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141.
Rudolf Beran 《Annals of the Institute of Statistical Mathematics》2008,60(4):843-864
The unknown matrix M is the mean of the observed response matrix in a multivariate linear model with independent random errors. This paper constructs
regularized estimators of M that dominate, in asymptotic risk, least squares fits to the model and to specified nested submodels. In the first construction,
the response matrix is expressed as the sum of orthogonal components determined by the submodels; each component is replaced
by an adaptive total least squares fit of possibly lower rank; and these fits are then summed. The second, lower risk, construction
differs only in the second step: each orthogonal component is replaced by a modified Efron-Morris fit before summation. Singular
value decompositions yield computable formulae for the estimators and their asymptotic and estimated risks. In the asymptotics,
the row dimension of M tends to infinity while the column dimension remains fixed. Convergences are uniform when signal-to-noise ratio is bounded.
This research was supported in part by National Science Foundation Grant DMS 0404547. 相似文献
142.
Achiya Dax 《Mathematical Programming》1986,36(1):72-80
Thekey problem of the Euclidean multifacility location (EMFL) problem is to decide whether a givendead point is optimal. If it is not optimal, we wish to compute a descent direction. This paper extends the optimality conditions of
Calamai and Conn and Overton to the case when the rows of the active constraints matrix are linearly dependent. We show that
linear dependence occurs wheneverG, the graph of the coinciding facilities, has a cycle. In this case the key problem is formulated as a linear least squares
problem with bounds on the Euclidean norms of certain subvectors. 相似文献
143.
Josef Bukac 《逼近论及其应用》2008,(4):330-335
Minimization of the weighted nonlinear sum of squares of differences may be converted to the minimization of sum of squares. The Gauss-Newton method is recalled and the length of the step of the steepest descent method is determined by substituting the steepest descent direction in the Gauss-Newton formula. The existence of minimum is shown. 相似文献
144.
E. Demidenko 《Journal of Optimization Theory and Applications》2008,136(3):375-395
We develop criteria for the existence and uniqueness of the global minima of a continuous bounded function on a noncompact
set. Special attention is given to the problem of parameter estimation via minimization of the sum of squares in nonlinear
regression and maximum likelihood. Definitions of local convexity and unimodality are given using the level set. A fundamental
theorem of nonconvex optimization is formulated: If a function approaches the minimal limiting value at the boundary of the
optimization domain from below and its Hessian matrix is positive definite at the point where the gradient vanishes, then
the function has a unique minimum. It is shown that the local convexity level of the sum of squares is equal to the minimal
squared radius of the regression curvature. A new multimodal function is introduced, the decomposition function, which can
be represented as the composition of a convex function and a nonlinear function from the argument space to a space of larger
dimension. Several general global criteria based on majorization and minorization functions are formulated. 相似文献
145.
146.
147.
Yutaka Kano 《Annals of the Institute of Statistical Mathematics》1986,38(1):57-68
Summary This paper is concerned with the consistency of estimators in a single common factor analysis model when the dimension of
the observed vector is not fixed. In the model several conditions on the sample sizen and the dimensionp are established for the least squares estimator (L.S.E.) to be consistent. Under some assumptions,p/n→0 is a necessary and sufficient condition that the L.S.E. converges in probability to the true value. A sufficient condition
for almost sure convergence is also given. 相似文献
148.
149.
This paper considers numerical simulation of time‐dependent non‐linear partial differential equation resulting from a single non‐linear conservation law in h, p, k mathematical and computational framework in which k=(k1, k2) are the orders of the approximation spaces in space and time yielding global differentiability of orders (k1?1) and (k2?1) in space and time (hence k‐version of finite element method) using space–time marching process. Time‐dependent viscous Burgers equation is used as a specific model problem that has physical mechanism for viscous dissipation and its theoretical solutions are analytic. The inviscid form, on the other hand, assumes zero viscosity and as a consequence its solutions are non‐analytic as well as non‐unique (Russ. Math. Surv. 1962; 17 (3):145–146; Russ. Math. Surv. 1960; 15 (6):53–111). In references (Russ. Math. Surv. 1962; 17 (3):145–146; Russ. Math. Surv. 1960; 15 (6):53–111) authors demonstrated that the solutions of inviscid Burgers equations can only be approached within a limiting process in which viscosity approaches zero. Many approaches based on artificial viscosity have been published to accomplish this including more recent work on H(Div) least‐squares approach (Commun. Pure Appl. Math. 1965; 18 :697–715) in which artificial viscosity is a function of spatial discretization, which diminishes with progressively refined discretizations. The thrust of the present work is to point out that: (1) viscous form of the Burgers equation already has the essential mechanism of viscosity (which is physical), (2) with progressively increasing Reynolds (Re) number (thereby progressively reduced viscosity) the solutions approach that of the inviscid form, (3) it is possible to compute numerical solutions for any Re number (finite) within hpk framework and space–time least‐squares processes, (4) the space–time residual functional converges monotonically and that it is possible to achieve the desired accuracy, (5) space–time, time marching processes utilizing a single space–time strip are computationally efficient. It is shown that viscous form of the Burgers equation without linearizing provides a physical and viablemechanism for approaching the solutions of inviscid form with progressively increasing Re. Numerical studies are presented and the computed solutions are compared with published work. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
150.