Gauss-Markov Estimator Research Articles

High Accuracy On-ground Attitude Reconstitution for the ESA Astrometry Mission Hipparcos

The ESA space - astrometry mission Hipparcos will provide a uniform and high-precision stellar Catalogue of the whole sky. During 2.5 years 100,000 selected stars will be observed by a special telescope mounted on a satellite placed in earth stationary orbit and their observations will be processed on-ground by two independent scientific Consortia. The data processing requires that the star coordinates in the telescope focal plane be transformed into a common inertial reference frame. To correctly perform this transformation, the satellite three-axes attitude must be reconstituted within a specified accuracy - rms error < 0.1 arcsec on each angle-for the mission lifetime, by processing the data of the on-board attitude sensors: star mapper and gyros. In this paper, after an overview of the Hipparcos mission, the attitude reconstitution is formulated as a Gauss-Markov estimation problem and solved by processing only star mapper data. The Gauss-Markov procedure takes advantage from a guaranteed error linear model describing the attitude evolution over the desired time interval.

High correlation among errors and the efficiency of ordinary least squares in linear models

This note compares ordinary least squares (OLS) and Gauss-Markov (GM) estimates of regression parameters in linear models when errors are homoscedastic but otherwise arbitrary. It is shown that the efficiency of OLS relative to GM depends crucially on the underlying regressor matrix. This extends and qualifies previous results (Kramer 1980), where errors were confined to be first-order autoregressive. In particular, whenever there is a constant in the regression, it is shown that OLS has limiting efficiency of 1 as correlation increases also in the general case.

Quasi-Likelihood Functions

The connection between quasi-likelihood functions, exponential family models and nonlinear weighted least squares is examined. Consistency and asymptotic normality of the parameter estimates are discussed under second moment assumptions. The parameter estimates are shown to satisfy a property of asymptotic optimality similar in spirit to, but more general than, the corresponding optimal property of Gauss-Markov estimators.

Some Classes of Optimality Criteria and Optimal Designs for Complete Two-Way Layouts

For a given class of linear models in standard form an optimal experimental design is to be chosen for estimating some linear functions of the unknown parameters. An optimality criterion is defined to be a real function of the covariance matrices of the Gauss-Markov estimators. Conditions which are imposed on the criteria are monotonicity, quasiconvexity or quasiconcavity, and invariance or order-invariance. A characterization of the $D$-criterion by order-invariance is included which strengthens a result of P. Whittle. In the main part of the paper optimal designs for the usual two-way layouts in ANOVA are computed for large classes of optimality criteria. Some related optimization problems are solved with the technique of majorization of vectors in the sense of Schur.

Inefficiency and correlation

SUMMARY Various extremal results have been proved concerning the efficiency of least squares estimates relative to Gauss-Markov estimates and concerning the canonical correlations between two sets of random variables. In this paper we show how these two types of result are related, and derive a new result from which many previous inequalities can be derived. Some key word8: Canonical correlation; Inefficiency; Least squares. The inequality for the generalized efficiency of least squares estimates relative to best linear unbiased estimates (Bloomfield & Watson, 1975; Knott, 1975) has found applications in other areas. A consequence was used by Venables (1976) in testing for sphericity of a Gaussian distribution, and by Khatri (1978) in proving extremal results for canonical correlations. We shall prove a slightly more general result concerning canonical

Finite Sample Efficiency of Ordinary Least Squares in the Linear Regression Model with Autocorrelated Errors

Abstract In the standard linear regression model y = X β + u, with errors following a first-order stationary autoregressive process, it is shown that the relative efficiency of the ordinary least squares (OLS) as compared with the Gauss-Markov estimator depends to a great extent on the X matrix observed. In particular, it is seen that the relative efficiency of OLS increases with increasing correlation for certain cases important in practice. Since this seems to run contrary to what one should expect on the basis of previous Monte Carlo studies, some additional sampling experiments are also briefly discussed that keep the X matrix fixed in repeated runs.

Small sample considerations in estimation from panel data

Recent interest in statistical inference for panel data has focused on the problem of unobservable, individual-specific, random effects and the inconsistencies they introduce in estimation when they are correlated with other exogenous variables. Analysis of this problem has always assumed the variance components to be known. In this paper, we re-examine some of these questions in finite samples when the variance components must be estimated. In particular, when the effects are uncorrelated with other explanatory variables, we show that (i) the feasible Gauss-Markov estimator is more efficient than the within groups estimator for all but the fewest degrees of freedom and its variance is never more than 17% above the Cramer-Rao bound, (ii) the asymptotic approximation to the variance of the feasible Gauss-Markov estimator is similarly within 17% of the true variance but remains significantly smaller for moderately large samples sizes, and (iii) more efficient estimators for the variance components do not necessarily yield more efficient feasible Gauss-Markov estimators.

