On estimating nonnegative definite quadratic forms

It is often required to estimate a quadratic form in survey sampling, especially when one has to estimate the mean squared error of a linear estimator of the population total. In this note we consider the problem of obtaining uniformly nonnegative quadratic unbiased estimators for nonnegative definite quadratic forms. The estimators considered here are necessarily quadratic. Received January 1997     

On the Distribution of General Quadratic Functions in Normal Vectors

A representation in terms of independent standard normal variables tor the general quadratic form in normal variables in the univariate case, obtained by DIK and DE GUNST (1985), is extended to the multivariate situation. A representation for the quadratic function in normal vectors X'AX , where X is a random matrix with normally distributed elements and A a real symmetric matrix, is given in terms of independent and identically distributed central normal vectors. The representation is valid only when the covariance structure of X is of a special form, but all known results, especially necessary and sufficient conditions for X'AX to have a Wishart distribution, can easily be derived from it.     

Adapting Productivity Theory to the Quadratic Cost Function. An Application to the Spanish Electric Sector

In this article we have adapted productivity analysis to the case of a cost model using a quadratic cost function and discrete data. The main theoretical result is a productivity index that can be decomposed into modified versions of the contribution of technical change and the effect of the variations in the scale of production. This framework has been applied to the study of the Spanish electric sector from 1985 to 1996, during which relevant regulatory changes were introduced in order to increase productivity. For this, a normalized quadratic cost function was estimated. The results show important productivity gains with both technical change and scale effect playing important roles.     

Designs for estimating an extremal point of quadratic regression models in a hyperball

This paper is devoted to studying optimal designs for estimating an extremal point of a multivariate quadratic regression model in the unit hyperball. The problem of estimating an extremal point is reduced to that of estimating certain parameters of a corresponding nonlinear (in parameters) regression model. For this reduced problem truncated locally D-optimal designs are found in an explicit form. The result is a generalization of the results of Fedorov and Müller (1997) for onedimensional quadratic regression function in the unit segment. Received February 2002     

On iterative procedures of asymptotic inference

Abstract  An informal discussion is given on performing an unconstrained maximization or solving non-linear equations of statistics by iterative methods with the quadratic termination property. It is shown that if a miximized function, e.g. likelihood, is asymptotically quadratic, then for asymptotically efficient inference finitely many iterations are needed.     

On the interpretation of cointegration in the linear–quadratic inventory model

The traditional formulation of the linear–quadratic inventory model with unit roots predicts cointegration between inventories and sales. That formulation implies that marginal production costs and the marginal benefits of inventories are both tending to ∞, and the cointegrating coefficient reflects the optimal trade-off between these competing factors. This paper suggests a reformulation of the problem in which marginal production costs and marginal inventory benefits are both stationary and in which the cointegrating coefficient is the same as the value that characterizes the target inventory level in the cost function.     

A conditional minimax approach in survey sampling

Summary Applying the usual minimax criterion in finite sampling theory yields complicated solutions except the parameter space has certain invariance properties. A conditional minimax criterion is suggested. After a sample is selected it is reasonable to seek an estimator that has good properties (e.g. minimaxity) for that sample. Explicit solutions are given in the case where the parameter space is described by quadratic forms.     

计算机辅助机场地势优化设计

本文在机场地势设计中建立以点的高程作为变量的优化数学模型,将机场地势设计处理成一个具有一组线性约束,二次目标函数的优化数学模型,并用Ｃ语言编出优化设计程序,和对ＡｕｔｏＣＡＤ进行二次开发,实现了计算机自动绘制地势优化设计结果。     

THE DISTRIBUTION OF GENERAL QUADRATIC FORMS IN NORMA

For a general quadratic form in normal variables a representation in terms of independently distributed standard normal variables is derived. The necessary and sufficient conditions for such a quadratic form to have a non–central chi–squared distribution can be found easily using this representation.     

柯西不等式的几种证明方法

如果某一知识跟很多学科或者一个学科的很多分支有着密切联系,那么这个知识肯定是很重要的,而二次型、欧式空间内积、詹森不等式都是高等数学中代数、实函、微积分的基本内容。本文运用二次型理论、欧式空间中内积性质和詹森(Jensen)不等式三种方法证明柯西不等式,并简要说明柯西不等式与高等数学之间的联系。     

Variance estimation for unequal probability sampling

In this paper, we discuss the optimality of the variance estimator of the Horvitz-Thompson estimator proposed by Kott (1988) in the class of model-unbiased quadratic estimators. We also propose some improved estimators over Kott's estimator in the class of general quadratic estimators. Received: February 1999     

二项式系列题目类解初探

二次函数是中学数学的重要内容之一,和它相关的内容,如:二次方程、二次不等式、二次三项式及二次函数的图像,构成了一个体系,称之为"二次式系列";本文主要在以下三个方面对这些题目的解法进行探讨:(一)求参数的范围;(二)根的存在情况的判别;(三)求最大、最小值;通过以上的讨论,以期提高中学生对此类题目的解题方法和解题技巧。     

