Rate optimal estimation with the integration method in the presence of many covariates

For multivariate regressors, integrating the Nadaraya–Watson regression smoother produces estimators of the lower-dimensional marginal components that are asymptotically normally distributed, at the optimal rate of convergence. Some heuristics, based on consistency of the pilot estimator, suggested that the estimator would not converge at the optimal rate of convergence in the presence of more than four covariates. This paper shows first that marginal integration with its internally normalized counterpart leads to rate-optimal estimators of the marginal components. We introduce the necessary modifications and give central limit theorems. Then, it is shown that the method apply also to more general models, in particular we discuss feasible estimation of partial linear models. The proofs reveal that the pilot estimator shall over-smooth the variables to be integrated, and, that the resulting estimator is itself a lower-dimensional regression smoother. Hence, finite sample properties of the estimator are comparable to those of low-dimensional nonparametric regression. Further advantages when starting with the internally normalized pilot estimator are its computational attractiveness and better performance (compared to its classical counterpart) when the covatiates are correlated and nonuniformly distributed. Simulation studies underline the excellent performance in comparison with so far known methods.     

An iterative algorithm for estimation of signals from analytical instruments in the presence of constraints

Translated from Pryamye i Obratnye Zadachi Matematicheskoi Fiziki, pp. 45–48, 1991.     

Lectures on the theory of estimation of many parameters

This article is based on the series of lectures given by Prof. Charles Stein of Stanford University at LOMI AN SSSR in the fall of 1976. The first three lectures are concerned with the estimation of the mean vector of a multivariate normal distribution with quadratic loss function. James-Stein estimators are considered and their relation to Bayesian estimators is discussed. The problem of estimating the covariance matrix of the normal distribution and the estimation of the entropy of a multinomial distribution are considered in the following two lectures. The final lecture discusses several problems related to the estimation of multivariate parameters and poses some unsolved problems.Published in Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 74, pp. 4–65, 1977.In conclusion of this series of lectures, I would like to acknowledge the assistance of M. Ermakov, A. Borodin, and A. Makshanov in translating my lectures into Russian.     

Strongly optimal algorithms and optimal information in estimation problems

This paper studies some aspects of information-based complexity theory applied to estimation, identification, and prediction problems. Particular emphasis is given to constructive aspects of optimal algorithms and optimal information, taking into account the characteristics of certain types of problems. Special attention is devoted to the investigation of strongly optimal algorithms and optimal information in the linear case. Two main results are obtained for the class of problems considered. First, central algorithms are proved to be strongly optimal. Second, a simple solution is given to a particular case of optimal information, called optimal sampling design, which is of great interest in system and identification theory.     

Nearly optimal stochastic approximation for online principal subspace estimation

Principal component analysis(PCA) has been widely used in analyzing high-dimensional data. It converts a set of observed data points of possibly correlated variables into a set of linearly uncorrelated variables via an orthogonal transformation. To handle streaming data and reduce the complexities of PCA,(subspace)online PCA iterations were proposed to iteratively update the orthogonal transformation by taking one observed data point at a time. Existing works on the convergence of(subspace) onli...     

New algorithm for optimal parameter estimation with linear constraints

This paper presents a new algorithm for optimal parameter estimation problems with linear constraints. The algorithm developed is based on least absolute-value approximations. The problem is solved first using a least-error-square technique, where we add to the cost function the equality constraints via Lagrange multipliers, to obtain a good estimate for the residuals of the measurements, having gained this information, we choose a number of measurements with the smallest residuals. This number equals the number of parameters to be estimated minus the number of constraints. Using these measurements together with the constraints, we obtain a number of observations equal to the number of parameters to be estimated. By using this technique, we show that there is no need to either iterate or use linear programming to obtain the estimation.This work was supported by the Natural Sciences and Engineering Research Council of Canada, Grant A4146.     

Reliable a posteriori error estimation for state-constrained optimal control

We derive a reliable a posteriori error estimator for a state-constrained elliptic optimal control problem taking into account both regularisation and discretisation. The estimator is applicable to finite element discretisations of the problem with both discretised and non-discretised control. The performance of our estimator is illustrated by several numerical examples for which we also introduce an adaptation strategy for the regularisation parameter.     

