MATRIX IMS-ILAS

,l

INTERNATIONAL

IMS-ILAS WORKSHOP ON

MATRIX ^{METHODS FOR}

STATISTICS

4

&

5 Decemb er

\992 University of Auckland

New Zealand

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INTERNATIONAI. IMS-ILAS WORKSHOP ON MATRIXMETHODS

FOR STAT'ISTICS

Auckland, New Zealand 4-5 December 1992

ABSTRACTS

Characterizing Invariant Convex Functions of Matrices James V. Bondar and Alastair J. Scott

Carleton University

ancl

University of Auckland

Some years ago, C. Davis (1957) showed that the invariant convex functions on the class of symmetric matrices are precisely the functions that are convex increasing functions of the eigenvalues. (Here "invariant" ^rneans invariant under the similarity transformation

O

^--+ OMO'where O is orthogonal.)

In

this

talk

we show

that

the invariant convex functions on the class of all matrices are precisely those func.tions of the singular values

that are both convex and weak Schur convex. (Here "invariant" means invariant under the transformation

M

-+ OtMOz where the O; are orthogonal.)

It

^{has been} known for some tirne that increasing symrnetrix convex func.tions of the squares of the singular values are rnatrix convex. Bondar (1985) strengthens Schwartz' results

to

increasing sylnmetric convex functions of the singular values. Here we show

that to be a convex, weak Schur convex function of the singular values is necessary and sufficient for matrix convexity. We also derive sufficient conditions

that

are simpler to apply in practice.

References

J. Bondar (1985). Convexity of acceptance regions of multivariate tests.

Linear ^Algebra Appl.,67: 195-199.

C. Davis (1957). Al1 convex invariant functions of Hermitian matrices. Arch. Math.,

8:

276-278.

Application of a Generalizecl Matrix Schwarz hrequality Evaluation of Biasecl Estimation in Linear Regression

J. S. Chiprnan

unive

^rs

i\o

l}fi['..' ^{", "}

In the general linear regression rnoclel, three estiurators are considered: (1) the Gauss-

Markoff estimator; (2) the Gauss-Marlioff estirnator subject to a set of linear restrictions on the regression-coefficient vector; (3) the Theil-Goldberger estirnator with uncertain linear restrictions corresponding to the certain ones of (2). This third estimator is styied tlre generalized ridge estimator.

It

^{is shown,} generaliztng a result of Toro-Vizcarrondo and Wallace,

that

^a sufficient condition for both estimators (2) and (3)

to

have lower matrix-mean-square error than estimator (1) is that the noncentrality parameter arising

in

the F-test for the restrictions

in

(2) should be less than

1.

A generalization is also obtained of the Hoerl-Kennard result that for sufficiently small but positive values of their l.-parameter, estimator (:3) has lower matrix-mean-square error than estimator ^(1).

References

.L S. Chipmarl, "On least ^squares

with

insufficient observations,"

J.A.S.A.59

^(1964),

1078-1111.

A. E. Hoerl and R. W. Kennard, "Ridge regression: biased estimation for nonorthogonal problerns," Techtrometric-s 12 (1970), 55-67.

C. Toro-Vizcarrondo and

T.

D. Wallac€,

"A

^{test of the} rnean square error criterion for restrictions in linear regression," J.A.S.A. 63 (1968), 558-572.

Statistical Contribr.rtions to Matrix Methocls Richard Wiiliani Farebrother

University of Manchester

The Formalisation of Matrix Algebra in the 1850s and 1860s was preceded by a number of significant developments, sorne of which

(*)

are to be found in statistics works:

1. Cauchy's use of rnodem subscript notatiou for matrix eiements.

2. Gauss's use of matrix rnultiplication when trausforming variables

in

a quadratic form.

3.*

Mayer's prototype of the elirnination ^procedure.

4.*

Gauss's identification of a iinearly dependent systern of equations.

5.*

Gauss's elirnination procedure and the implicit LU decomposition of a square matrix.

6.*

Clauss's definition of a generalised inverse of a tnatrix.

7.*

Clauss's derivation of a LDL' decomposition of a symmetric matrix.

8.*

Gauss's least squares updating fortuulas.

