KEYNOTE ADDRESS Canonical analysis and reduced rank regression in autoregressive models
T. W. ANDERSON, Stanford University, USA

When the rank of the autoregression matrix is unresticted, the maximum likelihood estimator under normality is the least squares estimator. When the rank is restricted, the maximum likelihood estimator (reduced rank regression) is composed of the eigenvectors of the effect covariance matrix in the metric of the error covariance matrix corresponding to the largest eigenvalues (Anderson, 1951). The asymptotic distribution of these two covariance matrices under normality is obtained and is used to derive the asymptotic distribution of the eigenvectors and eigenvalues. These distributions are different from those for independent (stochastic or nonstochastic) regressors. The asymptotic distribution of the reduced rank regression is the asymptotic distribution of the least squares estimator with some restrictions; hence the covariance of the reduced rank regression is smaller than that of the least squares estimator. This result does not depend on normality although the asymptotic distribution of the eigenvectors and eigenvalues does.
 

Keywords: Canonical correlations and vectors, eigenvalues and eigenvectors, asymptotic distributions.
 
 

ILAS LECTURE

Reconstruction of a polygon from its moments
Gene H. GOLUB*, Stanford University, USA

Computation of certain kinds of numerical quadratures on polygonal regions of the plane and the reconstruction of these regions from their moments can be viewed as dual problems. In fact, this is a consequence of a little-known result of Motzkin and Schoenberg. In this talk, we discuss this result and address the inverse problem of (uniquely) reconstructing a polygonal region in the complex plane from a finite number of its complex moments. Algorithms have been developed for polygon reconstruction from moments and have been applied to tomographic image reconstruction problems. The numerical computations involved in the algorithm can be very ill-conditioned. We have managed to improve the algorithms used, and to recognize when the problem will be ill-conditioned. Some numerical results will be given.
 
 

* Joint work with Peyman MILANFAR (University of California, Santa Cruz, USA) and James VARAH (University of British Columbia, Vancouver, Canada).
 
 

INVITED TALKS

Arithmetic-geometric mean inequalities for matrices
T. ANDO, Hokusei Gakuen University, Sapporo, Japan

The arithmetic-geometric mean inequality for positive numbers a, b > 0 or the Schwarz inequality for complex numbers x, y

$$\sqrt{ab} \; \leq\; \frac{a + b}{2} \quad {\rm or} \quad \vert xy\vert \; \leq \;\frac{\vert x\vert^2 + \vert y\vert^2}{2}$$

is the most fundamental inequality. One of its generalization is the Young inequality

$$a^tb^{1-t} \leq ta + (1-t)b \quad ( 0 \ < t <\ 1)$$

or

$$\vert xy\vert \leq \frac{1}{p}\vert x\vert^p+ \frac{1}{q}\ver......q \quad \left(p, q > 1;\frac{1}{p} + \frac{1}{q} = 1\right).$$

If scalars a, b are replaced with matrices A, B, there is no problem in understanding $A, B \geq 0$ as positive semi-definiteness of A and B. Correspondingly, the order relation, Löwner order, $X\geq Y$ between two Hermitian matrices is induced by the cone of positive semi-definite matrices.

There are, however, several directions for matrix generalizations of the Schwarz inequality as well as the Young one. We take up two of them.

The first one is an approach using matrix inequalities, that is, to discuss the inequalities with respect to the Löwner order while the second one is to discuss the eigen (or singular) value inequalities and their variants.

In the direction of matrix inequalities a serious obstacle is in that for $A, B \geq 0$ the product AB is Hermitian (and positive semi-definite) only if A and B are commutative. If A and B are commutative, through simultaneous diagonalization, the situations are almost the same as the scalar case.

There are several natural ways of defining a positive definite geometric mean for (non-commuting) positive definite A, B > 0. One candidate is

$${\bf G}(A,B) \ \equiv \ \exp\left(\frac{\log A + \log B}{2}\right).$$

It is rather surprising to see that with this definition the arithmetic-geometric mean inequality is not valid in general. The only valid inequality is

$$\log {\bf G}(A,B) \; \leq\; \log\left(\frac{A + B}{2}\right). %$$

The other candidate, denoted by $A\char93 B$, is

$$A\char93 B \; \equiv \; A^{1/2}\cdot(A^{-1/2}BA^{-1/2})^{1/2}\cdot A^{1/2},$$

for which the arithmetic-geometric mean inequality is valid. This geometric mean$A\char93 B$ admits several nice characterizations. First $A\char93 B$ is proved to be the maximum of $X \geq 0$ for which

$$\left[\begin{array}{cc} A & X \\ X & B \end{array}\right] \; \geq \;0.$$

This implies that $A\char93 B$ is a unique positive definite solution of the matrix quadratic equation $XA^{-1}X \ = \ B.$It is seen from this that despite its asymmetric appearance $A\char93 B$ coincides with $B\char93 A$.

There are many evidences which support rightness of this definition of geometric mean. $A\char93 B$ becomes the midpoint of a geodesic curve connecting A and B with respect to a natural Riemannian metric on the cone of positive definite matrices. Also there is an iteration scheme leading to this geometric mean $A\char93 B$ using the arithmetic means and the harmonic means. Starting from $X_0 \ \equiv\ A$ and $Y_0 \ \equiv\ B$, define

$$X_n \ \equiv \ \frac{X_{n-1} + Y_{n-1}}{2}, \qquad Y_n \ \equ......-1} + (Y_{n-1})^{-1}}{2}\right\}^{-1} \qquad n= 1, 2, \ldots$$

Then it is proved that both X_n and Y_n converge to $A\char93 B$. In the same spirit a possible matrix generalization of a^tb^1-t may be

$$A\char93 _tB \ \equiv \ A^{1/2}\cdot(A^{-1/2}BA^{-1/2})^{1-t}\cdot A^{1/2}$$

for which the Young inequality is valid.

Let us turn to the eigen (or singular) value inequalities. For matrices X, Y the singular value inequalities $\sigma_k(X) \; \leq \; \sigma_k(Y) \ k = 1,2, \ldots,$where the singular values are arranged in non-increasing order, is equivalent to the existence of a unitary matrix U such that $\vert X\vert\\leq\ U^*\vert Y\vert U$. In this connection the Schwarz inequality or more generally the Young inequality is valid in the following form ; for general matrices X, Y and for $1 < p, q < \infty$ with 1/p + 1/q = 1 there is a unitary matrix Usuch that

$$\vert XY\vert \; \leq \; U^*\left\{\frac{1}{p}\vert X\vert^p + \frac{1}{q}\vert Y\vert^q\right\}U.$$

A weaker comparison is based on (weak) majorization relation among singular values, defined as

$$\sum_{j=1}^k\sigma_j(X) \; \leq \; \sum_{j=1}^k\sigma_j(Y) \qquad k =1,2,\ldots$$

This is equivalent to the existence of a finite number of unitary matrices $U_j\ (j = 1,2,\ldots, N)$ and a finite number of non-negative numbers $\alpha_j\ (j = 1,2, \ldots, N)$ with $\sum_{j=1}^N\alpha_j = 1$ such that

$$\vert X\vert \; \leq \; \sum_{j=1}^N\alpha_jU_j^*\vert Y\vert U_j.$$

It is, however, more convenient to formulate this condition as $\vert\vert\vert X\vert\vert\vert\\leq \ \vert\vert\vert Y\vert\vert\vert$ for all unitarily invariant norms $\vert\vert\vert\cdot\vert\vert\vert$. There is the Young inequality of the form; for $A, B \geq 0$and for general matrix X

$$\vert\vert\vert A^tXB^{1-t} + A^{1-t}XB^t \vert\vert\vert \; ......\; \vert\vert\vert AX + XB\vert\vert\vert \qquad (0 < t< 1).$$

Up to this point, for both matrix and singular value inequalities, the most basic fact is that we are treating a pair of matrices. We shall discuss, when $n \geq 3$, what is a reasonable definition of geometric mean of $A_j \geq 0 \ (j = 1,2,\ldots, n)$ and what kind of arithmetic-geometric mean inequalities are expected.
 

Checking the independence assumption in linear models by control charting residuals
Ronald CHRISTENSEN, University of New Mexico, Albuquerque, USA

Fisher's lack of fit test was based on having exact replicates in the rows of the model matrix of a linear model. In recent years, considerable work has been done on using clusters of near replicates to test for lack of fit. If, instead of restricting ourselves to using clusters of near replicates, we more generally create rational subgroups of the data in which there is reason to believe that observations in a subgroup are more similar than observations in different subgroups, the tests used for detecting lack of fit have been shown to be sensitive to lack of independence. The original motivation for this application was the observation that Shewhart's means charts based on rational subgroups are sensitive, not only to nonconstant mean values, but also are senstivie to lack of independence. To develop graphical procedures for checking lack of independence (as well as lack of fit) we returned to the idea of creating means charts. An obvious, but naive and little referenced, method for creating means charts for linear models is simply to create rational subgroups, average the residuals within the rational subgroups, and chart these means of residuals. We discuss the weaknesses of this naive approach, suggest two alternatives, examine some theory for the alternatives, and explore the usefulness of all three approaches.
 

Matrix computations in Survo
Seppo MUSTONEN, University of Helsinki, Finland

Survo is a general environment for statistical computing and related areas (Mustonen 1992). Its current version SURVO 98 works on standard PC's and it can be used in Windows, for example. The first versions of Survo were created already in 1960's. The current version is based on principles introduced in 1979 (Mustonen 1980, 1982).

