Asymptotic normality of cumulant spectral estimates

The paper considers higher-order cumulant spectral estimates obtained by directly Fourier transforming weighted cumulant estimates. Such estimates computationally are different from those based on the finite Fourier transform. These estimates can be looked at continuously as well as directly on submanifolds. The estimates of cumulants are based on unbiased moment estimates. Asymptotic normality is obtained for these estimates and is based on a strong mixing condition and only a finite number of cumulant summability conditions.     

Universal potential estimates

We prove a class of endpoint pointwise estimates for solutions to quasilinear, possibly degenerate elliptic equations in terms of linear and nonlinear potentials of Wolff type of the source term. Such estimates allow to bound size and oscillations of solutions and their gradients pointwise, and entail in a unified approach virtually all kinds of regularity properties in terms of the given datum and regularity of coefficients. In particular, local estimates in Hölder, Lipschitz, Morrey and fractional spaces, as well as Calderón–Zygmund estimates, follow as a corollary in a unified way. Moreover, estimates for fractional derivatives of solutions by mean of suitable linear and nonlinear potentials are also implied. The classical Wolff potential estimate by Kilpeläinen & Malý and Trudinger & Wang as well as recent Wolff gradient bounds for solutions to quasilinear equations embed in such a class as endpoint cases.     

Resolvent estimates in homogenisation of periodic problems of fractional elasticity

We provide operator-norm convergence estimates for solutions to a time-dependent equation of fractional elasticity in one spatial dimension, with rapidly oscillating coefficients that represent the material properties of a viscoelastic composite medium. Assuming periodicity in the coefficients, we prove operator-norm convergence estimates for an operator fibre decomposition obtained by applying to the original fractional elasticity problem the Fourier–Laplace transform in time and Gelfand transform in space. We obtain estimates on each fibre that are uniform in the quasimomentum of the decomposition and in the period of oscillations of the coefficients as well as quadratic with respect to the spectral variable. On the basis of these uniform estimates we derive operator-norm-type convergence estimates for the original fractional elasticity problem, for a class of sufficiently smooth densities of applied forces.     

Interior estimates for second-order elliptic differential (or finite-difference) equations via the maximum principle

Second-order elliptic operators are transformed into second-order elliptic operators of a higher dimensionality acting on differences of functions. Applying the maximum principle to the resulting operators yields various a-priori pointwise estimates to difference-quotients of solutions of elliptic differential, as well as finite-difference, equations. We derive Schauder estimates, estimates for equations with discontinuous coefficients, and other estimates.     

Window estimates of location and scale with applications to the cauchy distribution

Location and scale parameters are estimated via “window estimates”. The consistency and asymptotic normality of the estimates are established. The special case of the Cauchy distribution is considered, where the estimates are shown to have the same asymptotic distribution as the maximum-likelihood estimates. Additional applications are given for the Pearson type-VII distributions. The estimates have the advantages of ease of computation and high asymptotic efficiencies for certain heavy-tailed distributions.     

Error analysis of variable degree mixed methods for elliptic problems via hybridization

A new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree Raviart-Thomas method for second order elliptic problems. The approach gives error estimates for the Lagrange multipliers without using error estimates for the other variables. Error estimates for the primal and flux variables then follow from those for the Lagrange multipliers. In contrast, traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the Lagrange multipliers. The new approach not only gives new error estimates for the new variable degree Raviart-Thomas method, but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates. The error analysis is achieved by using a variational characterization of the Lagrange multipliers wherein the other unknowns do not appear. This approach can be applied to other hybridized mixed methods as well.

     

Linear estimates of accuracy for approximate solutions of inverse problems

In this paper, we explore the question of which non-linear inverse problems, which are solved by a selected regularization method, may have so-called linear a priori accuracy estimates – that is, the accuracy of corresponding approximate solutions linearly depends on the error level of the data. In particular, we prove that if such a linear estimate exists, then the inverse problem under consideration is well posed, according to Tikhonov. For linear inverse problems, we find that the existence of linear estimates lead to, under some assumptions, the well-posedness (according to Tikhonov) on the whole space of solutions. Moreover, we consider a method for solving inverse problems with guaranteed linear estimates, called the residual method on the correctness set (RMCS). The linear a priori estimates of absolute and relative accuracy for the RMCS are presented, as well as analogous a posteriori estimates. A numerical illustration of obtaining linear a priori estimates for appropriate parametric sets of solutions using RMCS is given in comparison with Tikhonov regularization. The a posteriori estimates are calculated on these parametric sets as well.     

Error estimates of the DtN finite element method for the exterior Helmholtz problem

A priori error estimates are established for the DtN (Dirichlet-to-Neumann) finite element method applied to the exterior Helmholtz problem. The error estimates include the effect of truncation of the DtN boundary condition as well as that of the finite element discretization. A property of the Hankel functions which plays an important role in the proof of the error estimates is introduced.     

Lipschitz Estimates in Almost‐Periodic Homogenization

We establish uniform Lipschitz estimates for second‐order elliptic systems in divergence form with rapidly oscillating, almost‐periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded C^1,α domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the results for the Neumann conditions are new even in the periodic setting, since we can treat nonsymmetric coefficients. We also obtain uniform W^1,p estimates.© 2016 Wiley Periodicals, Inc.     

Error estimates for linear systems with applications to regularization

In this paper, we discuss several (old and new) estimates for the norm of the error in the solution of systems of linear equations, and we study their properties. Then, these estimates are used for approximating the optimal value of the regularization parameter in Tikhonov’s method for ill-conditioned systems. They are also used as a stopping criterion in iterative methods, such as the conjugate gradient algorithm, which have a regularizing effect. Several numerical experiments and comparisons with other procedures show the effectiveness of our estimates.     

