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.     

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 construction of admissible quadratic estimators

A random linear model for spatially located sensors measured intensity of a source of signals in discrete instants of time is considered. A basis of a quadratic subspace useful in quadratic estimation of a function of model parameters is given. Received: December 1999     

A Note on Properties of Iterative Procedures of Asymptotic Inference

This note deals with the article 'On iterative procedures of asymptotic inference' by K.O. DZHAPARIDZE (1983), in which 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 discusses the theorem that if a maximized function, e.g. the likelihood function, is asymptotically quadratic, then for asymptotically efficient inference finitely many iterations are needed.
It is argued here that the theory still applies if certain well specified inexact (hence computationally cheaper) line searches are used in the optimization.     

The solution of the infinite horizon tracking problem for discrete time systems possessing an exogenous component

In this paper we derive an algorithm that yields, for a discrete-time system, a control minimizing a quadratic cost functional. The system considered is linear and possesses an exogenous component. The cost functional is a quadratic tracking equation over an infinite time horizon with positive semi-definite weighting matrices such that a weighted sum of these matrices is positive definite. The infinite planning horizon Minimum Variance cost criterion and the Linear Quadratic regulator are special cases. For stabilizable systems we give a characterization of the asymptotically admissible reference trajectories.     

Local minimax pointwise estimation of a multivariate density

We consider the problem of the nonparametric minimax estimation of a multivariate density at a given point. A concept of smoothness classes in nonparametric minimax estimation problems is proposed. The smoothness of a function is characterized by the approximability of the function at a point by an integral of the product of this function with an approximate identity. We propose a singular integral estimator, an integral of this approximate identity with respect to the empirical distribution function. Under some assumptions on the approximate identity, the bias of the estimator is shown to be of smaller order asymptotically than the variance, and the estimator itself is shown to be asymptotically locally minimax with respect to the quadratic risk in a proper topology.     

An efficient GMM estimator of spatial autoregressive models

In this paper, we consider GMM estimation of the regression and MRSAR models with SAR disturbances. We derive the best GMM estimator within the class of GMM estimators based on linear and quadratic moment conditions. The best GMM estimator has the merit of computational simplicity and asymptotic efficiency. It is asymptotically as efficient as the ML estimator under normality and asymptotically more efficient than the Gaussian QML estimator otherwise. Monte Carlo studies show that, with moderate-sized samples, the best GMM estimator has its biggest advantage when the disturbances are asymmetrically distributed. When the diagonal elements of the spatial weights matrix have enough variation, incorporating kurtosis of the disturbances in the moment functions will also be helpful.     

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.     

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     

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.     

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。     

Jackknife model averaging

We consider the problem of obtaining appropriate weights for averaging M approximate (misspecified) models for improved estimation of an unknown conditional mean in the face of non-nested model uncertainty in heteroskedastic error settings. We propose a “jackknife model averaging” (JMA) estimator which selects the weights by minimizing a cross-validation criterion. This criterion is quadratic in the weights, so computation is a simple application of quadratic programming. We show that our estimator is asymptotically optimal in the sense of achieving the lowest possible expected squared error. Monte Carlo simulations and an illustrative application show that JMA can achieve significant efficiency gains over existing model selection and averaging methods in the presence of heteroskedasticity.     

Optimal estimation under nonstandard conditions

We analyze optimality properties of maximum likelihood (ML) and other estimators when the problem does not necessarily fall within the locally asymptotically normal (LAN) class, therefore covering cases that are excluded from conventional LAN theory such as unit root nonstationary time series. The classical Hájek–Le Cam optimality theory is adapted to cover this situation. We show that the expectation of certain monotone “bowl-shaped” functions of the squared estimation error are minimized by the ML estimator in locally asymptotically quadratic situations, which often occur in nonstationary time series analysis when the LAN property fails. Moreover, we demonstrate a direct connection between the (Bayesian property of) asymptotic normality of the posterior and the classical optimality properties of ML estimators.     

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.     

Distribution-free tests for time series models specification

We consider a class of time series specification tests based on quadratic forms of weighted sums of residuals autocorrelations. Asymptotically distribution-free tests in the presence of estimated parameters are obtained by suitably transforming the weights, which can be optimally chosen to maximize the power function when testing in the direction of local alternatives. We discuss in detail an asymptotically optimal distribution-free alternative to the popular Box–Pierce when testing in the direction of AR or MA alternatives. The performance of the test with small samples is studied by means of a Monte Carlo experiment.     

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.     

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.     

Distributional and Inferential Properties of the Estimated Precision Cp Based on Multiple Samples

Process precision index C_p has been widely used in the manufacturing industry to provide numerical measures on process potential. Pearn et al. (1998) considered an unbiased estimator of C_p for one single sample. They showed that the unbiased estimator is the UMVUE. They also proposed an efficient test for C_p based on one single sample, and showed that the test is the UMP test. In this paper, we consider an unbiased estimator of C_p for multiple samples. We show that the unbiased estimator is the UMVUE of C_p, which is asymptotically efficient. We also consider an efficient test for C_p, and show that the test is the UMPtest for multiple samples. The practitioners can use the proposed test on theirin-plant applications to obtain reliable decisions.     

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.