General Gauss-Markov Model Research Articles

Sufficient and admissible estimators in general multivariate linear model

The notion of linear sufficiency in general Gauss–Markov model is extended to a general multivariate linear model for any specific set of estimable functions. A general formula of the difference between the dispersion matrix of the BLUE in the original model and that in the transformed model is provided, which brings some further contributions to the theory of linear sufficiency. Moreover, a general formula of the change of BLUE due to transformation is obtained. The analysis here leads to some results, some of which are known in the literature. Besides linear sufficiency, the admissibility of a linear statistic is also extended to the multivariate case.

The general Gauss-Markov model with possibly singular dispersion matrix

Linear models with possibly singular dispersion (variance-covariance) matrix of the vector of disturbances have been considered in the literature since the late sixties. In this paper we give a survey of important results and consequences without a claim to completeness. Proofs are given for the results in Sections 2 to 5.

A new approach to the concept of a strong unified-least-squares matrix

For given matrices A , B , C , and D of appropriate sizes, a criterion for the range of the product AB − C to be a subspace of the range of D irrespective of the choice of a generalized inverse B − of B is established. This result is crucial in a new approach to the concept of the strong unified-least-squares matrix, which plays an essential role in the theory of estimation in the general Gauss–Markov model.

04.2.1. A Range Equality for Block Matrices with Orthogonal Projectors

When considering the general Gauss–Markov model Y = Xβ + e, where E(e) = 0, Cov(e) = V, one often needs to know whether the range inclusion range(X) ⊆ range(V) holds.

An elementary development of the equation characterizing best linear unbiased estimators

Puntanen et al. [J. Statist. Plann. Inference 88 (2000) 173] provided two matrix-based proofs of the result stating that a linear estimator By represents the best linear unbiased estimator (BLUE) of the expectation vector X β under the general Gauss–Markov model M={ y , X β,σ 2 V } if and only if B ( X : VX ⊥)=( X : 0 ) , where X ⊥ is any matrix whose columns span the orthogonal complement to the column space of X. In this note, still another development of such a characterization is proposed with reference to the BLUE of any vector of estimable parametric functions K β . From the algebraic point of view, the present development seems to be the simplest from among all accessible in the literature till now.

Linear sufficiency in the general Gauss-Markov model

This paper provides further contributions to the theory of linear sufficiency in the general Gauss-Markov model E(y)=Xβ, Var (y)=V. The notion of linear sufficiency introduced by Baksalary and Kala (1981) and Drygas (1983) is extended for any specific estimable function c′β. Some general results with respect to the extended concept are obtained. An essential result concerning the former notion is a direct consequence of this paper.

Nonnegative-definite covariance structures for which the blu, wls, and ls estimators are equal

For the general Gauss–Markov model with E( Y )= X β and Var( Y )= V , we give a concise proof of an explicit characterization of the general nonnegative-definite covariance structure V such that the best linear unbiased estimator, weighted least-squares estimator, and least-squares estimator of X β are identical.

On pre-test estimation of parametric functions in the general Gauss-Markov model with quadratic risk

The approach to preliminary test estimation based on comparing the weighted quadratic risk function of two competing estimators of β under the linear regression model {y,Xβ, σ2I} is extended to the case when a given vector of parametric functions κ=Kβ is to be estimated under the general Gauss-Markov model.

A note on the concepts of linear and quadratic sufficiency

Under the general Gauss-Markov model y = Xβ + ϵ, linear transformations Fy of the observable random vector y which preserve enough information to release the best linear unbiased predictor/estimator of ϵ as a linear function of Fy are considered. Transformations with this property are called ‘linearly error-sufficient’. It is shown how linear error-sufficiency is connected with known concepts of linear and quadratic sufficiency and several new results are delivered.

A note on equality of MINQUE and simple estimator in the general Gauss-Markov model

Under the general Gauss-Markov model { y, Xβ, σ 2 V}, equality of MINQUE and so called ‘simple’ estimator (1/ƒ)y T (I − XX +)y with ƒ = rank(X : V) − rank(X) for σ 2 is considered. It is revealed that this equality holds if and only if the quadratic form (1/ σ 2) y T( I − XX +) y admits a χ 2-distribution under normality of y.