9.*

Gauss's proof that the inverse of a symrnetric matrix is itself syrnmetric.

10. Cauchy's latent value decomposition of a 3

x

3 rnatrix.

11.* Laplace's derivation of an orthogonalisation procedure subsequently rediscovered by Gram and Schmidt.

12.* Donkin's definition of an orthogonal basis for the orthogonal courplement of a rna-

trix.

13. Jacobi's characterisation of the least squares solution as a weighted sum of subset estimates. This ciraracterisation of the Least Squares solution as a weighted sum of subset estimates by C)laisirer and Subrahrnaniarn.

Later Contributions Include:

14.* Thiele's discussion of the canonical form of the linear model.

15.* The Singular value decomposition derived by Eckart and Young.

i6.*

The so-callecl Gur.r-Mu.kov theorem is clue to Ciauss alone.

17.* The generalisation of this theoren usually attributed to Aitken would seem to be due to Plackett.

Stationary Distributions and Mean First ^Passage Times in Markov Chains using Generaltzed Inverses

Jeffrey J. Hunter

Massey University

The determination of the mean first passage tirnes in finite irreducible discrete tirne Markov chains requires the computation of a generalized inverse of .I

-

^{P, where P is the}

transition matrix of the Markov chain, and also knowledge of the stationary distribution of tlie Markov chain. Generalized inverses can be used to find stationary distributions.

Tire linking of these two procedures and the cornputation of stationary distributions ^using various multi-condition generalized inverses is investigated.

Cornbining Inclependent Tqsts For A Cotllmon Mean

-lA"

Ap'irlication of the Parailel ^Surn of Matrices Thomas Mathew

Departrnent of Mathematics and Statistics ' UniversitY of ^MarYlancl Baltimore

^CountY

Baltimore,

^{MarYl an}

d

21228

U.S.A.

I.

the co.text

of

the recovery

of

inter-block inforrnation

in a BIBD,

the problem ^of conrbining two inclepenclent tests is addressecl in cohen ^and sackrowitz (1989, Journ'al of

the.

Ameico,

Stotirtical Association). Apart from the intra- and inter-block ^F-tests, their cornbi.ecl test uses a correlation type statistic, ^and this statistic ^has a positive expected value whe' the null hypothesis of equality of the treatment ^{effects is} not true' We show

that a similar statistic

.or b"

^{definecl in the context of} hypotiresis testing for a common lnean

i'

several indepenclent linear models. ^Using the properties of the parallel ^{surn of} matrices, we also sirow

that

this statistic has a positive expected value wheu the null hypothesis is not true.

Invariant Preorcleriirgs of Matrices ancl Approximation Problerns

in N4ultivariate Statistics ancl MDS Renate Meyer

Institute of

Startistics

and Docurnentation Aachen University of Technology,

German.y

Tlre singular value decomposition of a complex n x k -matrix

A :

Dl=, og4(A)u;vf, with

o1r1(A)

2 _"pt(A)

the Hennitian matrix A

* A,

plays an important role in various multivariate descriptive statistical nethods, ^as for exarnple in principal components and analysis, canouical corre-

lation analysis, discriminant ^analysis, and correspondence analysis.

It

is applied to solve the key problem of approximating a given rz

x

k-rnatrix (1

<

k

< ,)

by a matlix of lower rank

r < k.

Define A1,;

:

Di=rop4(A)u;vf

.

The minimum norm rank

r

approximation result

,b@

- ^{At,l) <} _'i(A - G)

^{for all G} with rank (G)

(

_{r, (0} 1) was obtainecl by Eckart ancl Young (1936) for the Eucliclean norm

$(A:

{tr,O.e.1}. ^The solution A1,; turned out to be "robust" with respect to the choice of the approxiuration criterion, as

it

was extended by Mirsky (1960) to the class of unitarily invariant norms.

The objective in ^the first part of this paper is to generalize this result to an even larger class of real-valued loss functions, encolrlpassing the unitarily invariant norurs. Certain preorderings

{

of complex rnatrices, that occur naturally in the context of multivariate statistics, will be considering in proving uniuersal optimality of A1,;, i.e.