According to these principles, all tasks in Survo are handled by a general text editor. Thus the user types text and numbers in an edit field which is a rectangular working area always partially visible in a window on the computer screen. Information from other sources - like arrays of numbers or textual paragraphs - can also be transported to the edit field.

Although edit fields can be very large giving space for big matrices and data sets, it is typical that such items are kept as separate objects in their own files in compressed form and they are referred to by their names from the edit field. Data and text are processed by activating various commands which may be typed on any line in the edit field. The results of these commands appear in selected places in the edit field and/or in output files. Each piece of results can be modified and used as input for new commands and operations.

This interplay between the user and the computer is called `working in editorial mode'. Since everything is directly based on text processing, it is easy to create work schemes with plenty of descriptions and comments within commands, data, and results. When used properly, Survo automatically makes a readable document about everything which is done in one particular edit field or in a set of edit fields. Edit fields can be easily saved and recalled for subsequent processing.

Survo provides also tools for making new application programs. Tutorial mode permits recording of Survo sessions as sucros (Survo macros). A special sucro language for writing sucros for teaching and expert applications is available. For example, it is possible to pack a lengthy series of matrix and statistical operations with explanatory texts in one sucro. Thus stories about how to make statistical analysis consisting of several steps can be easily told by this technique. Similarly application programs about various topics are created rapidly as sucros.

Matrix computations are carried out by the matrix interpreter of Survo (originated in its current form in 1984 and continuously extended since then). Each matrix is stored in its own matrix file and matrices are referred to by their file names. A matrix in Survo is an object having the following attributes:

1.

Internal name of matrix, for example, INV(X'*X)*X'*Y,

2.

# of rows and columns,

3.

Type of matrix (general, symmetric, diagonal),

4.

Labels of rows and columns, each up to 8 characters,

5.

Matrix elements, real numbers in double accuracy.

A matrix has always an internal name, typically a matrix expression that tells the `history' of the matrix. The internal name is formed automatically according to matrix operations which have been used for computing this particular matrix. Also row and column labels - extremely valuable for identifying the elements - are inherited in a natural way. Thus when a matrix is transposed or inverted, row and column labels are interchanged by the Survo matrix interpreter. Such conventions quarantee that resulting matrices have names and labels reflecting the true role of each element. Assume, for example, that for linear regression analysis we have formed a column vector Y of values of regressand and a matrix X of regressors. Then the command $${\hbox{MAT }} B=INV(X'*X)*X'*Y$$

gives the vector of regression coefficients as a column vector (matrix file) B with internal name INV(X'*X)*X'*Y and with row labels which identify regressors and a (single) column label which is the name of the regressand. All statistical operations of Survo related e.g. to linear models and multivariate analysis give their results both in legible form in the edit field and as matrix files with predetermined names. Then it is possible to continue calculations from these matrices by means of the matrix interpreter and form additional results. When teaching statistical methods it is illuminating to use the matrix interpreter for displaying various details step by step.

The website of Survo (www.helsinki.fi/survo/eindex.html) will also contain more information about the topic of this talk.
 

References

Mustonen, S. (1980). SURVO 76 Editor, a new tool for interactive statistical computing, text and data management. Research Report No. 19, Dept. of Statistics, University of Helsinki.

Mustonen, S. (1982). Statistical computing based on text editing. Proceedings in Computational Statistics, 353-358, Physica-Verlag, Wien.

Mustonen, S. (1992). Survo - An Integrated Environment for Statistical Computing and Related Areas. Survo Systems, Helsinki.
 

Kiefer ordering of simplex designs for second-degree mixture models with four or more ingredients
Friedrich PUKELSHEIM*, University of Augsburg, Germany

For mixture models on the simplex, we discuss the improvement of a given design in terms of increasing symmetry, as well as obtaining a larger moment matrix under the Loewner ordering. The two criteria together define the Kiefer design ordering. The Kiefer ordering can be discussed in the usual Scheffé model algebra, or in the alternative Kronecker product algebra. We employ the Kronecker algebra which better reflects the symmetries of the simplex experiment region. For the second-degree mixture model, we show that the set of weighted centroid designs constitutes a convex complete class for the Kiefer ordering. For four ingredients, the class is minimal complete. Of essential importance for the derivation is a certain moment polytope, which is discussed in detail. Kiefer ordering of simplex designs for second-degree mixture models with four or more ingredients.
 
 

* Joint work with Norman R. DRAPER (University of Wisconsin, Madison, USA) and Berthold HEILIGERS (Otto-von-Güricke-University, Magdeburg, Germany).
 

An extension of Richards' lemma with applications to a class of profile likelihood problems
Alastair J. SCOTT*, University of Auckland, New Zealand

Suppose that $\ell(\mbox{\boldmath$\theta$ }, \mbox{\boldmath$\tau$ })$ is a log-likelihood function, where $\mbox{\boldmath$\theta$ }$ is the $p\times 1$ vector of parameters of interest and $\mbox{\boldmath$\tau$ }$ is a$q\times 1$ vector of nuisance parameters. Set $\mbox{\boldmath$\delta$ }= (\mbox{\boldmath$\theta$ }^T,\mbox{\boldmath$\tau$ }^T)^T$. We let $\widehat{\mbox{\boldmath$\delta$ }}= (\widehat{\mbox{\boldmath$\theta$ }}^T, \widehat{\mbox{\boldmath$\tau$ }}^T)^T$ denote the maximum likelihood estimator of $\mbox{\boldmath$\delta$ }$ and write the observed information matrix, $\mbox{\boldmath$\cal I$ }(\mbox{\boldmath$\delta$ }) = -\frac{\partial^2\ell}{\partial\mbox{\boldmath$\delta$ }\partial\mbox{\boldmath$\delta$ }^T},$in the partitioned form

$${\mbox{\boldmath$\cal I$ }}(\mbox{\boldmath$\delta$ }) \left[\begin{array}{cc}-\frac{\partial^2\ell}{\partial\mbox{\bol......ial\mbox{\boldmath$\tau$ }\partial\mbox{\boldmath$\tau$ }^T}\end{array}\right]$$ =

$$\left[\begin{array}{cc} {\mbox{\boldmath$\cal I$ }}_{\theta\theta......I$ }}_{\tau\theta} & {\mbox{\boldmath$\cal I$ }}_{\tau\tau} \end{array}\right].$$ = (1)

The profile likelihood, $\ell_M(\mbox{\boldmath$\theta$ })$, is defined by setting $\ell_M(\mbox{\boldmath$\theta$ })= \sup_{\mbox{\boldmath$\tau$ }}\ell [\mbox{\boldmath$\theta$ }, \mbox{\boldmath$\tau$ }]$. Richards (1961) showed that, under suitable regularity conditions:

(i)

$$\left.\frac{\partial\ell_M(\mbox{\boldmath$\theta$ })}{\partial\m......\boldmath$\theta$ }}\right\vert _{\widehat{\mbox{\boldmath$\theta$ }}} = {\bf0}$$; and

(ii)

$$-\frac{\partial^2\ell_M(\mbox{\boldmath$\theta$ })}{\partial\mbox......ath$\cal I$ }}_{\theta\tau}^T)\bigg\vert _{\widehat{\mbox{\boldmath$\delta$ }}}$$.

The result means that, if we work with $\ell_M(\mbox{\boldmath$\theta$ })$ in place of $\ell(\mbox{\boldmath$\theta$ }, \mbox{\boldmath$\tau$ })$ and treat it as an ordinary likelihood function, then we get the correct results, i.e.,

1.

The $\widehat{\mbox{\boldmath$\theta$ }}$ which maximizes $\ell_M$ is the MLE of $\mbox{\boldmath$\theta$ }$;

2.

$$\left[-\frac{\partial^2\ell_M({\theta})}{\partial\mbox{\boldmath$\......}\right]^{-1}_{\mbox{\boldmath$\theta$ }=\widehat{\mbox{\boldmath$\theta$ }}}$$ is the asymptotic covariance matrix of $\widehat{\mbox{\boldmath$\theta$ }}$, i.e. it is equal to the appropriate block of the inverse information matrix for the full likelihood;

3.

$\ell_M(\widehat{\mbox{\boldmath$\theta$ }}) = \ell (\widehat{\mbox{\boldmath$\theta$ }}, \widehat{\mbox{\boldmath$\tau$ }})$ is the maximized value of the log likelihood for constructing deviances etc.

 
 

In this paper we use an extended version of Richards' result to produce semi-parametric maximum-likelihood estimators of regression parameters for a wide class of study designs and data structures in which some units are not fully observed. (Case-control studies are the most common examples of such studies; see Scott and Wild (1999) for a more complete list of examples.) The nuisance ``parameter" here is the joint distribution function of the covariates, something that is rarely of interest in its own right and which is often impossible (or at least very difficult) to model.
 

References

Richards, F. S. G. (1961). A method of maximum likelihood estimation. J. Roy. Statist. Soc. B 23, 469-476.

Scott, A. J. and Wild, C. J. (1999). Semi-parametric maximum likelihood inference for generalized case-control studies. J. Statist. Plann. Inf. 69, in press.
 
 

* Joint work with Chris WILD (University of Auckland, New Zealand).
 