Large-sample estimation strategies for eigenvalues of a Wishart matrix

The problem of simultaneous asymptotic estimation of eigenvalues of covariance matrix of Wishart matrix is considered under a weighted quadratic loss function. James-Stein type of estimators are obtained which dominate the sample eigenvalues. The relative merits of the proposed estimators are compared to the sample eigenvalues using asymptotic quadratic distributional risk under loal alternatives. It is shown that the proposed estimators are asymptotically superior to the sample eigenvalues. Further, it is demonstrated that the James-Stein type estimator is dominated by its truncated part.     

The locally MIMSQE of nonnormal error variance in quadratically balanced models

Without normality assumption, an explicit form of the locally minimum mean square error translation-invariant quadratic estimator for the error variance in a quadratically balanced design is obtained. The estimator depends on the kurtosis of the random error. Under the normality the estimator becomes globally optimal.     

The distribution function of a statistic for testing the equality of scale parameters in two gamma populations

Summary The exact distribution function of the ratio of two sums of gamma variates is derived in this paper. The result applies to ratios of quadratic forms and to a statistic used for testing the equality of scale parameters in two gamma populations.     

Convexity and duality properties of a quadratic intraregional location model

This paper has two main purposes. The first one is to analyse the convexity and duality properties of a quadratic intraregional location model that has been developed for long-term indicative planning in the Stockholm region. The second one is to review the results of Koopmans and Beckmann (1957) about the inadequacy of a linear price system in sustaining an optimal assignment of plants to locations when the costs of transporting intermediary commodities are taken into consideration. At the outset a model is formulated which is a transposition of a continuous Koopmans-Beckmann model into the urban scene. It is shown that this quadratic programming model is non-convex in all practical cases of interest, due to the simple fact that transportation costs increase with distance. A modification of the model is proposed in which the centralising transportation cost criterion is traded of against a decentralising so called congestion cost which penalizes over-exploitation of urban space. It is shown that the modified model tends to be convex. In the light of these results Kuhn- Tucker theory is used to derive a set of conditions that will ensure that the optimal solution is stable relative to all potential moves by individual decision-makers. This result forms the basis for the conclusion that the failure of the price system in the Koopmans-Beckmann model is rather due to properties of the quadratic criterion function than the integral restrictions on the variables.     

Interview design for revealing preferences of policy makers

A policy maker is asked a few simple questions about his preference. Then the model represents it by a quadratic utility function, which can be made monotonic and quasi-concave (= to provide the convexity of the preference). The design of the interview with a policy maker is aimed at attaining the following goals: (a) no ambiguous output (= degeneration of the model), (b) ordinal approach to preferences (= asking questions about ordinal preferences and providing the uniqueness of the ordinal preference at the model output, regardless of its representation by a quadratic utility function), (c) stability of the model (= the model's input–output transformation is continuous). We also describe briefly the implementation of our model in a user-friendly interface to a corresponding computer program.     

Nonparametric covariance estimation in multivariate distributions

The estimation problem of the unknown covariance matrix of a multivariate distribution with the known mean is studied under a matrix-valued quadratic loss function. The conditions on the sample sizes for the best unbiased estimator to have a smaller risk than the sample covariance matrix is established. The former estimator is completely (without exceptional sets of Lebesgue measure zero) characterized by its expectation in the class of all multivariate distributions with zero mean and finite fourth moments. Received: November 1998     

An optimization model for the allocation of resources in family planning programs

In this paper we present a quadratic programming model for the allocation of resources in a family planning program. A quadratic cost function is minimized subject to three sets of linear constraints: one for a diffusion model of the acceptance, switching and dropout behavior of the target population; one for manpower limitations; and one for incorporating government policies on desired levels of birth rate. There are two important features to be noted. First, a recruitment cost parameter is introduced to account for the effects of market saturation on new acceptance. Second, by treating the problem as one of cost minimization rather than one of birth rate minimization, the model can be utilized to evaluate the feasibility and the minimum cost requirement of government policies on desired reductions in the birth rate. Computational considerations are also discussed.     

Finite sample properties of maximum likelihood estimator in spatial models

We investigate the finite sample properties of the maximum likelihood estimator for the spatial autoregressive model. A stochastic expansion of the score function is used to develop the second-order bias and mean squared error of the maximum likelihood estimator. We show that the results can be expressed in terms of the expectations of cross products of quadratic forms, or ratios of quadratic forms in a normal vector which can be evaluated using the top order invariant polynomial. Our numerical calculations demonstrate that the second-order behaviors of the maximum likelihood estimator depend on the degree of sparseness of the weights matrix.