A-Ar,r ^< A-G forall Gwith

rank (G)

Sr.

^(0.2)

Of course, this irnmediately entails (1) for ail real-valued {-monotone functionsTy'.

The ^second part treats the problem of approximating a Herrnitian

n x n

matrix by a positive-sernidefinite rnatrix of given rank, which is of major relevance

in

the context of multidimensional scaling (MDS). Thereby, the hitherto rnost general result of Mathar (1985) is extended and further universally optirnal properties of the classical MDS solu-

tion are provided.

Iiey

uords:

singular value decornposition, group induced preorderings, invariant order- ings, weak majorization, principal components analysis, rnultidimensional scaling.

References

Eckart, C., Young,

G.

(1936). The approximation of one matrix by another of lower rank.

P.sy ch o m etrik a L, 2Il -218.'

Jensen, D.R.

(1984).

Invariant ordering Irtequalities in,9tat'istics ^an d Probability.,

Ca.,26-ii4.

and order preservation.

In: Y.L.

Tong, Ed., histitute of Mathematical Statistics, ^Hayward,

6

Mathar,

R.

(1985). The best Euclidean

fit

to a given distance matrix in prescribed dimen- sions. Linear Algebra Appl. 67, 7-6.

Meyer,

R.

(1991) Multidirnensional Scaling as a frarnework for Correspondence Analysis and its extensions,

in:

M. Schader, Ed., Analyzirtg and Mod.elin.g Data and l{nowledge, Springer, 63-72.

Mirsky,

L.

^(1960. Symmetric ^gauge functions and unitarily invariant norms. Quart. ^J.

Math. Oxford, ,9er. 2,11, 50-59.

A Cornpact Expression For Variance of Sample Second-order Moments in Multivariate Linear Relations

Heinz Neudecker

Department of Actuarial

Science

and Econometrics University of

Amsterdam"

The Netherlands

Satorra (1992) considered the random

(prl)

vector

, :iBr;; _*

_tr,

i=7

where the random

(n;rl)

vectors 51 are independent with rnean

E(6) :

0 and variance D.(5;), and

B;

are _LL are coustants

(i:t...nr).

He derived the variance of u(zz'), where u(.) is the short version of vec(.).

It is the aim of this note to give a compact expression for the variance and subsequently derive Satorra's result.

Matrix ^{Tiicks Related} to Deleting an Observation in the General

Linear Model

Markliu Nurhonen and ^Simo Puntanen Departrnent of Mathematical - UniversitY of

^{TarnPereSciences}

Consicler the linear motlel U,

X0,V,

^where

^X

^has

^full

^{colurnn rank and}

^V

is positive definite.

when

estimating the pararneters of the model,

it

^{is natural}

^to

consider the consequences of sorne changes or perturbations

in

the data on the estimates: ^regression diag'ostics ^{rneasure these} consequencies. One fundamental perturbation is omitting one

o, ,",r"rul of the observations from the model. In this paper our interest ^focuses oll some helpful matrix forrnulas while studying regression diagnostics.

The full CS-cleconrposition of a partitionecl orthogonal tnatrixl

Chris C. Paige

Sc;hool

of Computer

Science,

McGill University Montr6al, _Qu6bec,

Canacla

H3A 2A7

The C,9-decomposition (CSD) of a 2-block by 2-block partitioned unitary matrix Q

: (fl:rt, 3::r)

^{reveais the} relationships betwee' the si*gular value c{ecornpositions (svDs) of eaclr of the 4 subblocks of Q. The ^CSD shows each SVD has the form Ql;

:

LI;D;iVf ,

lor i,

j :7,2,

where each [.I; and I/i is unitary, and each D;1 is essentially diagonal. Here we give a simple proof of this which has no restrictions ou the dirnensions of _Q11.

The CSD was originally proposed by C. Davis and

W.

Kahan, and is important in finding the principal angles between subspaces (Davis ancl Kahan, Bjorck and Golub), such as

in

cornputing canonical correlations between two sets of variates.