On estimation and detection of a function of infinitely many variables

We observe an unknown function of infinitely many variables f = f(t), t = (t₁, ..., t_n, ... ) ∈, [0, 1]^∞, in the Gaussian white noise of level ε > 0. We suppose that in each variable there exists a 1-periodical σ-smooth extension of the function f(t) to IR ^∞. Taking a quantity σ > 0 and a positive sequence a = {a_k}, we consider the set that consists of functions f such that . We consider the cases a_k = k^α and a_k = exp(λk), α > 0, λ > 0. We would like to estimate a function f ∈ or to test the null hypothesis H₀: f = 0 against the alternatives f ∈ , where the set consists of functions f ∈ such that ∥f∥₂ ≥ r. In the estimation problem, we obtain the asymptotics (as ε → 0) of the minimax quadratic risk. In the detection problem, we study the sharp asymptotics of minimax separation rates f _ɛ ^* that provide distiguishability in the problems. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 91–113.     

Adaptive optimal estimation control strategies for systems of simultaneous equations

The choice of control strategies to improve estimation of the parameters in a model of a simultaneous equations system with time-varying parameters is considered. Open-loop feedback (OLF) sequential procedures for handling nonlinear restrictions on reduced form parameters implied by the structural form are suggested, and the combination of sequential estimation and design control strategies feature a marked improvement in the behavior of estimates over the nonsequential [open-loop (OL)] formulation. The maximum accuracy control problem considered in this paper can also be treated as an initial phase of a forecasting and/or stochastic control problem. This will avoid solution to a more difficult problem such as the dual control problem.     

二阶数乘问题的一个最优算法

研究一个有趣的组合优化问题——二阶数乘问题.问题描述如下：给定n≥2个正整数a_1,a_2,…,a_n,设π为{1,2,…,n}的一个置换,表示该问题的一个解,试图找到一个置换π以至∑_（i=1）~n a_（π_i）a_（π_（i＋1））最小,在这里π_（n＋1）=π_1.给出了一个算法复杂度为O（n log n）的最优算法.     

An algorithm for determining the optimal batch size

Translated from Issledovaniya po Prikladnoi Matematike, Kazan', No. 14, pp. 3–9, 1987.     

An optimal control problem for the Hoff equation

The sufficient and necessary conditions are given for existence of an optimal control in the bending problem for an I-beam.     

On the integral equations of optimal signal detection and estimation theory

The analysis of many optimal signal detection and estimation processes involves the solution of Fredholm integral equations. A New approach to these equations is presented. It consists of reducing the Fredholm integral equation to a Cauchy system which is well suited to numerical solution.This research was supported by the National Institutes of Health, Grant No. GM-16437-03.     

An optimal lower bound for the Frobenius problem

Given N?2 positive integers a₁,a₂,…,aN with GCD(a₁,…,aN)=1, let fN denote the largest natural number which is not a positive integer combination of a₁,…,aN. This paper gives an optimal lower bound for fN in terms of the absolute inhomogeneous minimum of the standard (N−1)-simplex.     

射频信道最优分配的数学方法

分了平面蜂窝区域的结构与避免射频干扰的给定规则之间的关系,找到了适应给定规则的基模块,把射频信道的最优分配问题转化为基模块的最优分配问题和基模块的适当平移问题,由之给出了平面上任何蜂窝区域射频信道的分配方案,并证明了所给方案的最优性,求得满足给定规则簇的分配方案所使用最大频率的最小值ｓｐａｎ作为给定规则簇中信道代号变量ｋ的函数表达式为     

Super optimal rates for nonparametric density estimation via projection estimators

In this paper, we study the problem of the nonparametric estimation of the marginal density f of a class of continuous time processes. To this aim, we use a projection estimator and deal with the integrated mean square risk. Under Castellana and Leadbetter's condition (Stoch. Proc. Appl. 21 (1986) 179), we show that our estimator reaches a parametric rate of convergence and coincides with the projection of the local time estimator. Discussions about the optimality of this condition are provided. We also deal with sampling schemes and the corresponding discretized processes.     

Some models for estimation of total of a study variable having many zero values

Design-based and design-based model-assisted estimator of total for a variable having many zero values has high variance. The censored regression (tobit) model-based estimators of a finite-population total have been proposed earlier. The aim of the current research is to apply the semiparametric model to a variable with many zero values, to estimate the population total by model-based and model-assisted estimators, and to compare them with other known estimators by simulation.