Minimum Norm Quadratic Estimation Research Articles

Isotropic Spectral Additive Models of the Covariogram

Summary A class of additive covariance models of an isotropic random process is proposed, motivated by the spectral representation of the covariance function. Model parameters are estimated by using a special case of the minimum norm quadratic estimation estimator, whose asymptotic moments have convenient expressions in terms of spectral densities. Fitting a model in this class is equivalent to fitting an additive model of the spectral density. The class of spectral additive models proposed is dense in the set of summable covariance functions having a spectral density, allowing approximately unbiased estimation of an arbitrary covariance function and its spectral density. Theoretical results are supported by numerical comparison with commonly used models. A procedure to assist model selection is proposed. The techniques are illustrated with an application to contaminant data.

Unbiased invariant minimum norm estimation in generalized growth curve model

This paper considers the generalized growth curve model Y = ∑ i = 1 m X i B i Z i ′ + U E subject to R ( X m ) ⊆ R ( X m - 1 ) ⊆ ⋯ ⊆ R ( X 1 ) , where B i are the matrices of unknown regression coefficients, X i , Z i and U are known covariate matrices, i = 1 , 2 , … , m , and E splits into a number of independently and identically distributed subvectors with mean zero and unknown covariance matrix Σ . An unbiased invariant minimum norm quadratic estimator ( MINQE ( U , I ) ) of tr ( C Σ ) is derived and the conditions for its optimality under the minimum variance criterion are investigated. The necessary and sufficient conditions for MINQE ( U , I ) of tr ( C Σ ) to be a uniformly minimum variance invariant quadratic unbiased estimator ( UMVIQUE) are obtained. An unbiased invariant minimum norm quadratic plus linear estimator ( MINQLE ( U , I ) ) of tr ( C Σ ) + ∑ i = 1 m tr ( D i ′ B i ) is also given. To compare with the existing maximum likelihood estimator ( MLE) of tr ( C Σ ) , we conduct some simulation studies which show that our proposed estimator performs very well.

NONNEGATIVE ESTIMATORS FOR THE ONE-WAY RANDOM EFFECTS MODEL

The relative merits of ten estimators for the variance component of the balanced and unbalanced one-way random effects models are compared. Six of the estimators are nonnegative, two of which are obtained by modifying the Minimum Variance Quadratic Unbiased Estimator (MIVQUE) and the Weighted Least Square Estimator (WLS), and two more from the positive parts of these estimators. The Minimum Norm Quadratic Estimator (MINQE), which is nonnegative, is adjusted for reducing its bias. The nonnegative Minimum Mean Square Error Estimator (MIMSQE), the Analysis of Variance (ANOVA) and Unweighted Sums of Squares (USS) estimator are also included.

Minimum Norm Estimation Under Parameter Constraints with an Application to Insurance

A Minimum Norm Quadratic Estimator is developed for situations where some fixed effects and variance components coincide. The study is motivated by a semiparametric latent variable model frequently encountered in insurance, where a Poisson assumption induces identity between the mean and the within-unit variance. The method is applied to authentic group life insurance data, and its performance is also illustrated by simulations.

Nonnegative estimation of variance components in an unbalanced one way random effects model

The unbalanced one way random effects model with unequal error variances is considered. Two different procedures of estimating the unknown variance components are presented. At first the minimum norm quadratic unbiased estimators are examined, whose realizations may become negative. Nonnegative minimum norm quadratic minimum biased estimators are also derived. The biases and the mean squared errors of these estimators are derived and compared.

Invariant nonnegative minimum norm quadratic estimator of variance components and its application

Minimum norm quadratic estimators were introduced by C. R. Rao in the 70's to attempt to give an unified approach to variance component estimation. One such estimator, which is called invariant nonnegative minimum norm quadratic, was proposed but not explicitly determined. In this paper, a fnormal solution to the invariant nonnegative minimum norm quadratic estimator is presented and then applied to some special cases.

A note on non-negative minimum bias MINQE in variance components model

In this note it is shown that the minimum bias non-negative minimum norm quadratic estimator of a variance component in a general variance components model considered by Hartung (1981) can be used to obtain estimators which consider minimizing weighted Euclidean norm considered by C.R. Rao (1972) in this context rather than just the Euclidean norm of a matrix. The latter arises more naturally for minimizing the Euclidean norm of the matrix which represents the discrepancy between the ‘natural’ and the proposed estimator. The latter norm also incorporates the problem of considering the a priori assigned values of variance components.

Estimation of Variance Components and Applications

Matrix Algebra. Asymptotic Distribution of Quadratic Statistics. Variance and Covariance Components Models. Identifiability and Estimability. Minimum Norm Quadratic Estimation. Pooling of Information for Estimation. Uniform Optimality of MINQE's. Computation of MINQE's for Variance-Covariance Components Models. Iterated MINQE and MLE. 1 symptotic Properties of Estimators. Minimum Variance Quadratic Estimation. Applications to Selection Problems. References. Indexes.