The infusion of matrices into statistics
Shayle R. SEARLE, Cornell University, Ithaca, USA

An attempt is made at tracing the history of the use of matrices and their algebra in statistics. Much of that history is quite recent, compared to the age of matrices themselves. They are said to have originated in 1858; but barely entered statistics until the 1930s. Even the 1950s, when multiple regression (a topic ideally and easily suited to matrix notation) was coming to be widely taught and written about, that was mostly without the use of matrices.
 

Nonnegative estimation of variance components in multivariate unbalanced mixed models with two variance components
Bimal Kumar SINHA, University of Maryland Baltimore County, Catonsville, USA

In this talk we address the problem of nonnegative estimation of variance components in unbalanced multivariate mixed models with two variance components. This generalizes the work in Mathew/Niyogi/Sinha (JMA, 1994). We also discuss REML estimation of two variance components in univariate unbalanced models. We illustrate our results with an example.
 

Revisiting Hua's matrix equality and
related inequalities, Schur complements,
and Sylvester's law of inertia
George P. H. STYAN*, McGill University, Montréal, Canada

We revisit the elegant matrix equality first established in 1955 by the distinguished Chinerse mathematician Loo-Keng Hua (1910-1985):

(I-B^*B) - (I-B^*A)(I_n-A^*A)^-1(I-A^*B) = -(A-B)^*(I_m-AA^*)^-1(A-B).

Here both A and B are $m\times n$ complex matrices, I is the identity matrix (I_n is the $n\times n$ identity matrix), ^* denotes conjugate transpose, and all the singular values of A are strictly less than 1.

We begin our paper by presenting Hua's original proof, which does not appear to be widely known [``Inequalities involving determinants" (in Chinese), Acta Math. Sinica 5 (1955), 463-470; English translation: Transl. Amer. Math. Soc. Ser. II 32 (1963), 265-272].

We then offer two new proofs, with some extensions, of Hua's matrix equality, and associated matrix and determinant inequalities. Our new proofs use a result concerning Schur complements, and an apparently new generalization of Sylvester's law of inertia. Each proof is useful in its own right, and we believe that these two proofs are new.

We continue by generalizing Hua's matrix equality and inequality, and provide an upper bound as strong as the lower bound in Hua's determinantal inequality. Some historical, biographical and bibliographic information is also included.
 
 

* Joint work with Christopher C. PAIGE (McGill University, Montréal, Canada), Bo-Ying WANG (Beijing Normal University, Beijing, China) and Fuzhen ZHANG (Nova Southeastern University, Fort Lauderdale, USA).
 
 

CONTRIBUTED PAPERS

Improved estimation of characteristic roots in principal analysis
S. E. AHMED, University of Regina, Canada

The problem of estimation of characteristic roots of the covariance matrix of the multivariate normal distribution is considered. Stein-type and pretest estimators are proposed for the population characteristic root vector.

The statistical objective here is to investigate the merits of the proposed estimators relative to the sample characteristic roots. It is demonstrated that the proposed estimators are asymptotically superior to the sample characteristic roots.
 

The examination and analysis of residuals for some biased estimators in linear regression
Fikri AKDENIZ, Çukurova University, Adana, Turkey

In the presence of collinearity certain biased estimation procedures like ridge regression, generalized inverse estimator, principal component regression, Liu estimator, or improved ridge and Liu estimators are used to improve the ordinary least squares estimates in the linear regression model. In this paper new biased estimator (Liu estimator), almost unbiased (improved) Liu estimator and their residuals will be analyzed and compared with OLS residuals in terms of mean squared error.
 

2D projections of 3D spatial distributions
Sofia AZEREDO LOPES, University of Sheffield, UK

The present poster aims to give a brief explanation of my research in spatial statistics.

The study of composite materials have been increased in recent years. Studies about their composition and manufacture are extremely relevant for obtaining good results.

These materials consist of a solid matrix where several particles of other substances were embedded. It is of great importance to know the way these particles are distributed inside the material as this is going to influence the materials' quality and strength. Most composite materials are also opaque which makes it difficult to observe the particles distribution. The only possible way is to make a cut and observe the pattern obtained by the particles that were intercepted by that plane. This process is based on the application of several statistical tests. I intend to give an introduction about the tests used: Nearest Neighbour, F, J and Ripley's K functions and also their implementation on both spaces.

The main aim of this research is to try to infer (using spatial statistical methods) the way particles are distributed in a three dimensional space when only information from two dimensional spaces obtained from the same material are available.
 

Internet survey by Direct Mailing
Thomas BENESCH, Graz University of Technology, Graz, Austria

The methods of the written, telephone and personal questionings have been established for many years. Here the method of Direct Mailing is introduced.

The method of Direct Mailing means that a letter of transmittal was sent by means of e-mail, which refers to a questionnaire, which is at the WWW. Therefore representativity is ensured. Further the structure of the questionnaire can be carried out by the usual method of the written questionnaire.

The investigation was limited to students of Graz University of Technology, Austria.

On Friday, 9 October 1998 at 16:45 an e-mail was coincidentally sent to 1000 students in which they were asked to click and fill out the questionnaire on the network.

The date Friday, 9 October 1998 (16:45) was selected because:

• The semester began on Thursday, 1 October 1998, and therefore also most of the students were already on the Technical University;
• The accessibility should also be examined on the weekend;
• Of our hypothesis that says that we would get the most responses on Monday, 12 October 1998.

The e-mail, which was sent, was directly addressed in each case (no mass transmission). The sending was however already terminated on the same day, around 19 o'clock.

The questionnaire consists 27 questions about the internet and also 5 demographic questions. Our response rate was situated after 4 days when the e-mail was sent of approximately 25 per cent, i.e. 75 per cent of all responses, this increased after termination of the empirical study to approximately 34 per cent. On Monday, 12 October 1998 we received 123 responses, which corresponds to a response rate of approximately 13 per cent on this day. The overall distribution on the basis of the response arrival times was left-skewed. Separated the distribution per weekdays and weekend days showed that the response arrival times depended on them.
 

Statistical analysis of some multivariate models using quasi-least squares
N. Rao CHAGANTY, Old Dominion University, Norfolk, USA

The method of quasi-least squares has been introduced and developed recently in a series of three papers: Chaganty (1997), Shults and Chaganty (1998), and Chaganty and Shults (1999). These papers are concerned with estimating the parameters in longitudinal data analysis problems that occur in the framework of generalized linear models. In this talk I will discuss the application of the method to analyze growth curve models and some multivariate discrete models. I will also discuss some test statistics and their large sample properties. Finally, examples will be given to illustrate the method on real life data.
 

Statistical inference on special manifolds
Yasuko CHIKUSE, Kagawa University, Takamatsu, Japan

The special manifolds of our interest in this paper are the Stiefel manifold and the Grassmann manifold. Formally, the Stiefel manifold V_k,m is the space of k-frames in the m-dimensional real Euclidean space R^m, represented by the set of $m\times k$ matrices X such that X^'X=I_k, where I_k is the $k\times k$ identity matrix, and the Grassmann manifold G_k,m-k is the space of k-planes (k-dimensional hyperplanes) in R^m. We see that the manifold P_k,m-k of $m\timesm$ orthogonal projection matrices idempotent of rank k corresponds uniquely to G_k,m-k. This paper is concerned with statistical analysis on the manifolds V_k,m and P_k,m-k as statistical sample spaces consisting of matrices. For the special case k=1, the observations from V_1,m and G_1,m-1 are regarded as directed vectors on a unit sphere and as axes or lines undirected, respectively. There exists a large literature of applications of directional statistics and its statistical analysis, mostly occurring for m=2 or 3 in practice, in Geological Sciences, Astrophysics, Biology, Meteorology, Medicine and other fields. The analysis of data on the general Stiefel manifold V_k,m is required especially for $k\leq m\leq 3$ in practical applications, and the Grassmann manifold is a rather new subject treated as a statistical sample space.

Among population distributions uniform and non-uniform defined on the special manifolds, the matrix Langevin distributions are most commonly used for statistical analyses. The matrix Langevin distributions are exponential whose density functions are proportional to $exp[tr(\cdot)]$. The problems of estimation and tests for hypotheses of the parameters and of classifications of the matrix Langevin distributions and the related sampling distributions are expressed in terms of hypergeometric functions with matrix arguments. The solutions for these inference and distribution problems can be asymptotically evaluated for large sample size, for large concentration, or for high dimension.

By applying Procrustes methods we are led to the concept of Procrustes distance defined on the special manifolds as well as the usual Euclidean one. These distance measures are useful in constructing suitable (kernel) density estimators in the theory of density estimation and in defining measures of association on the manifolds. Statistical problems on the special manifolds are transformed to those on the Euclidean spaces by using the decomposition theorems of the manifolds; e.g., in evaluating multiple integrals and generating new population distributions on the manifolds.

A detail discussion will be given of those problems raised above which are of unique character as statistical analyses on our special manifolds.
 

An asymmetric property of the correlation coefficient
Yadolah DODGE, University of Neuchâtel, Switzerland

In this paper, we derive some simple formulae to express the correlation coefficient between two random variables in the case of a linear relationship. One of these representations, the cube of the correlation coefficient, is given as the ratio of the skewness of the response variable to that of the explanatory variable. This result, along with other expressions of the correlation coefficient presented in this paper, has implications for choosing the response variable in a linear regression modelling.
 

Keywords: Correlation, least squares, linear regression, response variable, direction dependence regression, skewness.
 

