Singular Dispersion Matrix Research Articles

Adjusting a 2D Helmert transformation within a Gauss–Helmert Model with a singular dispersion matrix where BQ is of smaller rank than B

The case of a singular dispersion matrix within the Gauss–Helmert Model has been considered before, most recently even allowing the rank of BQ to be smaller than the rank of B. In this contribution the emphasis is shifted towards an illuminating example, the 2D Helmert transformation.

A derivation of the multivariate singular skew-normal density function

We prove the existence of a multivariate singular skew-normal density function, derive its moment generating function, and demonstrate that the skewness parameter-vector is confined to the column space of the singular dispersion matrix.

On the Gauss–Helmert model with a singular dispersion matrix where [formula omitted] is of smaller rank than [formula omitted

The case of a singular dispersion matrix within the Gauss–Helmert Model has been considered before, usually assuming a sufficiently small rank deficiency in order to guarantee a unique solution for both the residual vector as well as the estimated parameter vector of type Best Linear Uniformly Unbiased Estimate (BLUUE). In this contribution the emphasis is shifted towards establishing necessary and sufficient conditions for a unique residual vector, along with a unique estimate of type Best Linear Uniformly Minimum Bias Estimate (BLUMBE) for the parameter vector. Should uniformly unbiased estimates exist, the BLUMBE obviously becomes the BLUUE.

Best affine unbiased representations of the fully restricted general Gauss–Markov model

This article completes and simplifies earlier results on the derivation of best linear, or affine, unbiased estimates in the general Gauss–Markov model with a singular dispersion matrix and additional restrictions under very general conditions. We provide the class of all linear representations of the best affine unbiased estimator with precise statements concerning its indeterminacy.

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.

Extending some results and proofs for the singular linear model

This note gives some new forms and shortened proofs for results on the linear model with singular dispersion matrix.

Testing for the mean vector of a multivariate normal distribution with a possibly singular dispersion matrix and related results

Let y ≈ N p ( μ, Σ) where Σ, possibly singluar, is unknown. We develop a test for H 0: μ = μ 0 against H 1:μ≠μ 0 based on a random random sample from y as an extension of Hotelling's T 2 test. We show that the same procedure can be used to compute the test statistic in both the cases of the singular and the positive definite dispersion matrices. This development involves the singular Wishart distribution. Motivated by Khatri (1968), we also obtain an expression for the density of the Wishart distribution W p ( k, Σ) in each o the following cases: (i) Σ positive definite, k < p, (ii) Σ positive semidefinite of rank r (< p) and k ⩾ r, (i of rank r (< p) and k < r.

ROPCA: a FORTRAN program for robust principal components analysis

Program ROPCA computes robust estimates of principal components based on the dispersion matrix given by one of the three robust methods: multivariate trimming, M-estimates using Huber weights or T-weights. The problem of a singular dispersion matrix is bypassed by using Euclidean distances between Q-mode loadings of principal components to replace Mahalanobis distances between raw data points. ROPCA has been tested on IBM PCs and compatible microcomputers. Its execution is simple and interactive. An example shows that the program successfully reduces the influence of outliers and yields reliable principal components.

Best linear unbiased estimation in the restricted general linear model2

Hallum, Lewis and Boullion [4] considered the problem of the best linear unbiased estimation of an estimable parametric function in the general linear model with a singular dispersion matrix when the vector of unknown parameters is subject to an a priori known system of linear restrictions. Unfortunately, the main result of [4] appears to be erroneous. In this paper, the correct formula for the Blue is given. Also, the problem of effects the restrictions have on the Blue of a parametric function is discussed.