It

^{aiso arises}

in, for ^{example, the} Total Least Squares (TLS) problern. The relationships betrveen the 4

subblock SVDs (in particular the way that each unitary matrix U; or V1 appears in 2 dif- ferent SVDs) has macle the CSD a powerful tool for providing sirnple and elegant proofs

of many useful results invoiving partitioned unitary matrices or orthogonal projectors.

Here we also show the CSD makes several nice rauk relations obvious, aud can be used

to prove some interesting results involving general nonsingular matrices.

I(e.y words: CS decornposition, unitary tnatrices, tank relations.

lJoint _{Research with}

Shanghai 200062, China.

Musheng Wei, ^Departmenl

10

of Mathematics, East China Normal University,

How Not to use Matric,es when Teacliing Statistics

D.J. Saville

NZ Pastoral Agriculture

^Research

lnstitute ^Ltcl

and

G.R. Wood

Department of Mathematics, university ^of canterbury

Hic,lclen in the appenclices of some statistical ^{texts is a} geometric apProaclt to analysis

of variance ^ancl

r"gr"rsion.

^{This approach} has been large un-used

in

teaching

for

^two

reasons. Firsi, the approach has been considered too hard, ^{and second,} this century has been a period in which algebraic methods have been dominant.

It

is the aim of this ^paper to show that the first reason offered is nonsense: the geornetric approach can make things easy.

How cioes the geometric methocl rnalie things easy? The matirematical ^framework

necessary for the geometric cievelopment of analysis of variance and regression can be reducecl

to

a halclful of straightforward ^vector ideas. Once these are learnt a concrete methocl is followecl i1 any situation. Three objects ^are isolated: the observation vector' the 'roclel space ancl hypothesis clirections. All lie in a finite-climensional Euclidearl space.

Two routine ^processes then ^serve to

fit

the rnodel and test hypotheses: to

fit

^{tire rnodel we} project the olservation vector onto the ^{model spac.e)} while to test hypotheses we cornpare averages of squared lengths of projections'

Why bother

to

clo

ii.ingr

clifferently? Conventional uretirods

fail to

convey a satis-

factory unclerstalcling of the principies which unify these basic statistical methods. The cookbook approach r"n ^uin, mysterious, while the matrix ^approach is accessible only to tlrose with lrathernatical rnaturity. The geometric ^approacir provides

at

elemen'tary b:ut rigorouspath through ^the rnaterial.

It

has been used successfuily by the atrthors to teach

,"'.ond-y"ur ^stuclelts in statistic.s as well as postgraduate stuclents in the applied ^sciences' These icleas will be cliscussecl, ancl an example of the method in actiou presented.

1l

Problerns with clirect soiritions of the normal eqrrations for rlon-paratnetric aclditive

^{uro clels}

M. G. Sc:hintek

Medical Biornetrics Group, University of Graz Medical

Schoois,

A-8036 Graz, Austria

To stucly the funciioual clependence between variables

of

a multivariate regression

problem

in

a data-driven fashion, non-pal'alletric techniques like Generalized Additive Models (Buja, A., Hastie,

T.

^and Tibshirani, R.: Linear smoothers and additive rnodels.

Artn. Statisf. 1989, 17,453-555) are very useful. Instead of solving the associated uonnal equations a Jacobi-type iterative procedure called backfitting is usually appiied. This is prirnarily done for the purpose of computational efficiency and

to

avoid singularity problens. But

little

is known from a theoretical and even less frorn a practical point of view about the quality of the obtained results.

Let

us observe (d

+

l)-dirnensioual data

(r;,X) with z; : (r0r,...,r;a).

^The

r1j,...tr,,.j

^represent indepenclent observations drawn frorn

a

randorn vector

X : (*r,,...,i0)anclther;anclXfuifilY;:g(r;)*e;forI<i(zr,wireregisallurl-

known smooth function from Rd to

ft,

and €1r... _)en are independent errors. An additive approximation to ^g

d

s6)veo+fsi,xi)

j=1

is aimed at, where ₉₆ is a constant and the

!;s

^{are any srnooth} functions. To make these functions g; identifiable

it

is required that

E(g16)):

^{0 for}

I

^<

i (

^{d. For instance we}

can apply thepopular cubicpolynomial srnoothing splines. Let

Sr:

(1+)A.1ifr)-1 bethe snroother matrix (operator) and 1(4 a penaity rnatrix of such a spline. For l.