Simultaneous SVD: Grillenberger's approach
Hilmar DRYGAS, University of Kassel, Germany

This paper deals with an approach due to Christian Grillenberger, late professor of mathematical stochastics at the University of Kassel, Germany. He passed away of a heart attack at the 25th of May, 1998.

The problem of jointly diagonalizing complex $n\times n$-matrices was dealt in many papers. The necessary and sufficient condition is that the involved matrices commute. Grillenberger gave a easy proof that also the projections appearing in the spectral decompositions commute. This results in simple simultaneous decompositions.

If A and B are rectangular matrices the hermiticity of both AB and BA is the necessary and sufficient condition for a joint singular decomposition. It is shown that this can also be obtained from Grillenberger's result. Generalizations to a finite number of matrices are obvious.
 

Optimality results for the method of averages
Richard William FAREBROTHER, Victoria University of Manchester, UK

Three distinct variants of the method of averages were proposed by Tobias Mayer in 1750, by Pierre Simon Laplace in 1788, and by Pierre Simon Laplace again in 1818. Even though the merits of the method of least squares became apparent in the 1820s and 1830s, these and other variants of the method of averages has continued to be employed by leading practical scientists until relatively recent times. In this paper, we outline some of the optimality results that have been established in respect of this fitting procedure.
 

Robust estimators in linear models
Johan FELLMAN*, Swedish School of Economics, Helsinki, Finland
Kenneth NORDSTRÖM, University of Oulu, Finland

In applied experimental design studies and in the theoretical studies of linear models the efficiency and the unbiasedness of the estimates have been considered but the studies have followed different paths. This study is an attempt to bridge the gap between these two research traditions.

If there are regressors of minor interest then a central problem is if the corresponding nuisance parameters should be included in the linear model or not. If the true model contains the nuisance parameters but we ignore them when estimate the main parameters then usually the estimates are biased. On the other hand if the nuisance parameters can be left out but we include them in the model then the estimates of the main parameters may be inefficient.

In applied research the interest has mainly been concentrated on the effect of these model mis-specifications on the estimates of the individual parameters. However, there may be parametric functionals which are robust against these mis-specifications. Within a more general framework Nordström and Fellman (LAA 127, 341-361) have discussed these problems. If the reduced model and the large model are nested then the subset of parametric functionals, which are ``bias-robust" with respect to underspecification, is equal with the subset of parametric functionals, which are ``variance-robust" with respect to overspecification. The coefficient vectors of such functionals belong to the orthogonal complement of the range of the alias (bias) matrix. If the inclusion or exclusion of block effects are considered then the models are not nested but using Khuri's generalized orthogonal block condition similar results can be obtained.

We present some theoretical results and elucidate them with examples from the literature. In this study we mainly consider bias-robustness of the estimates.
 

Estimation of variance components by maximum likelihood method in certain mixed linear models
Stanisaw GNOT, Pedagogical University, Zielona Góra, Poland

The main problem considered in the paper is studing properties of maximum likelihood estimators MLE's in certain mixed models with two variance components. It is shown how much MLE's can be biased. Some propositions is given how to improve them to make the biasedness smaller. Two way layouts models are considered in details, for which results of simulations are presented.
 

The Moore-Penrose inverse of a partitioned nonnegative definite matrix
Jürgen GROß, University of Dortmund, Germany

A formula for the Moore-Penrose inverse of an arbitrary partitioned symmetric nonnegative definite matrix is derived, and some necessary and sufficient conditions for the coincidence of our formula with the so called generalized Banachiewicz formula are given. Eventually, a Löwner ordering relationship between a generalized Schur complement in the matrix and a generalized Schur complement in it's Moore-Penrose inverse is revealed.
 

On some matrix inequalities and their
statistical applications
Jan HAUKE, Adam Mickiewicz University, Poznan, Poland

Let A and B be nonnegative definite matrices. It is known that the Löwner partial ordering ${\bf A} \leq {\bf B}$ implies $f({\bf A}) \leq f({\bf B})$ only for some classes of functions [see Löwner (1934)] and it is true in the case of ${\bf A}^2 \leq {\bf B}^2$ not for all matrices [see Stepniak (1987)], but is true for commuting matrices. Mathias (1991) analysed conditions for the equivalence of the above inequalities. We are interested in the implication: ${\bf A}^2 \leq {\bf B}^2$ implies ${\bf A}^2 \leq c({\bf C}+{\bf B})^2$. The aim is to characterize matrices C and scalars c for which the implication is true. Taking matrix C as an identity matrix we obtain special case of inequalities analysed by Nordström and Mathew (1996). Some statistical applications of the inequalities are pointed out.
 

References

Löwner, K. (1934). Über monotone Matrixfunktionen. Math. Z. 38, 177-216.

Mathias, R. (1991). The equivalence of two partial orders on a convex cone of positive semidefinite matrices. Linear Algebra Appl. 151, 27-55.

Nordstrom, K. and Mathew, T. (1996). Inequalities for sums of matrix quadratic forms. Linear Algebra Appl. 237/238, 429-447.

Stepniak, C. (1987). Two orderings on a convex cone of nonnegative definite matrices. Linear Algebra Appl. 94, 263-272.
 

Regression methods to model under registration problems in population data
Maged ISHAK, University of Newcastle, Australia

When reliable population data are available from the vital registration system, they are eventually sufficient to be used for analysing the demographic characteristics of the country and their important related implications. In fact, the reliability of the vital registration data in developing countries as well as some communities in developed countries (indigenous communities for example) is seriously affected by the problems of under-registration and age misreporting. Although the problem of misreporting of population data in developed countries has been tackled through many studies in literature (Keyfitz 1984, Ishak 1987 and Sivamurthy 1989), the more influential problem of under-registration is still a real obstacle towards a more valid estimates for population figures in these countries. Mortality data, especially of the childhood span, are known to be the mostly affected data by under registration. In fact, between one-third and two-third of the infant deaths in developing countries are escaping registration (International Institute of Statistics 1987). In this paper, some mathematical and regression techniques were presented to model the under registration behaviour, where only limited information about the magnitude of the problem in the country is available. The derived model was then combined with a regression model for infant mortality to finally present an improved model that is free from the effects of both the under registration and age misreporting problems. To verify the applicability of the model, this was applied to the infant mortality data of Egypt. The model could reflect an acceptable pattern of infant mortality in the country which is significantly higher in reliability than the pattern reflected by the officially published data.
 

Keywords: Direct/Indirect survival estimation, demographic estimations from incomplete data, mortality models, under registration model.
 

Recursive formulae for the average efficiency factor in alpha designs
J. A. JOHN*, University of Waikato, Hamilton, New Zealand
E. R. WILLIAMS, CSIRO Forestry and Forest Products, Kingston, Australia

Alpha designs are a class of resolvable incomplete block designs and are widely used in practice, particularly in plant and tree breeding trials. An alpha design for r replicates and block size k can be generated from a $k\times r$ array of numbers modulo s, where v=sk is the number of treatments in the design. Alpha designs are available for a wide range of parameter combinations. For some of these combinations they correspond to the well-known square and rectangular lattice designs and hence are optimal, i.e. they have maximum average efficiency factor, E. In general, however, computer search methods are used to choose the best alpha design, i.e. the design with E closest to a known theoretical upper bound. The calculation of E involves matrix inversions and so more easily obtained objective functions are used in the search algorithms. Typical surrogate objective functions include the trace of the square and cube of the design information matrix. There is, however, the danger that such alternatives can lengthen the computer search and perhaps even prevent the optimal design being found. This is particularly the case for computer search procedures based on exchange and interchange algorithms, e.g. the design generation package CycDesigN carries out random interchanges on the alpha array.

This talk presents a recursive method that allows an efficient updating procedure for E as part of the exchange algorithm operating on the alpha array. The method leads to some interesting results involving Hermitian matrices. The new updating procedure for E compares very favourably with the existing approaches based on surrogate objective functions and not only speeds up the search for near-optimal alpha designs, but often results in the generation of designs with improved E. Some comparisons will be presented.
 

An exact joint confidence region and test in multivariate calibration
Subramanyam KASALA, University of North Carolina, Wilmington, USA

Consider a vector valued response variable related to a vector valued explanatory variable through a normal multivariate linear model. The multivariate calibration deals with statistical inference on unknown values of the expanatory variable. The problem addressed is the construction of joint confidence regions for several unknown values of the explanatory variable. The problem is investigated when the variance covariance matrix is a scalar multiple of the identity matrix and also when it is a completely unknown positive definite matrix. An exact confidence region is constructed by generalizing the results in Mathew and Kasala (1994, The Annals of Statistics).
 

A generalization of Whittle's formula
André KLEIN, University of Amsterdam, The Netherlands

In a pioneering paper (1953) Whittle developed a formula for expressing Fisher's information matrix of multivariate time series models. It is described in function of the spectral density of the time series process. The existing relationship is extended to the whole matrix instead of one element and is related with an alternative expression. The Hermitian property of the matrices under study allows us to formulate the link in a theorem which is further illustrated with an example.
 

Keywords: Spectral density, Hermitian matrix, permutation matrix.
 