Relationships Between Some Representations of the Best Linear Unbiased Estimator in the General Gauss–Markoff Model

For the general Gauss–Markoff model $( {{\bf y},{\bf X}{\bf \beta },{\bf V}} )$ with a singular dispersion matrix ${\bf V}$, Watson [11] has developed a general representation of operators providing the best linear unbiased estimator for ${\bf \beta }_1 $, the orthogonal projection of the vector $\beta $ of unknown parameters on $\mathcal{C}( {{\bf X}'} )$, the column space of ${\bf X}'$. The form of every such operator as developed by him is ${\bf W} = {\bf W}_1 + {\bf W}_2 + {\bf W}_3 $, where ${\bf W}_1 $ is unique with $\mathcal{C}( {{\bf W}'_1 } ) = \mathcal{C}( {\bf X} )$, ${\bf W}_2 $ is also unique but with $\mathcal{C}( {{\bf W}'_2 } ) \subset \mathcal{C}^ \bot ( {\bf X} )$, the orthogonal complement of $\mathcal{C}( {\bf X} )$, while ${\bf W}_3 $ is arbitrary with $\mathcal{C}( {{\bf W}'_3 } ) \subset \mathcal{C}^ \bot ( {\bf X} ) \cap \mathcal{C}^ \bot ( {\bf V} )$. Since for any ${\bf y}$ for which model $( {{\bf y},{\bf X}{\bf \beta },{\bf V}} )$ is consistent ${\bf W}_3 {\bf y} = {\bf 0}$, i...

Choice of best linear in the guass-markoff model with a singular dispersion matrix

In this paper we obtain the complete class of representations and useful subclasses of MV-UB-LE and MV-MB-LE (minimum variance unbiased and minimum bias linear estimators) of linear parametric functions in the Gauss-Markoff model (Y,Xβ, σ 2V) when V is possibly singular.

Singular gauss‐markov models

AbstractFor the general linear model Y = X$sZ + e in which e has a singular dispersion matrix $sG2A, $sG &gt; 0, where A is n x n and singular, Mitra [2] considers the problem of testing F$sZ, where F is a known q x q matrix and claims that the sum of squares (SS) due to hypothesis is not distributed (as a x2 variate with degrees of freedom (d. f.) equal to the rank of F) independent of the SS due to error, when a generalized inverse of A is chosen as (A + X'X)–. This claim does not hold if a pseudo‐inverse of A is taken to be (A + X'X)+ where A+ denotes the unique Moore‐Penrose inverse (MPI) of A.

Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix

In the general Gauss-Markoff model ( Y, Xβ, σ 2 V), when V is singular, there exist linear functions of Y which vanish with probability 1 imposing some restrictions on Y as well as on the unknown β. In all earlier work on linear estimation, representations of best-linear unbiased estimators (BLUE's) are obtained under the assumption: “ L′Y is unbiased for Xβ ⇒ L′X = X.” Such a condition is not, however, necessary. The present paper provides all possible representations of the BLUE's some of which violate the condition L′X = X. Representations of X for given classes of BLUE's are also obtained.

Theory and Application of Constrained Inverse of Matrices

The concept of a constrained inverse of a matrix, introduced by Bott and Duffin [1], is extended so as to bring together the various generalized inverses and pseudoinverses studied in literature under a common classification scheme. Applications of this concept are indicated in obtaining restricted solutions of consistent linear equations, E-approximate solutions of inconsistent linear equations, stable values of a quadratic function and generalized least squares analysis of a linear model with a possible singular dispersion matrix.

Some distance properties of latent root and vector methods used in multivariate analysis

SUMMARY This paper is concerned with the representation of a multivariate sample of size n as points P1, P2, ..., PI in a Euclidean space. The interpretation of the distance A(Pi, Pj) between the ith andjth members of the sample is discussed for some commonly used types of analysis, including both Q and R techniques. When all the distances between n points are known a method is derived which finds their co-ordinates referred to principal axes. A set of necessary and sufficient conditions for a solution to exist in real Euclidean space is found. Q and R techniques are defined as being dual to one another when they both lead to a set of n points with the same inter-point distances. Pairs of dual techniques are derived. In factor analysis the distances between points whose co-ordinates are the estimated factor scores can be interpreted as D2 with a singular dispersion matrix.