:1,2,...,d

the normal equations form a (nd)

x

(n d) system

1) [) ^:I

:::

,9a ,94 ^,94

L9,

^,91

szlS, ^Sfl

^Sza

Sau

denoted by

Pg:

QA, where P and

Q

^are block matrices of smoothing operators _^9r.

One way to solve the system wouid be to apply the rnethod of successive overrelaxatiotl, but for d

>

2 we do not l<now how to choose the relaxation parameter. Instead we propose

taking advantage of the specific block structure (especially the position of the /s) of

P. It

allows us to clerive a recursive scheme for the calculation of

P-1.

IJnder non-singularity a direct solution ^{can be} obtained by 0

:

P-t Qy. This ^{approach is cheap} but depends on

tlre chosen scatterplot

smoolher.

12

In case of singularity we propose another approach based on a specific type of Tichonow regularization assuming measurement errors of tire dependent variable. Let us have the singular systern

Pr :

_{Qa. Let a} be a regularization parameter, then ihe disturbed system takes tlre forrn (P*P

* ^al)r - ^P*Qy.The

^deviation

lli - ^rll

^can be estimated in terms of 7/a ^{and the} measurernent error of the right-hand side of the equation. The system can be solved by standard techniques but is cotnputationally expensive.

Research supported by Austrian Science Research Fund trant P8153-PIIY,

i3

On the Efficiency of a Linear Unbiased Estimator and on ^a

Matrix Version of the Cauchy-Schwarz Inequality

George P.H. Styan

Department of Mathematics and Statistics, McGill lJniversity

Montr6a1, Qu6bec, Canada H3A 2A7

We consider the efficienc.y of a linear unbiased estirnator in the general linear model and

its

conlection

to

a determinant version of the Cauchy-Schwarz inequality; ^we illustrate our results with an example from simple linear regression.

I4

Alexander Craig Aitken: ^1895-1967

Garry J. Tee

Departrnent of Mathernatics

ancl

Statistics University of Aucklancl

Alexander Craig Aitken was bom at Dunedin on 1895 April 1, and he attended Otago Boys High Scirool. On holiday

at

his grandparents dairy falm on Otago Peninsula in 1904, he discoverecl the norv-famous breecling colony of the Royal Albatross at Taiaroa Head. His Calvinist grandparents punished hirn for telling ^{such. an} unlikely tale but later o1e of his uncles was appointed as a Ranger to protect the colony. After 2 years at the University of Otago ^he joined the army and was severely wounded at the Battle of the Somme.

.He completed his studies at the University of Otago, and graduated NI.A.

in

1919 ^.

In

1920 he rnarriecl Mary Betts, who lec.tured in Botany at Otago l-lniversity. He taught

at

Otago Boys High School

until

^{1923, when Professor R. J.}

T.

Bell ^persuaded him to study at the tiniversity of Edinburglt, ^{where he spent} the rest of his life. Professor E. T.

Whittaker asigned hint the problem of smoothing of data, which had practical iurportance i1 actuarial work. His thesis was of suc.h merit that he was awarded the degree of Doctor of Science, rather than Ph.D.

Aitken s mathentatical rvork was devoted mainly to nurnerical ^analysis, statistics and linear algebra. He founded the renowr ed Oliver & Boyd series of textbooks and wrote the firsb tlvo himseif: both

Determinants

and

Matrices

and

Statistical

Mathematics

are recognized as classic textbooks. In numerical analysis he devised many methods which exploited the capabilities of the calculating machines which were then available, and which have proved

to

be funclamental

to

rnuch later work

in

scientific cornputing. He gairrecl wide fame as the greatest mental calculator for whom detailed and reliable records exist.

lVhittaker retired

in

1946, and Aitken, without any move on his part, was elected to the Chair of Vlathematics at Edinburgh. He was elected Fellow of the Royal Societies of London and of Edinburgh, and Honorary ^Fellow of the Royal Society of New Zealand, of the Society of Engineers, and of the Paculty of Actuaries of Edinburgh. He was awarded Honorary Degrees by the Universities of New Zealand and of Glasgow, and he was awarded several rnajor prizes in mathematics. He was an inspiring lecturer ^and an extremely gentle person, intensely devoted to music, and ire wrote sorne poetry of distinction.