Efficiencies for window estimates of the parameters of the Cauchy distribution

Higgins and Tichenor [Appl. Math. and Comp. 3 (1977), 113-126] considered “window estimates” of location and reciprocal scale parameters for a general class of distributions and showed them to be asymptotically efficient for the Cauchy distribution. In this study, efficiencies of these estimates for the Cauchy distribution are investigated for small and moderate sample sizes by Monte Carlo methods. For n?40, window estimates of location are nearly optimal, and for n?20, they compare favorably with other easy-to-compute estimates. Window estimates of reciprocal scale are very good even for small samples and are nearly optimal for n?10. Thus, window estimates appear to have high efficiency for moderate as well as large sample sizes. Approximate normality is also investigated. The estimate of location converges rapidly to normality, whereas the estimate of reciprocal scale does not.     

Convex functions on Grassmannian manifolds and Lawson-Osserman problem

We derive estimates of the Hessian of two smooth functions defined on Grassmannian manifold. Based on it, we can derive curvature estimates for minimal submanifolds in Euclidean space via Gauss map as in [Y.L. Xin, Ling Yang, Curvature estimates for minimal submanifolds of higher codimension, arXiv: 0709.3686; 24]. In this way, the result for Bernstein type theorem done by Jost and the first author could be improved.     

Bochner–Weitzenböck formula and Li–Yau estimates on Finsler manifolds

We prove the Bochner–Weitzenböck formula for the (nonlinear) Laplacian on general Finsler manifolds and derive Li–Yau type gradient estimates as well as parabolic Harnack inequalities. Moreover, we deduce Bakry–Émery gradient estimates. All these estimates depend on lower bounds for the weighted flag Ricci tensor.     

二维区域上椭圆均匀化问题的收敛率（英文）

In this paper, we study the convergence rates of solutions for second order elliptic equations with rapidly oscillating periodic coefficients in two-dimensional domain. We use an extension of the "mixed formulation" approach to obtain the representation formula satisfied by the oscillatory solution and homogenized solution by means of the particularity of solutions for equations in two-dimensional case. Then we utilize this formula in combination with the asymptotic estimates of Green or Neumann functions for operators and uniform regularity estimates of solutions to obtain convergence rates in L~p for solutions as well as gradient error estimates for Dirichlet or Neumann problems respectively.     

Impacts of random noise and specification on estimates of capacity derived from data envelopment analysis

Data envelopment analysis (DEA) is widely used to estimate the efficiency of firms and has also been proposed as a tool to measure technical capacity and capacity utilization (CU). Random variation in output data can lead to downward bias in DEA estimates of efficiency and, consequently, upward bias in estimates of technical capacity. This can be particularly problematic for industries such as agriculture, aquaculture and fisheries where the production process is inherently stochastic due to environmental influences. This research uses Monte Carlo simulations to investigate possible biases in DEA estimates of technically efficient output and capacity output attributable to noisy data and investigates the impact of using a model specification that allows for variable returns to scale (VRS). We demonstrate a simple method of reducing noise induced bias when panel data is available. We find that DEA capacity estimates are highly sensitive to noise and model specification. Analogous conclusions can be drawn regarding DEA estimates of average efficiency.     

A posteriori error estimates with the finite element method of lines for a Sobolev equation

A posteriori error estimates for semidiscrete finite element methods for a nonlinear Sobolev equation are considered. The error estimates are obtained by solving local nonlinear or linear pseudo‐parabolic equations for corrections to the solution on each element. The ratios of these estimates and the true errors are proved to converge to 1, implying that the estimates can be used as indicators in adaptive schemes for the problem. Numerical results underline our theoretical results. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Landau-Fermi-Dirac方程的整体经典解

研究了周期区域上平衡态附近Landau-Fermi-Dirac方程的Cauchy问题.利用宏观-微观分解以及局部的守恒律得到一致空间能量估计.接着结合对非线性碰撞算子的细致估计,推导了包含随时间演化的等价瞬时能量的非线性能量估计,进而得到一致的先验估计.最后通过局部存在性、一致的先验估计以及连续性技巧,得到了Landau-Fermi-Dirac方程平衡态附近整体光滑解的存在性.     

Two-sided estimates for constants in the Marcinkiewicz inequalities

For symmetric independent random values, sharp estimates for constants in the Marcinkiewicz inequalities are derived. Estimates of the left-hand and right-hand constants are roughly equivalent. Similar estimates for selfnormalized sums and new comparative estimates of increasing for some symmetric polynomial are derived as well. Bibliography: 14 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 361, 2008, pp. 45–56.     

On the estimation of nonlinear time series models

In the present paper, a framework for parametric estimation in nonlinear time series is developed. Strong consistency and asymptotic normality of minimum Hellinger distance estimates for a determined class of nonlinear models are investigated. The main Interest for these estimates is motivated by their robustness under perturbations as it has been emphazized in Beran [2]. The first part of the paper is devoted to the study of some probabilistic properties which ensure the existence and the optimal properties of the estimates     

A Bidding Simulation for Training Estimators

A simulation for training estimators and managers is described. The game simulates a bidding situation in the construction industry, with the participants split into teams which bid against each other for government contracts. The contracts are differentiated in terms of workload implications and location, and the teams are encouraged to use discriminating bidding strategies. The cost estimates provided include substantial uncertainty, and the teams can purchase more accurate estimates as well as buying competitive information. As a result of playing the game, participants appreciated the importance of expenditure on estimates, keeping good records, and using simple bidding models.