Stability of invariant linearly sufficient statistics in the general Gauss-Markov model

Necessary and sufficient conditions are derived for the inclusions \(J_0 \subset J\) and \(J_0^* \subset J^* \) to be fulfilled where \(J_0 ,J_0^* {\text{ and }}J,J^* \) are some classes of invariant linearly sufficient statistics (Oktaba, Kornacki, Wawrzosek (1988)) corresponding to the Gauss-Markov models \(GM_0 = (y,X_0 \beta _0 ,\sigma _0^2 V_0 ){\text{ and }}GM = (y,X\beta ,\sigma ^2 V)\), respectively.

Sensitivity analysis in Gauss-Markov models

The estimated parameters from a Gauss-Markov model have varying sensitivity to the individual observations included in the model. Similarly, the redundancy contribution (number) of any observation is associated differently to all the other observations in the model. Evaluation of the sensitivity of parameters to observations, and the sensitivity of the redundancy contribution of one observation to the others are useful to gain more insight into Gauss-Markov models. Such analysis has found practical applications in survey network design and in multiple outlier detections. This paper presents some quantitative sensitivity measures for general Gauss-Markov models. The application of the concept in surveying network design is also discussed.

On a class of estimators in the general gauss-markov model

Under the general Gauss-Markov model {y, Xβ, σ2V}, Where X and V can be deficient in rank, a new class of estimators for the unknown parameter vector β is considered. Necessary and sufficient conditions for admissibility and linear sufficiency are given, thus establishing a subclass of general ridge estimators.

Admissible linear estimation in the general Gauss-Markov model with respect to an arbitrary quadratic risk function

In the first part of this paper, we give a complete characterization of the class, A W( Kβ) of all estimators that are admissible for any given vector of parametric functions Kβ, not necessarily identifiable, among the set of linear estimators under the general Gauss-Markov model M = { Y, Xβ, σ 2 V}, with both the model matrix X and the dispersion matrix V possibly deficient in rank, when the criterion for comparing estimators is the W-weighted quadratic risk function, with any nonnegative definite weight-matrix W. In the second part, we investigate the robustness of the class A W( Kβ) with respect to W, with the conclusion that, except for special situations only, there is full freedom in choosing W within the set of matrices having the same range.

Preliminary test estimation of a vector of parametric functions in the general Gauss-Markov model

The theory of preliminary test estimation based on comparing the matrix risks of two competing estimators of β or Xβ under the general Gauss-Markov model { Y, Xβ , σ 2 V }, is extended to the situation where the interest is focused on a given vector of estimable parametric functions. Applicability of this extension is shown in the context of a model in which Xβ=Wγ+Zδ , where γ is the vector of main parameters and δ is the vector of nuisance parameters. As an illustration, preliminary test estimation of a vector of treatment contrasts in a fixed-effect model corresponding to a block design is discussed.