In

1963 he published his mernoir

Gallipoli to the Somme:

Recollections

of

^a

New

Zealand

Infantryman,

^{which was} acclairned as a classic account of death and life i1 the trenches. The Royal Society of Literature elected hiur as a Fellow, in recognition of his achievernent in writing that ^memoir.

Aitken retired

in

¹⁹⁶⁵ (in poor health), and ire died at Edinburgh on 1967 November 3, aged 72.

15

Use of permatlettts ^ancl their analogues.in mr.rltivariate galnltta, binonlial and negative

the represeutatiou of binonrial distriltutious

D. Vere-,Jotres

Institute of Statistics and Operations

^Research

Victoria University of Wellington

Wellington, New

^Zealancl

Porvers of cleterminantal expressions of the form det[1

-TA],7:

diag(tt,t2,...,t,,) occur i1 expressiols for the moment or probability generating functions for several types of nultivariate clistribuiion

-

gamrna, negative binomial, binomial at least- The funda- 'rental

ideltity

relating the logarithm ^of the determinant to the trace of the logarithrn of the matrix allows the coefficients in the determinantal expressions to be characterised ^as multilinear ^forms in the elements of A, reducing to the detemrinant of A rvhen the power

is

*1, ^{to its}

^{permanent when tire power is}

-1,

^and

^to

^{an extension of both concepts}

(,,alpha-permalents" in the tenninology of Vere-Jones (1988)), for general porvers. This pup"'. will review some properties of these representations and the associated multivariate ,lirtrib.rtions, inclucling extensions to nixtures of binornial or ^negative biuomial distribu- tio1s, alcl to stochastic processes. Some open questions wiil ^be urentioned, including ^the problem of developing effective numerical techniques.

Reference

Vere-Jones, D., "A generalisation of pertlauents and detertninants,"

Li,near Algebra an'd lts Application-s 111 (1988)' 199-124'

16

Andrew

Balemi

^{Richard William} Farebrother

Mathematics and

Statistics

Department of Econometrics and Social Statistics University of

Auckland

University of Manchester

38 Princes

St

^Manchester

Private Bag

92019

^{M13 gPL}

Auckland

United Kingdom

New Zealand

m s rbs rf @cms. mcc. ac. ^u k balemi@mat.aukuni.ac.nz

Rod

Ball

^Arthur R. Gilmour

Hort

Research

NSW Agriculture Centre

Hort Research lnstitute

NZ

Forest Road

P.O. Box

92169

^Orange

Auckland

^NSW2800

New

Zealand

^Australia

rod@marc.am.nz

John S.

Chipman

^{Harold V. Henderson}

Department of

Economics

Statistics and Computing Centre University of

Minnesota

^Ruakura Agriculture Centre 1035 Management and

Economics

^Hamiiton

271 19lh Avenue

South

^{New Zealand}

Minneapolis, MN 55455

U.S.A.

henderson@ruakura.cri.nz

jchipman@ uminnl.bitnet

Ronald

Christensen

^{Jefirey J. Hunter}

Department of Mathematics and

Statistics

Department of Mathematics & Statistics University oi New

Mexico

Massey University

Albuquerque

Palmerston North

NM

87131

New Zealand

U.S.A.

jhunter@massey.ac. _nz

f letcher@gopher.unm.edu OR

Peter

Clifford

Snehalata Huzurbazar

Oxford

University

Department of Statistics

Jesus

College

University of Georgia

Oxford

^Athens

ox1

3DW

GA 30602-1952

UnitedKingdom

USA

clifford@vax.oxf

ord.ac.uk

^lata@rolf .stat.uga.edu

Richard Jarrett

Department of Statistics University of Adelaide Department of Statistics GPO 498

Adelaide 5001 Australia

rjarrett@stats. adelaid ^{e. ed} u.au

Murray A. Jorgensen

Waikato Centre for Applied Statistics University of Waikato

Private Bag 3050 Hamilton

New Zealand maj@waikato.ac.nz

Zivan Karaman Limagrain Genetics B.P. 1 15

63203-Riom Cedex France

Alan J. Lee

Department of Mathematics and Statistics University of Auckland

Private Bag 92019 Auckland

New Zealand

lee@math.aukuni.ac.nz

W. Geoffrey MacClement

John Maindonald

Horticultural Research lnstitute Private Bag 92169

Mt Albert Auckland New Zealand

Thomas Mathew

Department of Mathematics and Statistics University of Maryland-Baltimore County Catonsville