Density expansions for the
sample correlation matrix
Tõnu KOLLO* and Kaire RUUL, Institute of Mathematical Statistics, Tartu, Estonia

The most commonly used methods for approximating density and distribution functions are Edgeworth type expansions. Usually it is assumed that the dimension of the approximating distribution is the same as the dimension of the distribution which has to be approximated. In this paper we are going to examine multivariate expansions where the dimensions of the distributions can be different. In many situations it seems natural to approximate some multivariate distribution with a distribution of higher dimensionality. The sample correlation matrix R as a function of the sample covariance matrix S represents a typical example of the case: one might be interested in approximating the joint distribution of $\frac12p(p-1)$ nonsymmetric random elements of R with the joint distribution of $\frac12p(p+1)$ different random elements of S.

In Kollo & von Rosen (1998) a general formula is obtained to represent a multivariate density through another (possibly higher-dimensional) density as a formal series expansion where derivatives of the approximating density and cumulants of both random vectors (matrices) appear in growing orders in the terms of the expansion.

This general result was applied for approximating of the density of the sample correlation matrix R. The first terms of the expansion include the cumulants of R up to the third order. Approximate expressions of the cumulants have been found in the paper using an expansion of the characteristic function of R. Later the cumulants were used in the density approximation formulae, based on Wishart distribution and multivariate normal distributions.

In the twodimensional case a simulation experiment was carried out i.e. the density of the sample correlation coefficient was approximated by the Wishart distribution and the multivariate normal distribution. These approximations were compared with the simulated empirical distributions and the classical univariate Edgeworth expansion.
 

Reference

Kollo, T. and von Rosen, D. (1998). A unified approach to the approximation of multivariate densities. Scandinavian Journal of Statistics, 25, 93-109.
 

Bounds for the spread of a matrix
Ravinder KUMAR, Dayalbagh Educational Institute, Agra, India

Given a square matrix A, the maximum distance between its eigenvalues is known as spread of A. A new characterization for the spread of a normal matrix is presented. New bounds for the spread have been obtained and compared. Some of our results are generalizations of known results. These bounds have application in Statistics.
 

Generating correlated categorical variates via Kronecker products
Alan LEE, University of Auckland, New Zealand

Analysis methods for longitudinal or clustered categorical data are currently a hot reserarch topic. Most of these rely on asymptotic theory for the justification of inferences. To assess the small-sample behaviour of these methods, we need to be able to generate random vectors having a variety of categorical distributions. A feature of several of these generation methods is the use of the inversion algorithm, which requires that the joint distribution of the correlated variates be completely specified.

Suppose we have a random n-vector of categorical variates, $Y=(Y_1, Y_2, \ldots$, Y_n), where Y_i may assume any one of the values $y_{i,1},\ldots,y_{i,I_i}$, and let $p_{i_1,i_2,\ldots,i_n}$$=Pr[Y_1=y_{1,i_1},\ldots,Y_n=y_{n,i_n}]$ be its joint probability function. In many cases the probabilities $p_{i_1,i_2,\ldots,i_n}$ may be quite difficult to specify analytically, but simple formulae may exist for the cumulative probabilities

$$P_{i_1,i_2,\ldots,i_n}=Pr[Y_1\leq y_{1,i_1},\ldots,Y_n\leq y_{n,i_n}].$$

In this talk we describe develop a simple, general recursive method of calculating the p's from the P's which is based on Kronecker products. If p and P are the N-vectors whose elements are $p_{i_1,i_2,\ldots,i_n}$ and $P_{i_1,i_2,\ldots,i_n}$, arranged in lexicographic order, then

$$P=(A_{I_1}\otimes A_{I_2}\otimes\cdots\otimes A_{I_n})\,p,$$ (1)

where $\otimes$ denotes the Kronecker product, and A_k is the $k\times k$matrix with ones on and below the diagonal, and zeroes above. To compute p from P, we invert (1) and get

$$p = (A_{I_1}^{-1}\otimes A_{I_2}^{-1}\otimes\cdots\otimesA_{I_n}^{-1})P.$$ (2)

The matrix inverse of A_k can be given explicitly: it is

$$A_k^{-1}=\left[\begin{array}{cccccc}1&0&0&\cdots&0&0\\-1......&\ddots&\vdots&\vdots\\0&0&0&\cdots&-1&1\end{array}\right ].$$

We describe a recursive S function to implement the method and give several examples.
 

On maximum likelihood estimation for
the VAR-VARCH model
Shuangzhe LIU* and Wolfgang POLASEK, University of Basel, Switzerland

We consider a general multivariate conditional heteroskedastic time series model and derive a representation for the information matrix of the maximum likelihood estimator by using the standard matrix differential calculus techniques. As a special case, we discuss the VAR-VARCH model, and demonstrate the maximum likelihood estimation of the information matrix in an example with simulated data.
 

Robust linear estimation in an incorrectly
specified restricted linear model
Augustyn MARKIEWICZ, Agricultural University of Poznan, Poland

Admissible and linearly sufficient estimators in restricted linear model are considered. These estimators are found in Heiligers and Markiewicz (1996) to be special general ridge estimators. A preliminary study of their robustness (validity) was made in Markiewicz (1998) in the context of the correct model and the assumed model, resulting in specification errors. Robustness properties of the estimators are studied regarding misspecification of the dispersion matrix of the errors vector, the model matrix, and parameters restrictions. Some robust estimators are characterized.
 

Keywords: Admissibility, linear sufficiency, general ridge estimator, restricted linear model.
 

AMS Subject Classification: Primary 62F10; secondary 62J05.
 

References

Heiligers, B. and Markiewicz, A. (1996). Linear sufficiency and admissibility in restricted linear models. Statist. Probab. Lett. 30, 105-111.

Markiewicz, A. (1998). Comparison of linear restricted models with respect to the validity of admissible and linearly sufficient estimators. Statist. Probab. Lett. 38, 347-354.
 

A geometric approach to the canonical correlations in two- and three-way layouts
John MAROULAS, National Technical University, Athens, Greece

In this paper we present some results concerning canonical correlations in two- and three-way layouts. A useful proposition relating the canonical correlations to the singular values of a specific matrix is proved and a geometrical explanation is given. Furthermore, some formulas concerning the numbers of unit canonical correlations in a three-way layout are given and even they are generalized for k-way layout.
 

Tolerance regions and simultaneous tolerance regions in multivariate regression
Thomas MATHEW, University of Maryland, Catonsville, USA

In the talk, we shall describe the construction of tolerance regions and simultaneous tolerance regions in a multivariate linear regression model. Of particular interest will be the problem of obtaining suitable approximations to the tolerance factor. Following the ideas given by John (Sankhya, 1963), we shall consider several approximate tolerance factors and report the accuracy of the approximations, numerically. The numerical computation of the simultaneous tolerance factor wil be explained and the results will be illustrated using an example.
 

Simple and good bounds for the Perron root
Jorma K. MERIKOSKI* and Ari VIRTANEN, University of Tampere, Finland

Let A be a nonnegative $n\times n$ matrix with row sums r₁, r₂, $\ldots\,$, r_n and Perron root $\lambda$. It is well-known that $\mathop{\min}\limits_i\sum\limits_ka_{ik}r_k/r_i\le\lambda\le\mathop{\max}\limits_i\sum\limits_ka_{ik}r_k/r_i$, where the minimum and the maximum are taken over all the i's with r_i > 0. Using a simple shifting technique, we find bounds for $\lambda$, which appear to be amazingly good in general. We also study the behaviour of these bounds applied to powers of A.
 

In the world of collinearity $\ldots$
Simo PUNTANEN, University of Tampere, Finland

This talk considers some difficulties - inspired by Belsley (1991) - related to interpretation of the relation between the correlation and corresponding cosine. We illustrate how carefully we must take steps in the world of collinearity. Numerical calculations are done using Survo - a statistical software developed by Professor Seppo Mustonen; for general features of Survo, see Mustonen (1992, 1999).
 

References

Belsley, David A. (1991). Conditioning Diagnostics: Collinearity and Weak Data in Regression. Wiley, New York.

Mustonen, Seppo (1992). Survo: An Integrated Environment for Statistical Computing and Related Areas. Survo Systems, Helsinki.

Mustonen, Seppo (1999). Matrix computations in Survo. Abstracts of the Eighth International Workshop on Matrices and Statistics. Dept. of Mathematics, Statistics & Philosophy, University of Tampere, Finland.
 

Information channels and comparison of linear experiments
Czesaw STEPNIAK, Agricultural University of Lublin, Poland

Any linear experiment can be presented as an information channel transforming a signal $\beta$ into the observation vector $X=A\beta+e$. Suppose that X is transformed again in this way and results in a new random vector Y. It seems natural that the initial vector X is at least as informative as Y. It would be interesting to converse the problem.

Many orderings of statistical experiments or random vectors, considered in the literature, include the phrase ``$\ldots$ is at least as informative as $\ldots$" or ``$\ldots$ is at least as good as $\ldots$". Which of the orderings can be presented as an information channel$\,$?

Our aim is to answer this question.
 

Some extremal properties of correlation matrices
Ene-Margit TIIT*, Hele-Liis VIIRSALU and Virgi PUUSEPP
University of Tartu, Estonia

A. Pattern of signs of first type is defined as a symmetric matrix S of order k having all elements equal to +1 or -1 and positive elements ones on the main diagonal. As it follows, the number m(k) of different patterns of signs of order k equals

2^0,5k(k-1).

Let $\Theta_k$ be the set of all correlation matrices of order k. We are interested in correlation matrices having given structure of signs and hence describing some special dependence structure.