Estimation of parameters in certain hierarchical genetic models

Some hierarchical genetic models are considered. Necessary and sufficient conditions for the existence of best linear unbiased estimators of fixed parameters and best quadratic unbiased estimators of variance components in these models are given. An extended concept of the Gauss-Markov estimator is used to construct best estimators.

The use of prior information in linear regression

This paper considers the incorporation of previously obtained data with present data from the model Y=Xp+e wnere e~N(o.e) and b N(b,V) to construct an estimate for the present value of cs. The estimator deiived from combining these two sets or data Is biased, hence its mean square error is compared with the covariance matrix of the Gauss-Markov estimator using only present data. The probability that the combined data yields a better estimator is evaluated. For special cases, che optimal design procedures are presented for obtaining both sets of data.

Gauss-Markov Estimation for Multivariate Linear Models with Missing Observations

In this note we discuss multivariate linear models from the coordinate-free point of view, as earlier done by Eaton (1970). We generalize the result of Eaton by allowing for missing observations. This leads to models of the kind $EY \in L, Cov Y \in\{P(I \otimes \sum)P'\}$ where $P$ is a diagonal mapping. The paper starts by deriving the conditions for existence of Gauss-Markov estimators (GME) of $EY$ in models where the covariance-mappings are not necessarily nonsingular. These conditions are then applied to the above models if $\Sigma$ runs either over all PSD-mappings or over all diagonal PSD-mappings. In the latter case $L$ must be of the form $L = L_1 \times \cdots \times L_p$ while in the general case some further conditions on the $L_i$ must be met. (If $P = I$, then $L_i = L_j$ must hold for all $i, j$; this is equivalent to the result obtained by Eaton). Examples show that these conditions are satisfied only under rather exceptional conditions.

Optimum signal processing for passive sonar range and bearing estimation

Optimum signal processing for passive sonar target range and bearing estimation is discussed for the case where the sonar array consists of an M-element linear array of hydrophone point detectors whole individual outputs are corrupted by sensor-to-sensor-independent self-noises of arbitrary power spectra. By comparing the measurement error covariance matrix to the Cramér–Rao matrix bound the system performance, relative to the theoretical optimum, is determined. Further, the Cramér–Rao bound is used to determine an optimum signal processor. The optimum processor is configured as a set of M (M-1)/2 cross-correlator delay estimators (one for each hydrophone pair), followed by a Gauss–Markov estimation of the array delay vector (target steering vector), which in turn is followed by a linear weighting of the estimated delay vector elements to determine a bearing estimate and a range estimate. The processor is shown to have the Cramér–Rao matrix bound for its measurement error covariance matrix. Subject Classification: 60.30; 30.82.

A note on least squares estimation and the blue in a generalized linear regression model

In a generalized linear regression model, least squares and Gauss-Markov estimators differ, in general, if the variance-covariance matrix of the disturbances is singular. In the present note it is shown that, nevertheless, the conventional least squares procedure leads to a Gauss-Markov estimator if it is applied to a modified model which results from adding dummy constraints to the original model. These constraints reflect the effects of the singularity of the variance- convariance matrix. As a consequence, a Gauss-Markov estimate may always be obtained by standard least squares minimization, which offers considerable computational advantages.

Optimal designs and large-sample tests for linear hypotheses

SUMMARY Moment conditions beyond those required for Gauss-Markov estimation are shown to yield error bounds on normal-theory approximations to type I error probabilities and confidence coefficients associated with variance ratio tests, Scheff6's (1953) bounds, and Dunnett's (1955) procedure for comparing k treatments with a control. These bounds depend on the experimental design. The error-minimizing designs are characterized and shown to be orthogonal.

Coefficient Errors Caused by Using the Wrong Covariance Matrix in the General Linear Model

A method is derived to place an approximate bound on the mean-square error incurred by using an incorrect covariance matrix in the Gauss-Markov estimator of the coefficient vector in the full-rank general linear model. The bound thus obtained is a function of the incorrect covariance matrix $\tilde{S}$ actually used, the Frobenius norm of $S - \tilde{S}$, where $S$ is the correct covariance matrix, and the basis matrix $\phi$. This estimate can therefore be computed from known or easily-approximated data in the usual regression problem. All mathematics related to the method is derived, and numerical examples are presented.

DRYGAS, H.: The Coordinate Free Approach to Gauss‐Markov Estimation (Lecture Notes in Operations Research and Mathematical Systems 40). Springer‐Verlag, Berlin‐Heidelberg‐New York 1970. VIII, 113 S., DM 12,‐.