Statistical Data Analysis and Inference

Prologue. C. Radhakrishna Rao's Contributions to Statistics (Y. Dodge). Nonlinear Estimation in Linear Models (F. Hampel). On the Maximum Number of Factors in Two Construction Methods for Orthogonal Arrays (A.S. Hedayat, J. Stufken). Some Geometrical Aspects of Data Analysis and Statistics (J.M. Oller). Inferences and Test of Hypotheses. On Bayes and Empirical Bayes Two-Stage Allocation Procedures for Selection Problems (S.S. Gupta, T.C. Liang). Multi-Sample Functional Statistical Data Analysis (E. Parzen). Discussing Truth or Falsity by Computing a Q-Value (W. Schaafsma, J. Tolboom, B. van der Meulen). Linear Pivotals and the Bayes-Non Bayes Compromise (G.A. Barnard). Bayes Modified Minimax Experimental Design (R.V. Canfield). A Simple Class of Tests Locally Better than the Score Test (T.K. Chandra). Comparison of Tests in the Presence of a Nuisance Parameter (R. Mukerjee). Tests for Redundancy of Some Variables in Multivariate Analysis (Y. Fujikoshi). Model Choice in the Context of Simultaneous Inference (T. Havranek, O. Soudsky). First Zeros of Empirical Characteristic Functions and Extreme Values of Gaussian Processes (J. Husler). Parameter and Variance Estimation. Comparison of Experiments with Weighted Distributions (M.J. Bayarri, M.H. De Groot). Optimal Estimation for Weighted Distributions: Semi-Parametric Models (V.P. Godambe, M.B. Rajarshi). On Efficiency for Quasi-Likelihood and Composite Quasi-Likelihood Methods (C.C. Heyde). Improved Estimators of Dispersion of an Inverse Gaussian Distribution (N.Pal, B.K. Sinha). Applications of Edgeworth Expansions to Bootstrap - A Review (G.J. Babu). Dual Poincare- Type Inequalities via the Cramer-Rao and the Cauchy-Schwarz Inequalities and Related Characterizations (Th. Cacoullos). Some Optimality Results on Stein's Two-Stage Sampling (J.K. Ghosh, R. Mukerjee). The Cramer-Frechet-Rao Inequality for Sequential Estimation in Non-Regular Case (Z. Govindarajulu, I. Vincze). Whither Delete-K Jackknifing for Smooth Statistical Functionals? (P.K. Sen). Distributions, Weighted Distributions and Characterization. Three Useful Expressions for Expectations Involving a Wishart Matrix and its Inverse (G.P.H. Styan). Characterizations of Distributions via Moments of Order Statistics: A Survey and Comparison of Methods (G.D. Lin). Constancy of Regression of a Polynomial of Sample Average of Positive Random Variables on their Ratios Characterizes Gamma Distribution (A. Kagan). Probing Encountered Data, Meta Analysis and Weighted Distribution Methods (G.P. Patil, C. Taillie). Further Results on Identification when the Parameters are Partially Unknown (S. El Khattabi, F. Streit). Linear Models and Matrices . Weighted-Least-Squares Estimation in the General Gauss-Markov Model (J.K. Baksalary, S. Puntanen). Evaluating Pre-Test Predictors of Success in Linear Regression Models (A. Cohen, H.B. Sackrowitz). Optimum Invariant Tests in Mixed Linear Models with Two Variance Components (Th. Mathew).

Implied linear restrictions in the general Gauss-Markov model

The general Gauss-Markov model M = { Y, X, β, σ 2 V} with linear restrictions Rβ = r , denoted by M r , is compared with the model M with implied linear restrictions ARβ = Ar , denoted by M ∗ r . Necessary and sufficient conditions are derived under which (i) the consistency condition, (ii) the minimum dispersion linear unbiased estimator of a given vector of estimable parametric functions of β , and (iii) the minimum norm quadratic unbiased estimator of σ 2 for the two restricted models M r and M ∗ r are identical. The results obtained cover those concerning comparison of the restricted model M r to the unrestricted model M.

Spectrum and trace invariance criterion and its statistical applications

Necessary and sufficient conditions are established for the set of nonzero eigenvalues of the product B - A to be invariant with respect to the choice of a generalized inverse B - . It turns out that the same conditions constitute a criterion for the invariance of trace( B - A ). The algebraic result derived is then applied to investigate some properties of canonical correlations in the context of the general Gauss-Markov model M = { Y , Xβ ,σ 2 V } . In particular, the results of Watson (1967) and Puntanen (1987) are generalized, and a new characterization of the best linear unbiased estimator of Xβ under the model M is established.

Admissible linear estimators of an arbitrary vector of parametric functions in the general Gauss-Markov model

Abstract This paper derives a complete characterization of estimators that are admissible for any, not necessarily identifiable, vector of parametric functions among the set of linear estimators under the general Gauss-Markov model M = { Y,Xβ,σ 2 V } with both the model matrix X and the dispersion matrix V possibly deficient in rank. This characterization is then applied to examine admissibility of several estimators of β proposed in the literature.

Characterizations of the best linear unbiased estimator in the general Gauss-Markov model with the use of matrix partial orderings

Under the general Gauss-Markov model { Y, Xβ, σ 2V }, two new characterizations of BLUE( Xβ) are derived involving the Löwner and rank-subtractivity partial orderings between the dispersion matrix of BLUE( Xβ) and the dispersion matrix of Y. As particular cases of these characterizations, three new criteria for the equality between OLSE( Xβ) and BLUE( Xβ) are given.