and L ² norm. Numerical results indicate that the Morley reconstruction/BPT does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.     

Sampling designs for regression coefficient estimation with correlated errors

The problem of estimating regression coefficients from observations at a finite number of properly designed sampling points is considered when the error process has correlated values and no quadratic mean derivative. Sacks and Ylvisaker (1966,Ann. Math. Statist.,39, 66–89) found an asymptotically optimal design for the best linear unbiased estimator (BLUE). Here, the goal is to find an asymptotically optimal design for a simpler estimator. This is achieved by properly adjusting the median sampling design and the simpler estimator introduced by Schoenfelder (1978, Institute of Statistics Mimeo Series No. 1201, University of North Carolina, Chapel Hill). Examples with stationary (Gauss-Markov) and nonstationary (Wiener) error processes and with linear and nonlinear regression functions are considered both analytically and numerically.Research supported by the Air Force Office of Scientific Research Contract No. 91-0030.     

Error Bounds for the Solution Sets of Quadratic Complementarity Problems

In this article, two types of fractional local error bounds for quadratic complementarity problems are established, one is based on the natural residual function and the other on the standard violation measure of the polynomial equalities and inequalities. These fractional local error bounds are given with explicit exponents. A fractional local error bound with an explicit exponent via the natural residual function is new in the tensor/polynomial complementarity problems literature. The other fractional local error bounds take into account the sparsity structures, from both the algebraic and the geometric perspectives, of the third-order tensor in a quadratic complementarity problem. They also have explicit exponents, which improve the literature significantly.     

A global error bound for quadratic perturbation of linear programs

We prove a global error bound result on the quadratic perturbation of linear programs. The error bound is stated in terms of function values.     

Statistical Analysis of the Fractional Ornstein–Uhlenbeck Type Process

We consider the fractional analogue of the Ornstein–Uhlenbeck process, that is, the solution of a one-dimensional homogeneous linear stochastic differential equation driven by a fractional Brownian motion in place of the usual Brownian motion. The statistical problem of estimation of the drift and variance parameters is investigated on the basis of a semimartingale which generates the same filtration as the observed process. The asymptotic behaviour of the maximum likelihood estimator of the drift parameter is analyzed. Strong consistency is proved and explicit formulas for the asymptotic bias and mean square error are derived. Preparing for the analysis, a change of probability method is developed to compute the Laplace transform of a quadratic functional of some auxiliary process. This revised version was published online in August 2006 with corrections to the Cover Date.