Biometrische ZeitschriftVolume 15, Issue 4 p. 294-294 Buchbesprechung DRYGAS, H.: The Coordinate Free Approach to Gauss-Markov Estimation (Lecture Notes in Operations Research and Mathematical Systems 40). Springer-Verlag, Berlin-Heidelberg-New York 1970. VIII, 113 S., DM 12,-. H. Toutenburg, H. ToutenburgSearch for more papers by this author H. Toutenburg, H. ToutenburgSearch for more papers by this author First published: 1973 https://doi.org/10.1002/bimj.19730150412AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume15, Issue41973Pages 294-294 RelatedInformation

Empirical Bayes on vector observations: An extension of Stein's method

SUMMARY The statistician is considering several independent normal linear models with identical structures, and desires to estimate the vector of unknown parameters in each of them. An estimator is constructed which dominates the usual Gauss-Markov estimator in terms of total squared error loss. This estimator is shown to have good efficiency in the Bayesian situation where the parameter vectors themselves have a normal prior distribution. A practical example is given.

Consistency of the least squares and Gauss-Markov estimators in regression models

Consistency of the least squares and Gauss-Markov estimators in regression models

Gauss-Markov Estimation for Multivariate Linear Models: A Coordinate Free Approach

The coordinate free (geometric) approach to univariate linear models has added both insight and understanding to the problems of Gauss Markov (GM) estimation and hypothesis testing. One of the initial papers emphasizing the geometric aspects of univariate linear models is Kruskal's (1961). The coordinate free approach is used in this paper to treat GM estimation in a multivariate analysis context. In contrast to the univariate situation, a central question for multivariate linear models is the existence of GM estimates. Of course, it is the more complicated covariance structure in the multivariate case that creates the concern over the existence of GM estimates. As the emphasis is on GM estimation, first and second moment assumptions (as opposed to distributional assumptions) play the key role. Classical results for the univariate linear model are outlined in Section 1. In addition, a recent theorem due to Kruskal (1968) concerning the equality of GM and Least Squares (LS) estimates is discussed. A minor modification of Kruskal's result gives a very useful, necessary and sufficient condition for the existence of GM estimators for arbitrary covariance structures and a fixed regression manifold. In Section 2, the outer product of two vectors and the Kronecker product of linear transformations is discussed and applied to describe the covariance structure of a random matrix. This application includes the case of a random sample from a multivariate population with covariance matrix $\Sigma > 0 ("\Sigma > 0"$ means that $\Sigma$ is positive definite). The question of GM estimation in the standard multivariate linear model is taken up in Section 3. This model is described as follows: a random matrix $Y: n \times p$, whose rows are uncorrelated and each row has a covariance matrix $\Sigma > 0$, is observed. The mean matrix of $Y, \mu$, is assumed to have the form $\mu = ZB$ where $Z: n \times q$ is known and of rank $q$, and $B: q \times p$ is a matrix of regression coefficients. For this model, GM estimators for $\mu$ and $B$ exist and are well known (see Anderson (1958) chapter 8). The main result in Section 3 establishes a converse to this classical result. Explicitly, let $Y$ have the covariance structure as above and assume $\Omega$ is a fixed regression manifold. It is shown that if a GM estimator for $\mu\in\Omega$ exists, then each element $\mu\in\Omega$ can be written as $\mu = ZB$ where $Z: n \times q$ is fixed and $B: q \times p$ ranges over all $q \times p$ real matrices. The results in Section 4 and Section 5 are similar to the main result of Section 3. A complete description of all regression manifolds for which GM estimators exist is given for two different kinds of covariance assumptions concerning $\Sigma$ ($\Sigma$ as above). In Section 4, it is assumed that $\Sigma$ has a block diagonal form with two blocks. Section 5 is concerned with the case when $\Sigma$ has the so-called intra-class correlation form.

Optimale vorhersage und schätznng in regulären und singulären regressionsmodellen

We consider arbit,rary regression models y = Xβ + e, where the covariance matrix from e can also be singulary. First we determine optimal homogeneous and inhomogeneous linear predictions for y ∗where the true equation for y ∗ be y ∗ = Xβ + e ∗ As the risk we use a mean quadratic (not necessary) distance. Then we study the possibilities for improving the best homogeneous unbiased prediction. We show, that this is possible, if there are additional informations about X β if for instance ∣X β∣ is bounded. After this me consider the improvement of a homogeneous prediction by admitting inhomogeneous predictions. In a further section we determine for (may be even singulary) models y = X β + e a GAUSS-MARKOV estimation for β and we get also statements of the question in which cases this estimation can be improved. Thereby the strict connection between unbiased predictions and G.-M. estimations is used.