MD 21228 U.S.A.

mathew@umbc2.u mbc. ed u

Renate Meyer

lnstitute of Medical Statistics and Documentation Aachen University of TechnologY

Germany Pauwelsstr. 30 51 00 Aachen Germany

renate@tolkien. imsd. rwth-aachen

Gita Mishra

Mathematics and Statistics University of Auckland 38 Princes St

Private Bag 92019 Auckland

New Zealand

mishra@mat.aukuni.ac.nz

Chris C. Paige McGill U niversity Mont16al

Montr6al

Simo

Puntanen

^{George A.G. Seber}

Department o{ Mathematical

Science

Mathematics and Statistics University of

Tampere

University of Auckland

P.O. Box

607

³⁸ Princes St

SF-33101

Private Bag 92019

Tampere

^Auckland

Finland

^{New Zealand}

sjp@uta.fi

(il

12t92 seber@mat.aukun i.ac. nz

Ken Russell

Department of Mathematics University oi Wollongong P.O. Box 1144

Wollongong NSW 25OO

Australia

kgr@its. uow.edu.au

Jai Pal Singh

Department of Agricultural Economics Haryana Agricultural University Hisar

Haryana 125 004 lndia

Garry J. Tee

Mathematics and Statistics University of Auckland 38 Princes St

Private Bag 92019 Auckland

New Zealand

tee@mat.aukuni.ac.nz

Michael G.

Schimek

^{Jon Stene}

Department of

Psychology

^{lnstitute ol Statistics}

McGill

University

University of Copenhagen

1205 Dr. Penfield

Avenue

Studiestrcede 6

Montreal

^{DK -1455}

Quebec

^Copenhagen

Canada H3A

181

^Denmark

ramsay@ramsay2.psych. mcgill.c

Alastair J.

Scott

^George P. H. Styan

Mathematics and

Statistics

Department of Mathematics and Statistics

University of

Auckland

McGill University

Private Bag

92019

Burnside Hall

Auckland

^{805 ouest, rue} Sherbrooke Street West

New

Zealand

^{Montreal, Quebec}

Canada H3A 2K6 scott

mt56@m usica.mcgill.ca

Shayle R. Searle Biometrics Unit Cornell University 337 Warren Hall Ithaca

NY 1 4853-7801 U.S.A.

btry@cornella.bitnet

John M. D.

Thompson

Graham R. Wood

Cot-death

Department of Mathematics

University of

Auckland

University of Canterbury

Private Bag

92019

Christchurch

Auckland

New

Zealand

grw@math.canterbury.ac.nz cotdeath@ccu 1 .auku n i.ac. nz

Christopher M.

Triggs

Thomas W. Yee

Mathematics and

Statistics

Mathematics and Statistics University of

Auckland

University of Auckland

38 Princes

St

³⁸ Princes St

Private Bag

92019

Private Bag 92019

Auckland

^Auckland

New

Zealand

New Zealand

triggs@mat.auku ⁿ i.ac.nz yee@mat.aukuni.ac. nz

Alain C. Vandal

Department of Mathematics and Statistics McGill Universiiy

Burnside Hall

805 ouest, rue Sherbrooke Street West Montreal, Quebec

Canada H3A 2K6

alain@zaphod. math. mcgill.ca

David J. Vere-Jones

lnstitute of Statistics and Operations Research Victoria University

Wellington New Zealand dvj@isor.vuw.ac. nz

Christopher J. Wild

Mathematics and Statistics University of Auckland 38 Princes St

Private Bag 92019 Auckland

New Zealand

wild@mat.aukun i.ac.nz