Asymptotic Distributions of Minimum Norm Quadratic Estimators of the Covariance Function of a Gaussian Random Field

Consider a continuous Gaussian random field $z(x)$ defined on a compact set $R \subset \mathbb{R}^d$ with covariance function of the form $\operatorname{cov}(z(x), z(x')) = \sum^k_{i = 1}\theta_iK_i(x,x')$, where the $K_i$'s are specified and $\theta = (\theta_1, \ldots, \theta_k)'$ is to be estimated. Let $\{x_l\}^\infty_{l = 1}$ be a sequence of distinct points in $R$. Based on $z(x_1), \ldots, z(x_N)$, minimum norm quadratic estimation can be used to estimate $\theta$. Suppose $K_1, \ldots, K_k$ are compatible covariance functions on $R$, which means that the Gaussian measures with means zero and covariance functions $K_1, \ldots, K_k$ are mutually absolutely continuous. Then, as the number of observations $N$ increases, the minimum norm quadratic estimator of $\sum^k_{i = 1}\theta_i$ is asymptotically normal with variance of order $N^{-1}$. The minimum norm quadratic estimator of any other linear combination of the $\theta_i$'s converges (in $L^2$) to some nondegenerate random variable. This limit is the same for any two dense sequence of points in $R$. Thus, a definition of a minimum norm quadratic estimator of $\theta$ when $z(\cdot)$ is observed everywhere in $R$ is obtained.

Minimum Norm Quadratic Estimation of Spatial Variograms

Abstract The estimation of spatial variograms, a measure of spatial correlation, is a critical problem in the implementation of kriging, a method for interpolating random fields. We consider the use of minimum norm quadratic estimators of the variogram when it is specified up to a finite number of linear parameters. We investigate the asymptotic behavior of such estimators for Gaussian processes as the number of observations within some bounded region increases. The basic conclusion is that we can estimate consistently those functions of the parameters, and only those functions, that have a nonnegligible impact asymptotically on the kriging procedure. In general, the behavior of the variogram over relatively short distances is the only aspect of the variogram that is asymptotically important. As an example, consider a Gaussian process z(·) on the real line with unknown constant mean and , where γ(·) is known as the semivariogram of the process and θ is a finite vector of unknown parameters. Suppose, for 0...

MINQE for the One-Way Classification

The minimum norm quadratic estimator (MINQE), without the condition of unbiasedness, is given for the effect variance of a one-way classification. The computational form is relatively simple with no iteration necessary, and the prior weight is given as a function of the harmonic mean of the numbers of readings per classification. A comparison is made of the mean squared error (MSE) of MINQE and the estimators of Swallow and Monahan (1984) for the layouts of that article. The MSE of MINQE is shown to be smaller than the MSE of these estimators when the effect variance is greater than the error variance. A discussion is also given illustrating a desirable property of a smaller MSE even in the presence of nontrivial bias.

A modification of minimum norm quadratic estimation of a generalized covariance function for use with large data sets

Marshall and Mardia (1985) and Kitanidis (1985) have suggested using minimum norm quadratic estimation as a method to estimate parameters of a generalized covariance function. Unfortunately, this method is difficult to use with large data sets as it requires inversion of an n × n matrix, where n is number of observations. These authors suggest replacing the matrix to be inverted by the identity matrix, which eliminates the computational burden, although with a considerable loss of efficiency. As an alternative, the data set can be broken into subsets, and minimum norm quadratic estimates of parameters of the generalized covariance function can be obtained within each subset. These local estimates can be averaged to obtain global estimates. This procedure also avoids large matrix inversions, but with less loss in efficiency.

Minimum norm quadratic estimation of components of spatial covariance

Quadratic estimators of components of a nested spatial covariance function are presented. Estimators are unbiased and possess a minimum norm property. Inversion of a covariance matrix is required but, by assuming that spatial correlation is absent, a priori, matrix inversion can be avoided. The loss of efficiency that results from this assumption is discussed. Methods can be generalized to include estimation of components of a generalized polynomial covariance assuming the underlying process to be an intrinsic random function. Particular attention is given to the special case where just two components of spatial covariance exist, one of which represents a nugget effect.

Minimum norm quadratic estimation of components of spatial covariance

Minimum norm quadratic estimation of components of spatial covariance

Estimators of covariances in time series models

The theory of Minimum Norm Quadratic Estimators for estimating variances and covariances is applied to show that some commonly used estimators of covariances in time series models are easily derived using the above principle. In applying the theory MINQE, it is observed that no unbiased estimator exists in the class of invariant quadratics.

Minimum Norm Invariant Quadratic Estimation of a Covariance Matrix in Linear Model

AbstractA general linear model with a known covariance structure is considered. The method of Minimum Norm Quadratic estimation extending RAO'S (1972) argument is outlined. This method is illustrated for a particular model where it is noted that MINQE used for estimating intraclass correlation coefficient yields the maximum likelihood estimate.

Minimum norm quadratic estimators of variance components

In this note we consider the classes of quadratic estimators ofLamotte [1973] for estimating the variance components and derive the forms of the minimum norm quadratic estimators in the classes of quadratics not considered byC.R. Rao [1971a, 1972].