Let S be a fixed pattern of signs and $\rho$ be a number from the interval $\lbrack-1,1\rbrack$. Then the matrix

$$R=(S-I)\rho+I$$ (1)

consists of ones on main diagonal and all other elements equal either $\rho$ or $-\rho$. Our question is - when is matrix R a correlation matrix?

More precisely, we are interested to find for each pattern of signsS the set $\Omega(S)$ of values $\rho$ in such a way that the following inclusion holds:

$$R_1=(S-I)\rho +I\in\Theta_k\Leftrightarrow \rho\in\Omega(S).$$

It has been proved that the set $\Omega(S)$ is an interval $\lbrack\rho_1,\rho_2\rbrack$ where the inequalities

$$-1\le\rho_1\le -1/(k-1);\ \ 1/(k-1)\le\rho_2<1$$

hold. The values $\rho_1$ and $\rho_2$ are limiting correlations for the pattern of signs S, and the corresponding correlation matrices (1) are extremal in the sense that they belong to the boundary set of the set $\theta_k$.

The structure of the set $\Omega(S)$ (for given k) is determined by the set of different limiting correlations. The extremal values -1 and +1 occur for very special patterns of signs only.
 
 

B. Pattern of signs of the second type is a symmetric matrix Q of order k having all elements equal to +1, -1 or 0 and positive elements on the main diagonal. As it follows, the number n(k) of different patterns of signs of the second type of order k is much more compared with the number of patterns of signs of the first type and equals

3^0,5k(k-1).

The next problem is to find for each pattern of signs Q the set $\Omega(Q)$ of values $\rho$ in such a way that the following inclusion holds:

$$R_1=(Q-I)\rho+I\in \Theta_k\Leftrightarrow \rho\in\Omega(Q).$$

In the report the results known for the patterns of signs of the first type will be expanded for the patterns of signs of the second type, too.
 

Quaternions: A matrix oriented approach
Jürgen GROß, Götz TRENKLER and Sven-Oliver TROSCHKE*
University of Dortmund, Germany

The set of quaternions has been introduced by Sir William Rowan Hamilton in 1843. By representing quaternions as 4-dimensional vectors and the multiplication of quaternions as matrix-by-vector-product we provide a matrix oriented approach to the topic. We will investigate properties of the fundamental matrix and use the approach to derive basic and advanced properties of the multiplication of quaternions. As an application of our approach we will examine a quaternion equation which has been frequently considered in the literature.
 

Least squares approximation of non positive (semi)definite matrices of correlation coefficients
Maurizio VICHI, Societa Italiana di Statistica, Rome, Italy

Data with missing values, some three-way data analysis models and some robust estimators of the correlation coefficients may induce correlation matrix A not positive definite (psd). In general, an approximation method for recovering the psd property should transform A (linearly or nonlinearly) into a correlation matrix R, according to a defined loss function, avoiding: (i) changes in the strict monotone relations among the correlation coefficients and (ii) opposite relations between the corresponding elements of the observed and the approximated correlation matrix. Neglecting (i) and (ii) generally induces the change in the ``true" correlation pattern of A and may lead to contradictory conclusions in the following multivariate analyses based on R instead of A.

In this paper monotone parametric and non-parametric constrained transformations of the matrix A are introduced to find the least squares approximation of A subject to constraints avoiding (i) and (ii). A parametric analytical local minimum is determined as well as the optimal solution obtained by solving quadratic constrained problems through optimization tools. Some examples are examined to show the proposed transformations.
 

Keywords: Correlation, positive definite and positive semi-definite matrices, numerical methods, least squares approximation.
 

Essential references

Browne, M. W. (1992). Circumplex models for correlation matrices. Psychometrika, 57, 4, 469-497.

Devlin, S., Gnanadesikan R. and Kettenring, J. R. (1975). Robust estimation and outlier detection with correlation coefficients. Biometrika, 62, 3, 531-545.

Giovagnoli A. and Romanazzi M. (1990). A group majorization ordering for correlation matrices. Linear Algebra and Its Applications, 127, 139-155.

Grone, R., Pierce S. and Watkins W. (1990). Extremal correlation matrices. Linear Algebra and Its Applications, 134, 63-70.

Knol, D. L. and ten Berge, J. M. F. (1989). Least-squares approximation of an improper correlation matrix by a proper one. Psychometrika, 54, 1, 53-61.
 

More on the linear aggregation problem
Fikri AKDENIZ, University of Çukurova, Adana, Turkey
Hans Joachim WERNER*, University of Bonn, Germany

This paper deals with the linear aggregation problem. For the true underlying micro relations, which explain the micro behaviour of the individuals, no restrictive rank conditions are assumed. We compare several estimators for the systematic part of the corresponding macro relations.
 

Distance optimality design criterion and stochastic majorization
Oleksandr ZAIHRAIEV, N. Copernicus University, Torun, Poland

The main subject of the talk is Distance optimality criterion that was put forward by B. Sinha (On the optimality of some designs, Calcutta Stat. Assoc. Bull. 1970, 20, 1-20) quite long ago but has not drawn much attention in the literature for the present. The essense of this criterion within the framework of linear regression model $\left(y,X\beta,\sigma^2I\right)$, where y is a vector of observations having normal distribution, $\beta$ is a vector of unknown regression parameters, $\sigma>0$ is a constant, I is a unit matrix and X is a design matrix of full rank, is the following. We would like to choose such a design that

$$P\left(\Vert\widehat{\beta}-\beta\Vert\le\varepsilon\right)\longrightarrow\max\quad\forall\varepsilon\ge 0,$$

where $\widehat{\beta}=\widehat{\beta}_X$ is the usual least squares estimator of $\beta$.

The close connection of Distance optimality criterion with the notion of stochastic domination turns out to be quite obvious. This connection allows to establish some useful properties of Distance criterion. We also introduce and discuss some new optimal design criterions based on other notions of stochastic majorization, which are weaker than stochastic domination.

Studying of Distance criterion via stochastic majorization has motivated us to formulate some interesting design criterions more. Firstly, it is worth noting that in the case of one unknown regression parameter, Distance criterion is defined via the peakedness of the distribution of $\widehat{\beta}$ according to Birnbaum (On random variables with comparable peakedness, Annals of Mathematical Statistics, 1948, 19, 76-81). It is also well known that Birnbaum's definition was generalized to the multivariate case by Sherman (A theorem on convex sets with applications, Annals of Mathematical Statistics, 1955, 26, 763-766). So, in the case when $\beta$ is a vector, one can take Sherman's definition as a base for optimality criterion, though this criterion turns out to be strong enough. Secondly, it is easy to note that Distance criterion is defined via the Euclidean error $\Vert\widehat{\beta}-\beta\Vert^2$ but one can take instead of that the generalized Euclidean error $(\widehat{\beta}-\beta)'D(\widehat{\beta}-\beta)$ as well for given symmetric positive definite matrix D.

We apply our considerations and solve optimal design problems mainly for the case of one-way first degree polynomial fit model.
 

Testing hypotheses for parameters in mixed
linear models
Roman ZMY´SLONY, Institute of Mathematics, Zielona Góra, Poland

In the paper we deal with the usual mixed linear models. Thus the natural parameter space make variance components and parameters corresponding to the fixed effects. In this paper we present a relationship between quadratic estimation and testing hypothesis for some linear functions of the model parameters. In the problem of testing hypotheses for fixed parameters is well known F-test which accepts null hypotheses if a ratio of the quadratic lengthes of residuals under hypothesis and model is less than a given critical value. In the paper Michalski and Zmyslony (1996) it is proposed a test for vanishing of single variance component in more general linear model. A null hypothesis is rejected if the ratio of the positive and negative part of the locally best quadratic unbiased estimator of this component is sufficiently large. In this paper it is proved that F-test can be derived in the same way it means as a ratio of the positive and negative part of a quadratic unbiased estimator for a corresponding quadratic function connected with null hypothesis. Thus we get unified theorem for constructing tests for hypotheses about both fixed effects and variance components. Testing hypothesis for independence of binormal distribution will be presented as an application of the result.
 

Keywords: Mixed linear models, variance components, quadratic unbiased estimation, testing hypotheses, one-way classification model.
 

AMS Subject Classification: 62F03, 62J10.
 

References

Gnot, S. and Michalski, A. (1994). Tests based on admissible estimators in two variance components models. Statistics, 25, 213-223.

Lehmann, E. L. (1959). Testing Statistical Hypotheses. Wiley, New York.

Michalski, A. and Zmyslony, R. (1996). Testing hypotheses for variance components in mixed linear models. Statistics, 27, 297-310.

Rao, C. R. (1973). Linear Statistical Inference and its Applications, 2nd ed. Wiley, New York.
 

On construction of a basis of Jordan algebra of matrices with applications to a special linear model
Leszek WOJTASIK, Pedagogical University, Zielona Góra, Poland
Stefan ZONTEK*, Technical University, Zielona Góra, Poland

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

PhD SESSION

Matrices in statistical procedures for valid hypothesis testing in regression models with autocorrelated errors
Gülhan ALPARGU*, McGill University, Montréal, Canada

Autocorrelated errors in regression models are responsible for biased estimation of the standard error of coefficient estimates. They also affect the type I error risk of classical F-test of the model and classical t-test of individual coefficients. Hence it is important to take into account the autocorrelated errors in the considered statistical test.

In this paper, we consider ten potential statistical procedures for valid hypothesis testing in regression models with autocorrelated errors. One of them is ordinary least-squares estimation. Three of them are based on generalized least-squares estimation of the slopes, combined with classical t-test. These procedures differ in the nature (constant or random) and the structure of the variance-covariance matrix used for generalized least-squares estimation; see Searle (1971), depending on whether the autocorrelation model and the parameter values are known, or not. Three other procedures are inspired from modified testing in repeated measures analysis of variance; see Greenhouse and Geisser (1959) and also Huynh and Feldt (1976). These three procedures differ in the nature and structure of the variance-covariance matrix used for modifying the number of degrees of the t-test, whereas parameters are estimated by ordinary least squares. The last three procedures considered here originate from modified t-test developed in correlation analysis; see Clifford et al. (1989) and Dutilleul (1993).

We investigate by simulation the type I error risk of all ten procedures in the case of simple linear regression with time series data and an AR(1) structure of the errors for positive and negative values of the autocorrelation parameter. The design matrix can be of three types, fixed following a trend, pseudo-random or random following a first order auto regressive process. Results are discussed in terms of validity. A test procedure is said to be valid if the probability of rejecting the null hypothesis when actually the null hypothesis is true is less than or equal to some specified value, say 0.05.
 
 

* Joint work with Pierre DUTILLEUL (McGill University, Montréal, Canada).
 

The investigation of the stability of two statistical parameter estimators to complex disturbances in the data
Mikhail V. BORISOV, Novosibirsk State University, Russia

In the report, we make a comparison of the orthogonal regression (OR) estimator and the least squares (LS) estimator of the parameters of the autoregressive system:

$$x_{t}^*=\sum_{i=1}^{n}a_{i}x_{t-i}^*+\sum_{i=0}^{n}b_{i}u_{t-i}^*,\quad t=n+1,\ldots,N,\quad (N>n+1).$$

The both ``proper'' and ``non-proper'' disturbances are considered, namely, those under which the estimators do not possess the estimate consistency, asymptotic unbiasedness, etc. We prove the theorem about the LS and the OR estimators identity as functions of data in the first order term of the Taylor-series expansion with respect to the ``true'' non-perturbed data value. Then, by Monte-Carlo analysis, we investigate the region of the estimates' linear dependence on the data under small disturbances. By the above theorem, in this region the LS and the OR estimators are identical. To implement more thorough Monte-Carlo simulations, the terms up to third order in the Taylor-series expansions for both the LS and the OR estimation criteria were analytically achieved. It was found that under circumstances considered as quite general, with non-proper disturbances, the OR estimator yields less the mean square deviation, than the LS estimator.
 

Some comments and a bibliography
on the Craig-Sakamoto theorem
Mylène DUMAIS*, McGill University, Montréal, Canada

We are interested in the following two theorems :

    THEOREM 1. Let the random vector x follow a multivariate normal distribution with mean vector$\mu$ and dispersion (variance-covariance) matrix V, not necessarily positive definite. Let A and B be symmetric non-random matrices. Then x'Ax and x'Bx are independently distributed if and only if

$\hfill \pmatrix{V \cr \mu'}AVB(V : \mu) = 0. \hfill {\rm (1)}$     THEOREM 2. Let A and B be real symmetric matrices. If for all real s, t $\hfill \vert I - sA - tB\vert = \vert I - sA\vert \cdot \vert I-tB\vert \hfill {\rm (2a)}$ then $\hfill AB = 0. \hfill {\rm (2b)}$

That (2b) implies (2a) is "obvious". We note that when V = I in (1) then it reduces to AB = 0, and so then x'Ax and x'Bx are independently distributed if and only if AB = 0.

Both Theorem 1 and Theorem 2 have been called the Craig-Sakamoto Theorem (or Lemma). In this talk we review several "proofs" of Theorems 1 and 2 that have been presented in the literature and present a new proof of Theorem 2. We also present some biographical and historical information, as well as an extensive bibliography.
 
 

* Joint work with S. W. DRURY and George P. H. STYAN (both of McGill University, Montréal, Canada). Part of an MSc thesis at McGill University.
 

On nonnegative quadratic estimation
in linear models
Mariusz GRZADZIEL, Agricultural University of Wrocaw, Poland

Consider the general linear mixed model $\{y,X\beta,V(\sigma)=\sum_{i=1}^{r}\sigma_{i}^{2}V_{i}\},$ where y is an n-dimensional normally distributed random vector with

$$E(y)=X\beta, Cov(y)=V(\sigma)=\sum_{i=1}^{r}\sigma_{i}^{2}V_{i},$$

X is a known $n\times p$ matrix, $\beta$ is an unknown p-dimensional vector of fixed parameters, V_i , i=1,2,...,r, are known nonnegative definite matrices while $\sigma = (\sigma_{1}^2,...,\sigma_{r}^2)'$ is a vector of unknown variance components.

For a given nonnegative definite $n\times n$ matrix F and a given vector f=(f₁,...,f_r)' with nonnegative components f_i, I am interested in estimation of the following nonnegative parametric function

$$\gamma(\beta,\sigma) = \beta'X'FX\beta + \sum_{i=1}^{r}f_{i}\sigma_{i}^{2}$$

by a quadratic estimator y'Ay, where the matrix Ais both symmetric and positive semidefinite.

For r=1 and V₁=I, i.e. in the case of the linear regression model, estimating of $\gamma(\beta,\sigma)$ was investigated by Gnot, Trenkler and Zmyslony (1995) (J. Multivariate Analysis 54). Applications to the problem of variable selection in linear regression were presented in Gnot, Knautz and Trenkler (1994).

The paper Gnot and Grz¸adziel (1997) (Preprint 46 of the Dept. Math. of the Agricultural Univ. of Wrocaw) is devoted to the study of the problem of estimating $\gamma(\beta,\sigma)$ when r>1. In this preprint the convex programming approach developed by Hartung (1981) (Annals of Statistics 9) for estimating variance components was used. Positive SemiDefinite Minimium Biased (PSDMB) estimator of $\gamma(\beta,\sigma)$was defined and characterized by a set of nonlinear equations.

Solving this set of equations could be reduced to a nonconvex, nonsmooth optimization problem which can be solved by global optimization procedures. During the PhD Students Session I am going to discuss the efficiency of the algorithm finding the unique solution of this problem. Finally I want to present some alternative estimators of $\gamma(\beta,\sigma)$ and compare them with the PSDMB estimator of $\gamma(\beta,\sigma)$.
 

Modeling of the fiber length distribution of mechanical pulp using generalized linear models
Sami HELLE, University of Tampere, Finland

The degree of grinding of mechanical pulp, which is used in papermaking, is usually described by the so-called Freeness-number. Besides Freeness meters used for analysing pulp, these have also appeared optical fiber-length meters. Nowadays it is unclear if there is any use for these optical fiber length meters in analysing the quality of pulp. Fiber length meters give results in the form of a histogram of fiber lengths in predefined classes.

In this paper generalized linear models were used to modeling the distribution of mechanical pulp fiber length. The results of a mechanical pulp's end grinder test drive were modeled with generalized linear models. Each pulp's samples were measured with Kajaani FS-200 fiber length meter. The fiber length distribution was modeled with generalized linear models by using the energy specific consumption (ESC) of the end grinder for the explanatory variable. The other quality properties of mechanical pulp can be further modeled with this modeled fiber length distribution.

From the results we can see that generalized linear models are very suitable for modeling the fiber length distribution of mechanical pulp and they explain very well the properties of mechanical pulp, even better than the usually used Freeness-number.
 

Some comments and a bibliography on the
von Szökefalvi Nagy-Popoviciu and Nair-Thomson inequalities, with extensions and applications
break in header to statistics and matrix theory
Shane Tyler JENSEN*, McGill University, Montréal, Canada

We examine a 1918 inequality of Julius von Szökefalvi Nagy [Gyula Szokefalvi-Nagy] and some 1935 extensions of Tiberiu Popoviciu concerning the standard deviation and range of a set of real numbers and some equivalent inequalities for the Studentized range due to Keshavan Raghavan Nair (1947, 1948) and George William Thomson (1955). We also examine some closely related inequalities due to Bhatia and Davis (1999), Guterman (1962), and Margaritescu and Voda (1983). We survey the literature and give various extensions and applications in statistics and matrix theory. We also include some historical and biographical information and present an extensive bibliography with over 225 entries.
 
 

* Joint work with George P. H. STYAN (McGill University, Montréal, Canada). Part of an MSc thesis at McGill University.
 

Optimal designs for second-degree K-model
mixture experiments
Thomas KLEIN, University of Augsburg, Germany

Models for mixtures of ingredients are typically fitted by Scheffé's canonical model forms. Draper and Pukelsheim (1998) have suggested an alternative class of so called K-models based on the Kronecker algebra of vectors and matrices.

The design problem for these models has been solved only partially so far. For the first-degree and second-degree cases, Draper and Pukelsheim (1998) and Draper, Heiligers, and Pukelsheim (1998) have identified a set of designs which is a minimal complete class with respect to the Kiefer ordering of moment matrices.

Starting from this result, the problem of finding $\phi_p$-optimal designs for second-degree K-models is explored. In the case of two or three ingredients, such $\phi_p$-optimal designs are already known. Current research is devoted to the case of four or more ingredients.
 

References

Draper, N. R., Heiligers, B. and Pukelsheim, F. (1998). Kiefer ordering of simplex designs for second-degree mixture models with four or more ingredients. Report No. 403, Institut für Mathematik, Universität Augsburg.

Draper, N. R. and Pukelsheim, F. (1998). Mixture models based on homogeneous polynomials. Journal of Statistical Planning and Inference, 71, 303-311.
 

Some matrix results related to admissibility in linear models and generalized matrix versions of Wielandt inequality
Chang-Yu LU, Northeast Normal University, Chang Chun City, P. R. China

This topic concerning three special subjects:

(A) Admissibility of linear estimators in linear models.

The admissibility of linear estimator is characterized in the model $(Y,X\beta,$$V\vert(\beta,V)\inT)$ in the cases: (i) $T=R^{p}\times \sigma^2 V,\sigma^2 >0$, while $(\beta,\sigma^2)$ with a ellipsoidal constraint $(\beta-\beta_0)'N(\beta-\beta_0) \leq \sigma^2$, where N is known nonnegative definite matrix, or (ii) $\beta$ with inequlity constraints, that is $T={\cal C}\times{\cal V}$, where ${\cal C}=\{\beta: R\beta\geq 0,R$ is a known $k\timesp$ matrix},${\cal V}$ is a given set of nonnegative definite matrices of order n. For two important special cases , the regression parameter set is unrestricted or bounded ellipsoid set, therefore, the original results due to Rao (1976) and Hoffmann (1977) respectively, are unified.

(B) Admissibility of nonnegative quadratic estimators in variance

components models.

The estimation of the vector of variance components under general variance components models with respect to the scale quadratic loss function is consider. A new method of constructing a better estimator and how to deal with the admissibility of quadratic estimator of variance components with the covariance matrix may be singular is presented. Using this method, a necessary condition for admissbility of nonnegatve quadratic estimator is given. For a special case, a necessary and sufficient condition is also given.

(C) Generalized matrices versions of the Wielandt inequality

with some statistical application.

We give an answer of Wang's (1998) conjecture about matrix version of the Wielandt inequality, and some applications are discussed.
 

Distance criterion in theory of optimal design
Arto LUOMA, University of Tampere, Finland

Distance criterion is an optimality criterion in design of experiments. A design $\xi_1$ is said to be at least as good as $\xi_2$ with respect to DS($\varepsilon$)-criterion if $P\bigl(\Vert\widehat{\beta}_1-\beta\Vert\le\varepsilon\bigr)\ge P\bigl(\Vert\widehat{\beta}_2-\beta\Vert\le\varepsilon\bigr)$ where $\widehat{\beta}_1$ and $\widehat{\beta}_2$ are the estimators corresponding to $\xi_1$ and $\xi_2$, respectively. If the design $\xi_1$ is at least as good as any competing design $\xi_2$ for all $\varepsilon>0$, then it is said to be distance optimal (DS-optimal). In other words, a design is DS-optimal, if the Euclidian distance between the estimator and the true value is stochastically less than in any competing design. We have studied the distance criterion in the context of the classical linear model assuming that the random errors are normally distributed. We found that in polynomial fit models, there does not usually exist DS-optimal designs. However, nonsymmetric designs can be improved symmetrizing them. In the first degree m-way polynomial fit models when the experimental region is an m-dimensional Euclidian ball or cube, optimal designs exist and they are orthogonal. We also studied the limiting behaviour of the DS($\varepsilon$)-criterion as $\varepsilon$ tends to 0 or $\infty$.
 

Matrices related to diagnostic methods in classification problems
Markku NURHONEN, University of Minnesota, USA

We apply the local influence approach to the identification of influential observations in classification problems. Our main interest is in the posterior probabilities of group membership and in the atypicalities corresponding to a set of unclassified observations. Different perturbation schemes to extract influential data are applied and several ways to measure influence on various aspects of the classification results are suggested. We present expected influence measures for situations where no particular set of unclassified data has been specified. Three ways to calculate such expectations are compared. The proposed methods can be applied to discriminant analysis based on predictive densities and to results calculated from estimative densities. The group dependent sampling distributions are assumed to be multivariate normal distributions with either equal or unequal covariance matrices. Covariate adjusted classification problems are also considered. Most of the results we discuss involve eigenanalysis of a matrix which can be written out in a compact and interpretable way.
 

Comparing traditional cluster methods to networks with large medicine data
Jyrki OLLIKAINEN, University of Tampere, Finland

In this paper I will compare three different grouping analyses by doing cluster analyses for data where there are 6700 patients who have risk of coronary heart disease. I have first performed the well known K-Means Cluster analysis; the results supported well the predictions made by the doctors. After that I have used two different network application for the data: Kohonen's Self-Organizing Maps and Bayesian network modelling. Bayesian modelling has been developed in the Helsinki University of the Department of Computer Science and Self-Organizing Maps in the Helsinki University of Technology.

In the Bayesian network modelling we select first the model structure which can, for example, be the structure of a neural network. Then we order the structure's parameters. When selecting structure and parameters we use as a learning criteria Bayesian posterior probability.

In the Kohonen's neural network R-dimensional input data is mapped onto two-dimensional neural level. With every node i a parametric reference vector m is associated. An input vector is compared with the m and the input is mapped to the best matched node i. Values of the reference vector m change towards the values of the input vector which is mapped onto m.
 

Some universal block-matrix factorizations
and their applications
Yongge TIAN*, Concordia University, Montréal, Canada

We present a group of universal factorization equalities for $2 \times 2$ and $4 \times 4$ block matrices, and then use them to establish a variety of equalities for ranks and determinants of matrices. Based on them, we also establish a universal factorization equality for real quaternion matrices.
 
 

* Joint work with George P. H. STYAN (McGill University, Montréal, Canada). Part of an MSc thesis at Concordia University.
 

When do matrix equations have
block triangular solutions?$^{\hbox{\footnotesize\rm 1}}$
Yongge TIAN, Concordia University, Montréal, Canada

Let AXB = C is a matrix equation over the field of complex numbers. We consider under what conditions this equation has $2 \times 2$ or $3\times 3$ upper block triangular solutions, and present some special cases related to generalized inverses of matrices. Moreover, we also consider block triangular solutions of matrix equations of the form AX - YB = C and AX - XB = C.
 
 

¹ Poster.
 

The Drazin inverse of a modified matrix
Yimin WEI, Fudan University, Shanghai, P. R. China

We derive the expression for the Drazin inverse of a modified matrix. The perturbation bound for the Drazin inverse is also established, i.e., the open problem posed by Campbell and Meyer [in Linear Algebra Appl., 10(1975) pp. 77-83] has been partially solved.
 

Some results on estimator of the mean vector and variance in the general linear model and estimating problems for location (scale) models from grouped samples under order restrictions
Bao-Xue ZHANG, Northeast Normal University, Chang Chun City, P. R. China

(1) In this paper, necessary and sufficient conditions for equalities between a²y'(I-P_X)y and $\tilde{\sigma}^2$ under the general linear model, where

$$\tilde{\sigma}^2=\frac{y'T^{+\frac{1}{2}}(I-P_{T^{+\frac{1}{2}}X})T^{+\frac{1}{2}}y}{n-{\hbox{\rm rank}}\, T},$$

a² is a known positive number, are derived. Further, when the Gauss-Markov estimators and the ordinary least squares estimators are identical, we obtain a relative simply equivalent condition. At last, this condition is applied to an interesting example.

(2) Consider the partitioned linear $A=\lbrace Y,(X_1\, :\, X_2)\pmatrix{\beta_1\cr\beta_2\cr},\sigma^2V\rbrace$ and its five reduced linear models, where $V\ge 0$ and $\Re (V)\supseteq\Re(X_1\, :\, X_2)$. In this paper, the formulas for the differences between the BLUEs of $M_2X_1\beta_1$ under A and its BLUEs under reduced linear models of A are given. Further, the necessary and sufficient conditions for the BLUEs of $M_2X_1\beta_1$ under reduced linear models to be the BLUEs of $M_2X_1\beta_1$ under A are established. Moreover, we also study the connections between the MINQUE of $\sigma^2$ under A and the MINQUE of $\sigma^2$ under its reduced linear models.

(3) The Gauss-Markov estimator of X₁BX₂' under the growth curve model $\lbrace Y, X_1BX_2', V_2\otimes V_1\rbrace$ is given. Necessary and sufficient conditions for equality between A₁YA₂' and B₁YB₂', where A_i=X_i(X_i'W_iX_i)⁺X_i'W_i, B_i=X_i(X_i'S_iX_i)⁺X_i'S_i, with W_i, S_i being any matrix, i=1, 2, are derived. Finally, a number of criteria for A₁YA₂' to coincide with the Gauss-Markov estimator of X₁BX₂' are established.

(4) This article deals with estimating problems for location (scale) models from grouped samples. Suppose the support region of a density function, which does not depend on parameters, are divided into some disjoint intervals, grouped samples are the number of observations falling in each intervals respectively. The studying of grouped samples may be dated back to the beginning of the century, in which only one sample location and/or scale models is considered. This article considers one sided estimating problems for location models. Some methods for computing the maximum likelihood estimates of the parameters subject to order restrictions are proposed and a numerical example by the method is given.

Some of the above research is joint work with Wei Gao (Northeast Normal University), Bei-Sen Liu (Tian Jin University), Ning-Zhong Shi (Northeast Normal University) and Xiang-Hai Zhu (Northeast Normal University).
 

1999-06-21